is the weekly meanreversion parameter, or the longterm mean level <1 ea
of instantaneous volatility,
i, (T) the nearby implied volatility with time to expiration T and
id (K) the distant implied volatility with time to expiration K>T. Under Stein's continuoustime
AR(1) setup, implied
volatility
is meanreverting.
This
implied by longerterm hypothesises the that volatility of move a option should structure also less than one percent in response to a onepercent move in the implied volatility of a shorterB(K, T, p) can be thought of as an elasticity relationship given a Consequently term option. implied in volatility nearby movement implied volatility
i,", there should be a smaller movement in distant
i°. The boundary condition for this elasticity requirement is as follows:
0
This condition imposes a stringent constraint on how the termstructure of implied volatility S&P implied daily 100 1983 from Using December to two volatility series change. can September 1987, Stein found that elasticity turned out to be larger than suggestedby the AR(1) structure, indicating that long maturity options might have attached disproportionate importance or risk premiums to changesin shortmaturity options. is directly testable by substituting i,, id and a into the relationship In our analysis elasticity The nearby implied volatility i, is calculated using the shortest atthemoney
Qýir  Q)
99
Chapter 3: A Report on the Properties of the TermStructure
group.
of S&P 500 Implied volatility
it' is calculated using one of the longerdated option
The distant implied volatility
or 220+. In addition, we use the averaged expectations
groups, i. e. 71120,121170,171220
of implied volatility for all call and put options in tables 5 and 6 as a proxy for the longrun mean level of instantaneous volatility
or. Whilst using annual data is not the most technically
investigate it the to elasticity relationship, nevertheless should still shed some way rigorous light on the consistency of the implied volatility termstructure because of the extensive span of our dataset.
Our estimated or is 16.39% whilst a similar average historical volatility
index daily by 500 Zhang in Shu (1999) is Results from S&P 15.87%. returns and estimated tables 7 and 8 show that there are times when the empirically estimated ß(K, T, p) can depart significantly
from the theoretical elasticity requirements.
The highlighted areas in tables 7
and 8 identify a number of maturity groups that are not bounded within a reasonable range /3(K, This 19831998. demonstrates T, p) is very that empirical evidence the period over frequently is boundary the violated. Notably, these violations are most restriction variable and 220+. longest In for the maturity group, addition to Stein (1989), Bates (1996) pronounced and Bakshi et al. (1997) questioned whether the volatility process implied by traded options implied in its the timeseries. properties with was consistent
Whilst not mathematically
ß's provide evidence that option prices are inconsistent with the rational rigorous, estimated expectations under a meanreverting volatility process.
Table 7: 8(K, T, p) for Calls Nearby Options: 2170 Call Options Year Distant Groups
83
84
87
88
89
0.71
0.84
0.96
0.90
0.45
=
0.80
0.45 0.09
86
85
71120
121170
0.27
171220
0.92
220+
0.62
0
0.99
0
Is
ý
90
97
98
91
92
93
94
95
96
ý
0.92
0.87
0.92
0.87
0.98
0.80
1.03
0.87
0.52
0.81
0.04
0.63
0.80
0.75
0.90
1,22
0.83
1,24
0.90
0.79
0.69
0.62
0.77
0.64
0.90
0.73
0.96
ý'.*
m
0.69
0.84
0.37
0.65
0.55
0.78
0.91
0.85
ý. 24
100
ROM
Chapter 3: A Report on the Properties of the TermStructure
of S&P 500 Implied volatility
Table 8: 8(K, T, p) for Puts Nearby Options: 2170 Put Options Year Distant Groups
83
84
71120 121170
0.27
171220
0.92
220+
0.62
85
86
87
88
89
J.02
1.%
0.71
0.84
0.96
0.90
0.45
0.80
0.45
m 0.99
0.90
0.09
90
91
92
93
94
95
96
97
98
0.92
0.87
0.92
0.87
0.98
0.80
1.Q3 0.87 .
0.52
0.81
0.04
0.63
0.80
0.75
0.90
1.22
0.83
0.79
0.69
1,44
0.62
0.77
0.64
0.90
0.73
0.96
1.41
0.69
0.84
0.37
0.65
0.55
0.78
0.91
0.85
1.23
3.4.4 Option Pricing Under Asymmetric Processes Having examined many important features of the termstructure of implied volatility
in
in is investigate 3.4.3, 3.4.2 this to section our goal what types of models would and sections be consistent with the observed biases in the S&P 500 futures options market. We apply the developed by Bates (1991,1997) technique skewness premiums
to inspect S&P 500 futures
during 19831998. irregularities options' pricing
3.4.4.1 Skewness Premiums
3.4.4.1.1 Underlying Concepts Bates (1991,1997) demonstrated that asymmetries of the riskneutral distribution embedded in an American options could be examined by using relative prices of outofthemoney call This judged distributional hypotheses. the thereby merits of alternative and put options, and if the underlying asset price follows geometric Brownian motion that hypothesises technique be formula, BlackScholes the the in should x% outofthemoney call options the case of as With European the than outofthemoney x% put options. approximately x% more expensive distributions however, divergences. skewed create systematic market asymmetry, Consequently, one could use the observed prices of call and put options to judge whether the from distribution derived the the skewness any of with riskneutral consistent are rules x% hypothesis  an exercise roughly comparable to looking at moneyness distributional specific biases. For example, a perceived market crash will lead to outofthemoney put options on higher is it than indicating being that outofthemoney futures priced S&P 500 call options,
101
Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied volatility
finish in likely for to the money than call options. The x% skewnessis options put more defined as the percentagedeviation of x% outofthemoney call prices from x% put prices: SK(x) = c(F, T; Xc)/p(F, T; XP) I [F/(1 + x)]
=
[F(1 +
x)], x>0, and F is the underlying forward price for
American futures options. For American options on futures, the skewness premium has the following properties for the distributions regardless of the maturity of the options if atthemoney skewness premiums are approximately equal to zero:
i)
0%:5 SK(x)5 x% for 1) Arithmetic and geometric Brownian motion; 2) Standardconstantelasticity volatility processes; 3) Benchmark stochasticvolatility andjumpdiffusion processes;
ii)
SK(x) < 0% only if 1) Volatility of returns increasesas the market falls, or 2) Negative jumps are expectedunder the riskneutral distribution;
iii)
SK(x) > x% if and only if 1) Volatility of returns increasesas the market rises, or 2) Positive jumps are expectedunder the riskneutral distribution. 3.4.4.1.2 Data Construction
CME's settlement records are again used for the skewnesspremiums analysis. The sample S&P 500 futures inception December in from March 1983 the begins of to option period 1998. Three exclusionary restrictions are applied to the data: i)
ii)
Only contracts of a single maturity are considered for any day, namely, contracts with days. Longer 2170 between maturities are too thinly traded and shorter maturities information; to too contain maturity near useful are maturities Exclude nontraded options to eliminate artificial trading behaviour;
102
Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied volatility
iii)
At least five strikes for call options and five strikes for put options are required everyday to enhancethe quality of interpolations.
Since options exist only for specific exercise prices, skewness premiums cannot be implemented directly. implied volatility
In contrast to the methodology employed by Bates32, we interpolate
for desired exercise prices from a cubic spline fitted through the implied
daily for interest As (put) the middle proxy we a riskfree rate, use options. volatility of call date from Datastream bills S. Treasury U. to the matching maturity closest expiration rates on desired interpolated by inserting Option the the with strikes are obtained the prices options. of 's (1987) filtering American The BaroniAdesi into the al. et option model. pricing volatility days being data from 1,600 3,789 in total the used out of a of records. result restrictions
3.4.4.2.2 Results of Distributional Hypothesis Skewness premiums from March 1983 to December 1998 for x= 0% and x= 4% are given in figures 17 and 18, respectively. priced identically,
Theoretically, atthemoney call and put options should be
largely 0%, fact is in premium skewness value of which a yielding
S&P 500 0% in 19831984. Over inception 19831984, the the of options observed except at in fluctuates 8%. ± the randomly range of skewness premium
From 19851998,0%
The implies 4% however, zero. around skewness premium plot, remains skewness premium is largely negatively correlated to the futures price and outofthethat volatility of returns The higher than consistently call options. priced outofthemoney money put options are 4% skewness premium shown in figure 18 indicates gradual downward shifts over time in is typically negative and in excess of the 4% benchmark. 1983, In the premium skewness. Between 1984 and 1985, the premium is largely positive and less than 4%, suggesting that the formula. BlackScholes the with observed prices are consistent
Starting from the late 1986,
begin downside the emerging more until and negatively risk growing of strong assessments 1998. less level to then a stabilising around 1994 negative and slowly returning and middle of S&P 500 been has the that market systemically pricing away the Skewness premiums show biases These during 1986. the formula are substantial persistent and since even BlackScholes increasingly 1987. by an since around premiums negative early years and are accompanied
32Options prices were interpolated from a cubic spline fitted through the ratio of options prices to futures prices in Bates' study.
103
Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied volatility
The fluctuations in the sign and magnitude of skewness premium in figure 18 imply that one needs models of timevarying
skewness to complement the lognormal distribution.
The
identify fit best technique the observed cannot which premium process would skewness options but negative skewness premiums suggest that stochastic volatility large negative correlation between volatility
processes with a
and market shocks or jumpdiffusion
processes
could best fit the observed option prices. Whilst more broadreaching in this analysis, our investigation accords with and does not contradict Bates' (1997) investigation of the S&P 500 market.
3.5 Summary This study is descriptive research and we have employed many models and techniques to investigate the S&P 500 implied volatility termstructure.
Since we have anaylsed in excess
16year inferences drawn from 250,000 this research must not a period, over prices option of be viewed as tentative. implied
volatility
Contrary to the basic assumption of the BlackScholes formula,
exhibits
both smile effects and termstructure
patterns.
We have
demonstrated that the termstructure of S&P 500 implied volatility follows some patterns:
i) Implied volatility tends towards a longterm mean of about 16%; ii) Put options have higher premiums and a larger range of fluctuation than call options; iii)
Shortmaturity options are more volatile than longmaturity options.
Smile effects are found to be strongest for shortterm options, indicating that shortterm by BlackScholes formula the mispriced the severely most and therefore present options are the greatest challenge to any alternative option pricing models.
Basing our results upon a
harmonic model, we find the rate of change of put implied volatility is faster than call's, thus "responsive" basis that to to a change of market put options are argue more a providing is evidence that options prices are not consistent with there Furthermore, we report sentiment. the rational expectations under a meanreverting volatility assumption. Finally, skewness in 3.4.3.1 500 the termstructure S&P that results section moneyness with premiums agree biases have been progressively worsening since around 1987. Results of the negative 4% demonstrate As falls. that increases volatility the of returns as market skewness premiums for diagnostics (1997) be Bates skewness, our and responsible agree with correlation may (stohcastic between large leverage volatility processes with a negative correlation suggest that 104
Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied volatility
volatility and market shocks) and jumpdiffusion
models with negativemean jumps are more
recommended for capturing the observed biases in S&P 500 futures options market
105
Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied Volatility
Figure 3: Call Maturity = 21  70 Days
SIX
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Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied Volatility
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Chapter 3: A Report on the Properties of the TermStructure of S&P 500 Implied Volatility
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Chapter 3: A Report on the Properties of the TermStructure
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Chapter 3: A Report on the Properties of S&P 500 Implied Volatility
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Chapter 3: A Report on the Properties of S&P 500 Implied Volatility
Figure 13: 2170 Calls with SixthOrder
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Chapter 4: An Empirical Comparison of APARCH Models
CHAPTER 4
An Empirical Comparison of APARCH Models
Abstract Modelling asset return dynamics using GARCHtype models is an integral part of empirical finance. The existing literature favours some rather complex volatility specifications but usually their empirical This little is chapter compares a group of prominent and welltheorised models explored. performance that can potentially account for the termstructure biases observed in the S&P 500 futures options daily S&P 500 futures Sixteen of series are used to examine the performance of the years market. APARCH models that use asymmetric parameterisation and power transformation on conditional for decay in No its to the account residual slow absolute returns autocorrelations. and volatility found APARCH loglikelihood be the relatively complex supporting models, and ratio can evidence tests confirm that power transformation and asymmetric parameterisation are not effective in dynamics 500 S&P A 3state APARCH the the within returns context of specifications. characterising is in identify to the "quiet" and "noisy" periods and estimated model order volatility regimeswitching results support the notion that the performance of conditional volatility models is prone to the state of In AIC in EGARCH is best "noisy" that the series. addition, statistics stipulate returns of volatility in is "quiet" GARCH for Overall, top the the performer periods. aggregated rankings periods whilst AIC metric show that the EGARCH model is the best. Finally, optionsbased volatility trading GARCH EGARCH in that and can generate one statistically significant exante profit exercises reveal however, it also exposes the insufficiency of a four transactions after costs, periods sample out of deltaneutral hedge in the event of large market moves. The consequence of this research is not only finance but for discretetime also potentially to meaningful continuoustime stochastic significant GARCH EGARCH HullWhite Wiggins (1987) literature to the the and converge since and volatility (1987) models in diffusion limit. When considering a stochastic volatility model, there seems to be little incentive to look beyond a simple model which allows for volatility clustering and a leverage (1993). Heston effect such as
4.1 Introduction Study the Background 4.1.1 of The poor empirical performance of the BlackScholes option formula is well documented (e.g. MacBeth et al., 1979; Rubinstein, 1985; Bollerslev
et al., 1992).
Contrary to the basic
formula, BlackScholes Chapter implied 3 has that the shown evidence assumptions of factors both termstructure Many and effects such as smile patterns. market volatility exhibits industrial taxing, cycles, serial correlated news arrival, market psychology leverage effect, the in As biases in these causing the roles a crucial observed have very marketplace. played etc is not adequate to specify the returns dynamics and researchers distribution result, normal kurtosis fattails excess deal and which form the basis of smile effects. have yet to with
114
Chapter 4: An Empirical Comparison of APARCH Models
Following the pathbreaking paper by Engle (1982), an alternative literature has focused on discretetime
autoregressive
conditional
heteroskedasticity
(ARCH)
models.
The
development of ARCH models is driven by three regularities of equity returns: 1) equity returns are strongly asymmetric, e.g. negative returns are followed by larger increases in volatility than equally large positive returns; 2) equity returns are fattailed; 3) equity returns are persistent; persistence refers to the volatility clustering. This class of discretetime models hypothesises that both smile effects and termstructure patterns, as evidenced in Chapter 3, can be explained by allowing the underlying asset's volatility to obey a stochastic process. There is a voluminous literature suggesting that discrete timevarying practical and insightful.
The usefulness of ARCH
implemented. and readily predictable
modelling
volatility
models are
is such that volatility
is
ARCH models assume the presence of a serially
data. As the and process require only use past arrival of such, ARCH models news correlated function time to change over as a variance of past conditional variance and allow conditional error, whilst leaving unconditional variance constant. Most of the early research efforts focused on conditional models that imposed symmetry on the conditional variance structure. In response to criticisms that symmetric model may not be for modelling stock returns volatility, appropriate
more recent research has considered other
features such as leverage effects, power transformation etc in the variance equation. There are, indeed, so many conditional
volatility
models in the finance literature that it is
cumbersome to provide a comprehensive survey of them all. Recently, the topics of long memory and persistence have attracted considerable attention in terms of the second moment of an asset returns process. The development of longmemory Ding "stylised facts"33. For the is based the et socalled of observations example, on models APARCH invented the (1993) al.
models that used the BoxCox
transformation
on
its for decay to the residual account and absolute slow of autocorrelations variance conditional in the returns process. Subsequently, many researchers have also developed different longmemory process (e. g. Baille, 1996; Bollerslev et al., 1996; Ding et for the specifications have given the impression that their models are capable of Several 1996). papers al., features in leptokurtosis for the, such as volatility clustering empirical and accounting
33See section 2.2.3 for discussionof the longmemory process.
115
Chapter 4: An Empirical Comparison ofAPARCH Models
distribution of returns. Despite the huge amount of effort researchers have put into modelling volatility,
it is clear that empirical issues remain unexplored for many of these more
"elaborate" models.
4.1.2 The Problem Statementand Hypotheses This chapter investigates the insample performance of APARCH models (Ding et al., 1993) that can potentially account for the slow decay in returns autocorrelations using daily S&P 500 futures series from 1983 through 1998.
The use of the APARCH
framework
different because loglikelihoodbased to model specifications evaluate convenient
is
statistics
for The directly (see A. 1). be test the to of robustness many nested models appendix used can in hypotheses this project are that: used main i)
If the APARCH specification is a good description of the process driving volatility, then hypothesis tests can be applied to reject the nested models in favour of the less restricted models: Ho : restricted A PARCH mod els H, : less restricted APARCH model
ii) If structural change of volatility can have an influence on the performance of better heteroskedastic have then models, asymmetric models should conditional in high than symmetric models volatility state, and vice versa. performance In this chapter our goals are: iii)
To investigate the effectiveness of asymmetric parameterisation and power transformation within the context of APARCH specifications using loglikelihood ratio tests;
iv)
To provide evidence that the insample performance of asymmetrical and symmetrical 3state by to the are prone models a state of volatility using conditional volatility regime switching volatility
conditional
model to separate high and low volatility
states;
insample EGARCH APARCH (Nelson, 1991) the performance of To with compare v) AIC based statistics; aggregate on models different forecasts by the the illustrate of conditional quality To volatility predicting vi) implied (outofsample) volatility of changes and conducting exante onestep ahead S&P 500 straddle trading exercises. 116
Chapter 4: An Empirical Comparison ofAPARCH Models
4.1.3 The Significance of the Study The existing literature favours some rather complex volatility specifications but usually their empirical performance is little explored.
Since the development of longmemory models in
the early 1990's, there has been little research about the significance of their specifications. In this chapter we investigate the importance
of power transformation
parameterisation within the context of APARCH
specifications.
and asymmetric
The consequence of this
research is not only significant to discretetime finance but also potentially meaningful for continuoustime
stochastic volatility
literature.
Since there is a direct linkage between
discrete GARCHtype models and bivariate diffusion models, if it can be shown that there is from beyond to moving a more parsimonious discrete specification such as gain not much EGARCH or GARCH, there seems to be little incentive to look beyond a simple stochastic for volatility allows model which
clustering and a leverage effect such as the HullWhite
(1993). Heston (1987) the model or model
4.1.4 Organisation The remainder of this chapter is organised as follows.
We discuss the in and outofsample
introduce APARCH the models and explain our experiment design in performance criteria, 4.2. section
Section 4.3 describes the S&P 500 dataset. Section 4.4 presents estimation
inthe and outofsample performance of different conditional volatility results and evaluates models under different statistical and economic metrics. Section 4.5 summaries the results.
4.2 Methodology This study uses several econometric methods to evaluate the insample performance of a in data S&P 500 The the conditional volatility models used welltheorised market. group of for estimations are drawn from the S&P 500 futures and its options markets from the period 19831998. In this section we first review the criteria and methods that are used in comparing in We this in discuss project. then models will that are used volatility strategies conditional Finally, the in we will explain models results. used our carrying out the study, analysing APARCH to the models. giving special emphasis
117
Chapter 4: An Empirical Comparison ofAPARCH Models
4.2.1 Performance Criteria A few performance metrics are used in this project to measure the in and outofsample performance of different conditional volatility models:
i) Loglikelihood tests,which are basedon the maximum likelihood estimation numbers, for to test the effectivenessof APARCH features; are employed ii) AIC metric, which penalisesthe use of less parsimonious models, is used to select the best insample model in each subperiod; iii)
The best overall insample model is chosen by the use of aggregate AIC ranking, for defined is the as sum of rank each model in each subperiod. The lower, the which better;
iv) The only criterion used for evaluation of outofsample performance of conditional is loss. The and profit models performance of volatility predictors is volatility based on their ability to predict volatility changes and generate exante evaluated profits from trading nearestthemoneyS&P 500 straddles in four nonoverlapping The higher is the rate of returns per trade, the better is the periods. outofsample model. 4.2.2 Analytical Procedures This chapter uses many numerical and econometrical techniques to measurein and outofdifferent timeseries volatility models. They are carried out using the of sample performance following procedures: i) Construct the S&P 500 futures series by rolling over sixtyeight S&P 500 futures 19831998. for The issues futures the to the period of relating contracts rollovers in full detail in described 4.3.1; section are contracts ii) Partition the constructed timeseries into four nonoverlapping segments. The data is by the the observation that the series do not exhibit motivated of partitioning homogeneousbehaviour over the entire 16year period; iii)
Estimate the parameters of APARCH models and apply likelihoodbased statistics to different in APARCH The the of performance models relative each subperiod. assess framework provides a general specification of the volatility dynamics that nests many loglikelihood for be directly tests the and test models, ratio to can used wellknown different Consequently, model specifications. of the effectiveness of robustness 118
Chapter 4: An Empirical Comparison of APARCH Models
be transformation can and power examined within the asymmetric parameterisation context of APARCH specifications; iv)
Study how structural change of volatility can have an influence on the performance of To support the asymmetrical and symmetrical conditional volatility models. hypothesis that performance of asymmetric and symmetric models are prone to the developed by Hamilton threestate the and a regimeswitching model samples, state of Susmel (1994) is employed to identify any structural breaks in volatility of the S&P 500 futures series from 1983 through 1998. This regimeswitching model stipulates that conditional variance is selected from a number of possible ARCH processes depends upon the state that eventuates; which
loginclusion EGARCH. In the Repeat to the the analysis with of above addition v) likelihood inferences, we explore the ability of additional statistical error functions that allow for asymmetry in the loss functions of investors to track the insample forecasting performance of conditional volatility models. functions used in this study are listed in appendix B. 2;
The statistical error
illustrate trading Conduct to the quality of various straddle exercises outofsample vi) forecasts. volatility conditional
4.2.3 Conditional Volatility Models 4.2.3.1 APARCH Specification The APARCH (Ding et al., 1993) family is an ideal specification to study the longmemory in general since it can nest many popular models in a
process and conditional volatility Adopting investigate the these framework. therefore to specifications allows one common keeping the specifications existing of whilst empirical analysis number a of performance manageable.
Seven models are included as special cases: APARCH,
TSGARCHI, TSGARCHII,
GJR and TARCH.
ARCH, GARCH,
Appendix A. 1 shows the functional forms of
A framework APARCH more general models. which can also nest a number of these nested in Hentschel including by APARCH family, is the the models GARCHtype models, given (1995)34:
details. 34See for 2.2.3.5 section
119
Chapter 4: An Empirical Comparison of APARCH Models
E, = e1ht
bi) b; I f; (e,) =1e1c(e, hs 1
°sYg =ao+arhr;.
fr (e, )+ý /=1
i=1
ha, 1 S
S>0. and where 1<_c51, v>0 According to Engle and Ng (1993)35,the b parameter controls the magnitude and direction of It, "rotations". in the the space whilst produces c et_1 a shift ý#> transformation
S>1 if and
S controls the shape of the
the transformation is convex; otherwise it is concave.
The
APARCH model is a special case of v=S, b = 0,1c<_ 1:
s, = eh, pq
ha = ao +I
s, ý (I E, a, _, c;
_J)S
J=1
+1ß, /=1
hrdJ
Using S&P 500 returns data, some of Hentschel's important results are: 1) 8 =1.5 when v=S;
2) the c parameterwas neither statistically nor economically significant in the model;
3) small shocks made more contributions to volatility, but not large shocks. The "shifting" of dominating factor in the impact was curve modelling asymmetry. As a result, the news The b function for APARCH(1,1) than significant more c. autocorrelation was of presence (1996): Granger by Ding derived and was ý=Eýe1
i5
ý(1a,
ßý
pi
a1 +
ý1
) +/3, Pl(al Pk =
ý )(l+al Q, +ßý) 1 +/312 +2a, /3)
k1
It is noted that autocorrelations of APARCH models decrease exponentially, not hyperbolically. Ding et al.'s results showed that the estimated power S was 1.43 and its long to equal which suggested memory and c significant parameter 0.373, asymmetric leverageeffects did exist in S&P 500 returns.
for details 2.2.3.3 35The Engle Ng (1993). to is section of and referred reader
120
Chapter 4: An Empirical Comparison ofAPARCH Models
4.2.3.2 Lag Structure of APARCH Models A substantial simplification in comparing models can be made if one imposes a fixed lag Moreover, Pagan by the to the nested models and order of p=q restricting structure =136. Schwert (1990) showed that loworder GARCH models could fit stock return volatility benefit beyond including Therefore, the of additional p+q=2 parameters extremely well.
is
(e. is found be In the that to special g. case suffice applications, most p=q small. =1 very Akgiray, 1989; Bollerslev et at., 1992; Lamoureux and Lastrapes, 1990; Poon and Taylor, 1992; Engle, 1993; Taylor, 1994; Kang and Brorsen, 1995; Antoniou and Holmes, 1995; Jorion, 1995; Antoniou et al., 1998; Duan and Wei, 1999).
Throughout this study,
APARCH(1,1) is the unrestricted model. In addition to APARCH(1,1), we will present but for nonnestedasymmetric models, EGARCH (Nelson, 1991)37to wellknown estimates a complement our analysis.
4.2.3.3 EGARCH The exponential GARCH (EGARCH) model was invented by Nelson (1991) in response to the criticisms volatility.
that the stock returns were negatively correlated with changes in return
EGARCH
considers asymmetry in the variance equation.
The EGARCH(1,1)
follows: be as modelled specification can r, =g(x, _,;a)+E, N(0,1) h, e, = e,
e, 11,  N(0, h, ) _,
log h0 = V+A. iz, +22(1z, 1(2/7l)os)+, _, _,
where z, =h
ßlogh,
x _,
is the normalisedresidual.
increases X1 implies that the conditional variance; it measures the shock a negative A negative sign effect.
An estimated positive X2 indicates that a shock greater than (2/t)°'S
also
it measures the size effect. This model accommodates the variance; increases the conditional between stock returns and volatility asymmetric relation
changes. The degree of asymmetry
36With the exception of ARCH and TARCH in which p=1, q=0. 37Since EGARCH is not nestedwithin APARCH, they cannot be compared with the loglikelihood test.
121
Chapter 4: An Empirical Comparison ofAPARCH Models
S=I++. by be the the absolute of or skewness can measured ratio value
In other words,
it can be said that a negative standardised innovation (bad news) increases volatility
S times
more than a positive standardised innovation of an equal magnitude. The use of logarithms also means that parameters can be negative without
the variance becoming negative.
Therefore, it is not necessary to restrict parameter values to avoid negative variances as in the ARCH and GARCH models.
4.2.4 Summary of the Methodology Sections 4.2.13 have reviewed the performance criteria, models and strategiesused for the EGARCH of and APARCH models. It should be noted that the performance comparison of investigate is 1) to: the effectivenessof asymmetric parameterisationand this study purposeof 2) impact the transformation; study of structural change of volatility on the power We and symmetrical models. asymmetrical assessthe performance of our of performance The intended inis both to trading use outofsample. of and outofsample primarily models illustrate the usefulnessof our conditional volatility forecasts. The next section discussesthe S&P 500 in the this study. of partitioning returns series and construction
4.3 Data Description The dataset comprises of daily settlement prices of S&P 500 futures and its options for the 3 We 1998. Chapter data described in 1983 the from through and use same options period We data identical 3.3.4. filters in that to the to those are options section outlined apply several for its 3 futures Chapter S&P 500 the to contract options. specifications and of refer the reader
4.3.1 Rollover of S&P 500 Futures Contracts In order to investigate volatility forecastability, a futures series is required. Sixtyeight futures January 1983 between December through 1998. Because the maximum studied are contracts is futures 500 two years, a continuous series of nearby daily futures S&P contract life span of is It be futures that wellknown the to constructed. contracts can rollovers of prices needs biases in the timeseries various properties of the artificial price series, generate significant
122
Chapter 4: An Empirical Comparison ofAPARCH Models
dependingupon the rollover method chosen. The necessarydecisions involved in rolling over contractsinclude: i) The point in time at which the current contract is rolled to the next; ii) The adjustmentof price level of the contract upon rollover. According to Ma et al. (1992), it appears that different conclusions can be drawn from the empirical
results estimated from timeseries generated from different
contract rollover
differences in results cannot be predicted. While adjusting Moreover, the most of methods. the differential
price levels at rollover dates reduces volatility,
some artificial but drastic
if The the created returns are computed are using adjusted price series. errors measurement large biases direction that the the the error can get so measurement size of and magnitude of from the different rollover methods is lost. Two subtle problems arise when the price levels different differenced times over ranges of the time series: multiple are
i) The leveladjust procedure effectively replaces the large positive/negative daily price dates Consequently, the the variance and rollover with zero price changes. at changes be biased; estimates may correlation serial ii) Negative price syndrome. Futures prices can become negative if they are differenced multiple times. There is no "best" method to rollover contracts. Despite the fact that rollover methods are long has be degrees timeseries to to of a constructed provide enough potentially problematic, freedom for any meaningful statistical inference. We avoid rolling over at the delivery date During it the maturity months, the nearby generates excessive volatility. always almost since futures prices are rolled over by the daily prices of first deferred contract. The rollover is futures day first Following the the trading this of maturity the month. method, occurred on datasetcontains 4,046 observations.
4.3.2 Partitioning
Statistics Descriptive for TimeSeries and
The S&P 500 futures timeseries constructed in section 4.3.1 are divided into four non19831986,19871990,19911994 19951998. The and partitioning of periods: overlapping by do behaviour that the the homogeneous is observation data series not exhibit the motivated S&P The 500 futures is 16year I(1) in period. series the each of these periods. entire over futures 500 levels are employed to calculate returns. Each S&P logs in of differences First 123
Chapter 4: An Empirical Comparison ofAPARCH Models
period contains about 1,000 observations. Tables 911 show the descriptive statistics for r, r2 Ir1. The DickeyFuller test rejects the null hypothesis that there is a unit root in the full and sample and each of the subperiods. The JarqueBera statistic also rejects the null hypothesis that r, r2 or IrI is normal in the full sample and all of its subperiods. The standard deviation of 19871990's return is 0.017467, which is highest among all subperiods. Skewness is negative in all periods except in 19911994, which is slightly positive. Therefore, it is more likely to have negative than positive returns. Excess kurtosis is 179 for the entire series. Excess kurtosis in subperiods 19831986,19871990,19911994 1998 are 2.5,148.8,3.1
and 8.2, respectively.
The 19871990 return series has the most
kurtosis. most positive and excess negative skewness futures returns are fattailed and not normal.
and 1995
Our preliminary statistics posit that
Figures 1933 are the sample autocorrelation
IrI 95% for their with confidence intervals +/and r2 plots r,
1.96
Figures 19,25 and 31 .
loworder there that some small negative are show return autocorrelations in 19831998, 19871990and 19951998. In addition, LjungBox statistics for r in table 9 are significant in 19831998,19871990 and 19951998,which also suggestthat is are serial correlated. An inspection of their correspondingautocorrelation plots, however, show that is are not related to many lags  an indication of short memory. This suggeststhat volatility in the distant future is insensitive to current information in subperiods. LjungBox
statistics for r2 in table 10 are significant in all periods except in 19831986.
Figures 20,23,26,29
and 32 are correlograms for r2. It is evident from figures 23 and 29
do 19911994 in 19831986 and not contain many lags of memory. Table 11 shows that r2 Ir1. for Although descriptive returns themselves contain little serial correlation, the statistics there is substantially more correlation in absolute returns.
LjungBox
IrI for statistics
are
in 19831986. On inspection 33, in figures 21,27 except periods we all and of significant have IrI high 70 lags; autocorrelations as they decay slowly and remain that can as observe significant
70,15 around until
and 75 lags in 19831998,19871990
long memory. of evidence respectively  an
and 19951998,
Figures 20,26, and 32 also display significant
first for in few lags the large peaks r2 of autocorrelations in 19831998,1987positive and
124
Chapter 4: An Empirical Comparison of APARCH Models
1990 and 19951998. However, they decay very rapidly and disappearcompletely within 10 lags. 4.3.2.1 Summary of Descriptive Statistics A number of observations can be drawn from the descriptive analysis in this section: i) Returns are not independent, although they are likely to be uncorrelated; ii)
Transformation of returns, i. e. I rI and r2 are more predictable.
These two series
have "longer memory" than returns; are statistically more "noisy" and correlated;
iii)
19831998,19871990,19951998
iv)
19831986 and 19911994 are statistically more "quiet" and less correlated.
4.4 Results & Analysis This section discusses our empirical results for the insample and outofsample tests. First, likelihood from for APARCH the the maximum estimation results of parameters we present investigate how influence have Second, structural on we change volatility of can an models. the performance of asymmetrical and symmetrical conditional volatility
models. Third, we
by introducing insample analysis the extend
and an additional
more loss functions
different Fourth, the volatility model. of we conditional evaluate performance asymmetrical by in trading conducting option models experiments a number of outofvolatility conditional sample periods.
4.4.1 Rationale forAR(1) Return Process 500 (1996), Tucker S&P in displayed Koutmos the the and serial correlation According to futures 'series could be a result from
thin trading
of some stocks, nonsynchronous
index or rollover of futures contracts. To the the stock prices component of of measurement in daily S&P 500 is the AR(1) to used series, the an autocorrelations process remove This AR(1) return process is given by: returns. formulate the conditional mean r=
ao +
125
Chapter 4: An Empirical Comparison ofAPARCH Models
where E, = v,h,, v,  i. i. d. studenttand h, is the conditional volatility. The AR(1) process simply states that returns are first order autocorrelated.
Among others,
Akgiray (1989), Hamilton (1989,1994), Heynen and Kat (1994) and Bracker et al. (1999) also in AR(1) the modelling the conditional mean equation. In practice, it is not use of suggested MA(1) (e. to returns using model g. Poon and Taylor, 1992; Ding et al., 1993). uncommon MA(1) periods.
500 S&P fit to the series but AR(1) is proven to be more suitable across all was Since the primary objective is to select a consistent conditional volatility
model
for S&P 500 index, the the than market microstructure of other specifications studying rather here38. not considered conditional mean are Empirical evidence frequently shows that normal distribution is not sufficient to remove fat tails from the empirical distribution of asset returns. Since nonnormal distributions usually density, better the than normal all models are estimated with tdistribution. results achieve The Berndt, Hall, Hall and Hausman (1974) algorithm is used to obtain parameter estimates loglikelihood the and maximises calculated numerically.
function in GAUSS program.
In addition, the gradient is
Our parameter estimates are insensitive to various initial conditions
for our sample, making it likely that global maxima are achieved.
4.4.2
InSample Analysis: Maximum Likelihood Estimations of APARCH
Parameters Section 4.4.2 has two goals: 1) to apply loglikelihood ratio tests to test for the "effectiveness" i. 2) features, to transformation APARCH power e. and asymmetric parameterisation; of investigate the insample performance of different APARCH models basedon loglikelihood likelihood 4.4.2.3. for in is Summary estimations maximum presented section statistics. The parameter estimatesfor the sevennested APARCH models in appendix A. 1 are obtained by maximisation of the loglikelihood function. The general APARCH framework is given by:
38For example, intraweek effects, such as Monday effect, are not to be studied. Mixon (2002) also argued that in by in implied S&P 500 be volatilities shortdated 8090% of the variation explained options could contemporaneousreturn.
126
Chapter 4: An Empirical Comparison ofAPARCH Models
rr = ao +a, r, +e , ha =ao+a, (ýE, _,
Tables 1216 present the estimates of APARCH models and their Akaike Information Criterion (hereafter AIC)39 and loglikelihood (hereafter LL) statistics for 19831998,19831986,19871990,19911994
and 19951998, respectively.
The 12th order LjungBox
2
EZ ' in for 17. Table 18 for table the the shown are shows model rankings and statistics h2 h,
AIC metric in eachof the periods. 4.4.2.1 InSample Results from Maximum Likelihood Estimations A number of observationscan be drawn from tables 1218. They are reported as follows: i) The intercept parameter ao in the conditional mean equation is positive but it is significant only in 19831998,19871990 and 19951998. The estimatedAR(1) term, be in likely 19831998, to is only suggesting returns are negative and significant a,, first order correlated in the long run. These results also suggest that the returns in 19831986 19951998; is process noise a white and process ii)
An observed trend for ARCH and TARCH in tables 1216 is that they have the lowest AIC and LL statistics in all sample periods. In addition, the 12`h order LungBox z
e2 in £ 17 for table and show that ARCH and TARCH are extremely poor statistics h1 ht
in capturing the first order and ARCH effects. AIC and LL statistics confirm that ARCH and TARCH are inferior models compared to other members of the APARCH family in every subsample. Because of this poor performance, one can safely disregardthe significance of ARCH and TARCH models; iii)
has the highest LL statistics in the full sample and its subperiods. This is not a surprising result since APARCH(1,1) is the least parsimonious model framework; APARCH the within APARCH(1,1)
S iv) The power parameter for APARCH(1,1) is estimated to be 0.9981 for the entire 16different is from in line is This significantly not with which one. result year period, the "Taylor effect" property.
The subperiods' results are mixed.
39AIC=LLR p wherep is the number of parameters.
127
The power
Chapter 4: An Empirical Comparison ofAPARCH Models
8 parameter is significant and close to one in 19871990and 19951998but it is close to two in 19831986and insignificant in 19911994; v) The asymmetric parameter y, is positive and significant for APARCH in 19831998, 19871990 and 1995199840but it is insignificant in 19831986 and 19911994;
vi) Model rankings for the AIC metric in table 18 indicate that the asymmetrical APARCH and TSGARCHII models are the top performers in 19831998,19871990 and 19951998; vii) Symmetrical GARCH ranks first for the AIC metric in 19831986 and 19911994, respectively. However, GARCH is only ranked fifth by the AIC metric in 19871990 and 19951998. 4.4.2.2 Are APARCH Specifications Effective? Estimation results obtained in section 4.4.2.1 lead us to cast doubt on the performance of APARCH specifications in subperiods because GARCH is found to have outperformed other in 19831986 19911994. and models more complex APARCH(1,1)
are often insignificant in subperiods.
In addition, estimated S and y, of In this section we use loglikelihood
LLR) (hereafter to examine the effectiveness of the power and asymmetric tests4' ratio APARCH in the context of subperiods. parameters within
4.4.2.2.1 LLR Test: Is Power Transformation Effective? The BoxCox power transformation is one of the most distinguishing features of APARCH S is believed in The be decay for long the to parameter the power responsible specifications. is it but in function improving insample fit? effective autocorrelation hypothesis following by the tests: answered conducting Ho : GJR vs HI: APARCH Hp : TSGARCH  II vs HI: APARCH
ý
y2
This question is
(1)
ýx42(1)
40 These results are consistent with Black's observation that negative shocks are weighed more heavily than in positive shocks modelling volatility. 41The idea behind the LLR test is that if the a priori restrictions are valid, the restricted and unrestricted logFormally be different. if model A, having n parameters,is nested within model B, not likelihood values should having m parameters,and the true parametersare within the parameterspacedefined by model A, then 2[ln(LB)ln(LA)] approximately follows a ;,'2 distribution with (mn ) degreesof freedom.
128
Chapter 4: An Empirical Comparison ofAPARCH Models
The following results are obtained for the hypothesis tests:
i) LLR tests can only reject GJR against APARCH(1,1)42at the 5% level in the entire sample,but not in any subperiods; ii) LLR tests cannot reject TSGARCHII (when S= 1) at the 5% level againstAPARCH in either the full sampleor any subperiods. Hence, it suggests that the incorporation of a free power parameter is less significant in subperiods.
The fact that LLR test cannot reject TSGARCHII
in favour of APARCH
also
in long is transformation the that of power usefulness sample questionable. signifies
4.4.2.2.2 LLR Test: Is Asymmetric Parameterisation Effective? Another fixture of APARCH specifications is what Engle and Ng (1993) term as "rotation" impact "news the curve"', when studying
in which y, is responsible for this "rotation" effect
framework. The help APARCH is the to use the of asymmetric y, supposed parameter within in leverage the underlying effects capture effective?
The following
asset but is this asymmetric parameterisation
hypothesis tests are conducted to test for the usefulness of
asymmetric parameterisation: Ho : GARCH vs H,: GJR Ho : TSGARCH I vs H,: TSGARCH  II Ho : GARCH vs H,: APARCH
x=(1) (1) x2 _

iL2(2)
The results are: i) LLR tests can only reject GARCH against GJR44 in full sample at the 5% level, but in 19951998. in except not other subperiods ii) LLR tests can only reject TSGARCHI (y, = 0) against TSGARCHII in full sample but in level, in 5% 19951998; subperiods not other except the at iii)
LLR tests cannot reject GARCH against APARCH45at 5% level in 19831986,19871990 and 19911994.
42APARCH(1,1) becomes GJR(1,1) when its power parameter S =2. for details Engle 4.2.1 43See Ng (1993). of 2.2.3.3 and and sections as GJR(1,1) becomes GARCH(1,1) when its asymmetric parameter y=0. 45ApARCH(1,1)
8=2 GARCH(1,1) becomes when
and Y, = 0.
129
Chapter 4: An Empirical Comparison ofAPARCH Models
Based on the above hypothesis tests, we conclude that the effectiveness of asymmetric parameterisation in subperiods is questionable within the APARCH
framework.
These
results are in agreement with Hentschel's (1995) findings that y, (rotation effect) is neither statistically nor economically significant.
4.4.2.3 Discussions for APARCH InSample Results Due to the complexity of our experiment design, it is necessary to restate the results of sections 4.4.2.12: models such as TSGARCHII and APARCH exhibit superior performance in the full period, 19871990 and 19951998 in terms of AIC and LLR 8, The it is is Therefore, to estimated power parameter, close not one. statistics.
i) Asymmetrical
TSGARCHII the that performance of surprising
is as robust as APARCH;
ii) Symmetrical GARCH outperforms other APARCH models during 19831986 and 19911994in terms of the AIC metric; iii)
Results from LLR tests question the usefulness of the power transformation in long APARCH in both the volatility within and context conditional short of modelling samples;
iv) Results from LLR tests cast doubt on the effectiveness of incorporating the leverage APARCH in the of context subperiods; within effect do TARCH incorporate lag ARCH models, which and not any conditional volatility, v) the times. model all every underperform In addition, table 18 also shows that the rankings across models are mixed in different sample indicate TSGARCHII that Our asymmetrical models as such results periods. in 19831998,19871990 fit better to tend provide
and APARCH
and 19951998 whilst
symmetrical
GARCH is favoured in 19831986 and 19911994. There is no evidence to confirm that there is a single model that will remain robust in every subperiod.
4.4.3 Are Conditional Volatility Models Prone to the State of Volatility? In this section we investigate whether structural change of volatility can have an influence on the performance of asymmetrical and symmetrical conditional volatility models.
130
Chapter 4: An Empirical Comparison ofAPARCH Models
4.4.3.1 Studentt SWARCH(3,2)L Model As mentioned earlier, the partitioning of the data is motivated by the observation that the futures series do not exhibit homogeneous behaviour over the entire 16year period.
To
support the hypothesis that performance of asymmetric and symmetric models are prone to the state of the samples, a 3state studentt Markovswitching
ARCHL(2)
developed by
Hamilton and Susmel (1994) is employed to model the S&P 500 futures series from 1/1/1983 to 31/12/1998 and identify any structural breaks in volatility. The specification of a leveraged studentt 3state, second orderARCH
Markovswitching
by: is model given
r, =a+Or, _, +u, Ss, 'uý
uý
ül=h1"vl
h2 = Ao +.ýü12, + /ý,
+
2u122
dlI
ü12l '
i. d. degrees unit variance with and studentt v of freedom, st = 1,2,3, g1=1 when where vt i. st =1, g2 =k when s, =2, and g3 =1 when st =3, Z>k>0,
dt_1=1 if ü, :50, dt_1=0 if _,
ü, >0 and 1<ý<1. _t This switching model postulates the existence of an unobserved state variable, denoted s, , that takes on the value of one, two or three.
This variable characterises the "state" or
"regime" that the process r is in at date t. When s, =1, the observed r is presumed to have been drawn from a low volatility state, when s, =2r
volatility
is presumed to have been drawn from a high
state, whereas when s, =3r
mid volatility
state.
is presumed to have been drawn from a
The transition probabilities
for the Markov
chain for evolution of the
is written as: variable unobserved state Pll
P=
P21
P31
P12 P22 P32 P13
i=j=1,2,3, where
P23
P33
p;, j =Prob(s, = jIs,
i), the transition probability from state i at time = _,
for The is j t. time process s, presumed to depend on past realisations of return t1 to state at
131
Chapter 4: An Empirical Comparison ofAPARCH Models
The inference about the particular state the process is in at r and state s only through s, ,. date t using the full sample of observations T can be used to construct the "smoothed probability",
AS, I rT, rr,,..., r3).
4.4.3.2 Detecting Structural Breaks in S&P 500 Futures Series The methods developed in Hamilton (1989) are used to estimate parameters for S&P 500 futures from 28/4/1982 to 31/12/1998 and make inferences about the unobserved regimes'. The estimated studentt SWARCHL(3,2)
specification with its standard errors are given by:
r, =0.06551+0.03369r,, +u, (0.01203) (0.01587) g, =1, g2 = 2.51152,g3 = 9.74334 (1.34630) (0.18513) h,' = 0.40248+ 6.68390. 104ü,, + 0.025455ü,2 + 0.078354d, "ü,?, 2 _, (0.01755) (0.03246) (0.029049) (0.01105) v=4.91416 (0.40128) 0.99242 0.00251 0.01594 P=0.00373 0.99493 0.01600 0.00375 0.00255 0.96801 4.4.3.2.1 Interpretations of Estimated SWARCH(3,2)L Parameters All coefficients, except ýi and '12, are significant. In addition, returns exhibit significant 22 its 1.4507 for Although level, 5% ttratio is insignificant the of the at serial correlation. line. degree from The freedom far is is 4.9142, is of out apart of which value not completely being normal. The conditional variance in states2 and 3 are estimated to be 2.512 and 9.743 e implying is 1, break in 3. 2 in large a subtle state and as of times as volatility regimes 5% level, leverage important the do that at suggesting an significant effects play and positive data. in our role
46We use GAUSS program downloaded from Hamilton's website at UCSD to estimate this model. We are thankful to JameHamilton's generosity.
132
Chapter 4: An Empirical Comparison of APARCH Models
The high transition probabilities (diagonal of P) indicate that if the system is in either state 1, 2 or 3, it is likely to remain in that state. Figure 34 plots the daily S&P 500 returns and smoothed probabilities
for state 3, the ultra high volatility
regions.
States 1 and 2 are
combined to display the milder volatility regions in figure 35. A 50% horizontal line is drawn in order to determine a switch of volatility state. Five short periods of highvolatility
episodes
can be identified characterising 4/198212/1982,3/19874/1987,10/19871/1988,8/199011/1990 and the twinpeak region between 10/199711/1997 and 7/199810/1998. inception of S&P 500 futures market in 1982, the market is extremely volatile.
At the
The October
1987 crash is likely to be responsible for the observed high volatility in 10/19871/1988. The market is judged to have been in the highvolatility
state in the second half of 1990 because of
the Gulf War. The surges of volatility in 10/199711/1997 and 7/199810/1998 coincide with the timing of the Asian Financial Crisis and Russian Debt Moratorium,
respectively.
The
origin of the 3/19874/1987 cannot be identified with any documented macroeconomic event.
4.4.3.3 Implications of Results from SWARCH(3,2)L Model The studentt SWARCHL(3,2) results confirm that: i)
19871990and 19951998are in the highvolatility state;
ii)
19831986and 19951998are in a more "subdued" state;
iii)
Studentt SWARCHL(3,2) is able to capture a number of economically important features of the data which may not otherwise be captured by standard conditional volatility models;
The studentt SWARCHL(3,2) models such as TSGARCHII
result has not only validated our assumption that asymmetric and APARCH
are more appropriate for volatile samples
(symmetric models such as GARCH are more appropriate for less volatile periods), but also lent credibility to our finding in sections 3.4.3 and 3.4.4 that the S&P 500 market has started behaving more volatile
and asymmetrically
since 1987.
Finally,
the multiple
volatility
breakpoints in S&P 500 futures series support the contention that perhaps there is no single APARCH model is rich enough to allow thorough assessment of asymmetry and structural effect at the same time.
133
Chapter 4: An Empirical Comparison ofAPARCH Models
4.4.4 Additional InSample Analysis: EGARCH and Statistical Loss Functions Evidence in sections 4.4.2 and 4.4.3 demonstrates that: 1) asymmetrical (symmetrical) models are superior to symmetrical (asymmetrical) models in more (less) volatile sample periods; 2) it is ineffective to incorporate power transformation and asymmetric parameterisation within the context of APARCH
specifications; 3) notably, multiple structural breaks in the S&P 500
futures series imply that no single APARCH model is rich enough to model volatility in the presence of asymmetry and structural change at the same time. In this section we extend our analysis by including (Nelson, 1991) model.
a popular asymmetrical EGARCH
In addition to the likelihoodbased
inferences, we also explore the
functions that allow for symmetry/asymmetry in the error statistical additional ability of eight loss functions of investors to track the insample performance of the conditional models.
4.4.4.1 Inclusion of EGARCH The EGARCH specification is not nested within the APARCH framework but it is important to study the performance of EGARCH with APARCH models because EGARCH is a more Wiggins limit. in diffusion to the converges which specification model parsimonious
A
EGARCH(1,1) can be written as: 6, = hy, 2
log h, = as + a, z1,+ Y,(I z, ,
1(217r)")+ß,
2 log hr,
d. i. degrees freedom. with unit studentt variance and v, v of i. where z, =hý, h, Estimates of the EGARCH model are displayed with APARCH models in tables 1216. The following are observed: i) All a, of EGARCH are negative, indicating that a negative shock increases the conditional volatility; ii)
All estimates of y, for EGARCH are positive and significant, suggesting that a shock increases the conditional volatility; (2/; r)''also than greater
iii)
Negative a, and positive v, are consistent with Black's leverage effect in equity returns;
EGARCH I< j ß, 1, that the processis stationary in each subperiod; All meaning iv) 134
Chapter 4: An Empirical Comparison ofAPARCH Models
IA
v)
is significantly
smaller during 19911994 and 19951998, which suggests that
the persistence of volatility clustering is relatively limited in the second half of the samples.
4.4.4.1.1 InSample Results for EGARCH Due to excess kurtosis and negative skewness in the S&P 500 futures returns series, the prior expectation is that asymmetrical models provide a better fit to the "noisy" periods as opposed to symmetrical models, and vice versa. Model ranking for AIC in table 19 shows that:
i) EGARCH is the best model in terms of AIC in 19831998and the predefined "noisy" subperiodsin 19871990and 19951998; ii) GARCH remains best model in terms of AIC in the predefined "quiet" subperiodsin 19831986and 19911994; iii)
EGARCH is only ranked fifth and fourth in the predefined "quiet" subperiods in 19831986 and 19911994.
iv) Overall EGARCH is best in terms of aggregateAIC score, followed by TSGARCHII GARCH APARCH GJR and are tied in fourth. whilst and Apparently our prior expectation that asymmetrical (symmetrical) models provide a better fit to the noisy (quiet) periods is upheld. In addition, theseresults demonstratethat EGARCH is APARCH 500 in S&P than to the capable and models capture consistent asymmetries a more market. 4.4.4.1.2 Discussion for InSample Results based on AIC
Our results indicate that the APARCH model performs poorly in the S&P 500 market. On the basis of the AIC metric, we find that EGARCH and GARCH provide the best insample fit for different The in data S&P500 subperiods. results obtained from the insample analysis the EGARCH both the since model measures sign and size effects: a unanticipated not are increases implies (sign that the shock a negative effect); an al conditional variance negative Al indicates that increases (2/i)°'S the a shock than greater also positive estimated Thus (the EGARCH the is size effect). model variance able to accommodatea conditional between stock returns and volatility changes. more complex asymmetric relation
135
Chapter 4: An Empirical Comparison ofAPARCH Models
4.4.4.1.3 Plausible Explanation for the Poor Performance of APARCH The APARCH model was originally designed to model the long memory property inherited in the power transformation of absolute returns.
There is, indeed, little evidence for long
memory in subperiods as evidenced in the autocorrelations plots in figures 23,24,26,27,29, 30,32 and 33 for r2 and Ir1.
Therefore, it is not surprising that the APARCH model is not
EGARCH GARCH large in to and even outperform a sample. able
4.4.4.2 Inclusion of Alternative Statistical Loss Functions In the previous analysis models are selected using the likelihoodbased inferences such as the AIC metric and LLR test. AIC and LLR statistics use information inferred from maximisation of loglikelihood
functions to give an indication of goodnessoffit of the model estimated,
from deviate the results of other more meaningful loss functions. which may likelihood
statistic selects the most appropriate model by maximising
The log
the probability
of
having the observed data given that the functional form of the probability density function is predetermined.
The AIC criterion, in turn, chooses the most parsimonious model by using
information from the loglikelihood
function plus a penalty adjustment involving the number
These it distributional Therefore, to criteria are subject parameters. assumption. of estimated is also very important to examine the ability of other distributionalfree
loss functions to track
the insample performance of conditional volatility models.
4.4.4.2.1 Procedures for Calculating InSample Statistical Errors In order to make complete our analysis, eight additional statistical loss functions are included in the insample study and their functional forms are shown in appendix A. 2. They are: i) Meansquare error (MSE); ii)
Mean absolute error (MSE);
iii)
Meanabsolutepercent error (MAPE);
iv)
Meanmixed error which penalises underpredictions (MMEU);
(MMEO); penalises Meanmixed which overpredictions error v) vi) vii)
Logarithmic loss function (LL); Heteroskedasticityadjusted meansquare error (HMSE);
136
Chapter 4: An Empirical Comparison of APARCH Models
viii)
Guassianquasimaximum likelihood function (GMLE).
MSE, MAE and MAPE are symmetrical loss functions whereas MMEO, MMEU, LL, HMSE loss functions. GMLE asymmetrical are and
Asymmetrical loss functions are included here
because investors do not necessarily attribute equal importance to both over and underpredictions of volatility
of similar magnitude.
The Performance of a conditional volatility
prediction model judged by its ability to predict future ex post volatility.
Following Bracker
and Smith (1999), the procedures in measuring the alternative insample statistical errors are:
i) Estimate the structural parametersfor the whole sample and each subperiod in our sample, i. e. 19831998,19831986,19871990,19911993 and 19951998; ii) Use e, as a volatitliy proxy esimatedfrom the structural mean equation at day t: rr =ao +a, r, +E, _,
iii)
Calculate the statistical error statistics according to table A. 2 where T is the number of is day h, the t. and per period expost conditional at predicted volatility observations 4.4.4.2.2 Results for Alternative Statistical Loss Functions
Table 21 exhibits the insample model rankings for MMEO and MMEU.
Table 22 shows the
insample rankings of models under HMSE, GMLE and LL statistics. Table 23 displays the insample rankings for MSE, MAE and MAPE.
Table 24 is the aggregated rankings for all
from Results 2123 functions. tables loss reveal that: statistical
i) Alternative model rankings are highly sensitive to the statistics used to assessthe is In but forecasts; EGARCH, MAE the by it first is the the case of of ranked accuracy MSE MAPE to and according statistics; model worst ii)
It is interesting to realise the exceptional performance of models ranked by some AIC in functions identified loss the that are previously as poor models statistical logby last ARCH TARCH For the example, and models are ranked analysis. likelihood inference statistics such as AIC, LL and GMLE but they are the best models MMEU the to statistic. according
4.4.4.2.3 Comments on Results for InSample Statistical Loss Functions is model clearly superior under alternative statistical criteria. that single no Our results show Consequently, it is not sensible to evaluate forecasting performance with only a single function. loss statistical
As suggested by Li (2002), these confusing results could be
137
Chapter 4: An Empirical Comparison of APARCH Models
introduced by the way volatility proxy was constructed using squared returns. Furthermore, the forecasting performance of different conditional volatility models may as well depend on the specific asset class under consideration.
The question remains what criteria should one
use to judge the superiority of any volatility forecast.
4.4.5 OutofSample Analysis: Trading S&P 500 Straddles
4.4.5.1 Background Whilst
investigations
insample
provide
useful
insights
into
volatility
basis the are selected only on models of expost information. performance,
forecasting For practical
forecasting purposes, the predictive ability of these models needs to be examined outofloss functions in Given the the results of conflicting competing statistical section sample. 4.4.4.2, it is recommended that the choice of error measure should depend on the ultimate The i. forecast. function forecasting the the the the procedure, e. utility user of of usage of AIC
metric
might
be more appropriate for selecting models when there is a given
distributional assumption. In the context of option trading, however, a call option buyer being MMEO the would prefer overpredictions, statistic47. with concerned
4.4.5.2 Volatility Forecasting Models The purpose of this section is to use an outofsample preferencefree approach to illustrate forecasting by different models predicting the onestep ahead changes of of the quality implied volatility
and conducting exante S&P 500 straddle trading.
investigation in this section are: i) EGARCH(1,1) ii)
GARCH(1,1)
iii)
ARCH(1)
iv)
A twostage predictor of conditional volatility.
47SeeBrailsford and Faff (1996) for details.
138
The models under
Chapter 4: An Empirical Comparison ofAPARCH Models
The forecasting models employed in our trading experiments are selected primarily based upon the results obtained in section 4.4.4. They range from naive models to the moderately complex class of ARCH models. Unlike other ARCHtype models that simultaneously estimate parameters from both the conditional mean and variance equations, a twostage regression model updates them independently. A twostage predictor models conditional volatility by first calculating the proxy for conditional volatility and then fitting a standard AR(1) model for this proxy. It is given by: r =ao+Sr
ý ý ýSý1= + Sr1 ao al h, is d. i. i. h,, the conditional volatility. and normal where e, = v, v, 
is a proxy for
expectedfuture volatility.
4.4.5.3 Trading Methodology As volatility
is unobservable, there is no natural metric for measuring the accuracy of any
however, Realised returns, allow one to test the performance of volatilitymodel. particular driven option trades and provide a test for market efficiency
with respect to volatility
forecasts. Many studies have used realised profits as a yardstick to assess the forecasting (1994)48. Engle (1993) Noh volatility models, conditional of e. g. al. et al. et performance and This section evaluates the performance of different volatility forecasting models by assessing be from 500 S&P trading generated can nearestthemoney49 straddles on whether profits futures with shortest remaining times to maturitySO.
4.4.5.3.1 Why Trading DeltaNeutral Straddles? According to Becker et al. (1991), the advantages of the use of nearestthemoney options are: i)
It reduces the nonsynchronous data problem because they have the greatest liquidity and represent accurate measures of exante market volatility;
48Seesection 2.5.6.1 for review of volatility trading. 49In the real world with limited supply of options we are more likely to trade nearestthemoneystraddles. 50It correspondsto region 11 in our database. They are nearestthemoneyoptions with maturities between 21 and 70 days.
139
Chapter 4: An Empirical Comparison of APARCH Models
ii) The purchase (sell) of a straddle is a simple strategy established as a volatility trade when a trader has a bullish (bearish) outlook on the volatility of the underlying futures; Nearestthemoneycall and put options have deltas close to 0.5 and 0.5, respectively, giving straddle a combined position of nearly zeros'. A nearzero delta would mean
iii)
that a small changeof the underlying futures in either direction would have little or no impact on the option price, thus making straddleessentially a volatility trade. 4.4.5.3.2 Trading Assumptions A few assumptions are needed in order to rationalise our trading strategy.
They are as
follows: i)
Conditional volatility is a reasonable proxy for atthemoney implied volatility. This is not unreasonable since many studies have found that the BlackScholes implied volatility
is empirically
indistinguishable
from
most stochastic and conditional volatility option pricing models when options are atthemoney and have short times to expiration;
ii) The changesin implied volatility is predictable in a statistical sense(e.g. Harvey et al., 1991,1992; Noh et al., 1994; Fleming et al., 1995; Bilson, 2002) but not the level of implied volatility, and profits depend on correct forecasts of the directional change of the underlying futures' volatility; iii)
Within this study, it is noted that the forecasting horizon matches the investment horizon, but not the remaining maturity of the straddles. The use of shortestmaturity impact the mitigate should of the maturity mismatch problem; straddles
iv) The forecasts made on week t for week t+1 are weekly instantaneousvolatility that tends towards the shortterm weekly mean volatility. An anticipated gain results from the expectedtendencyof options to increaseor decreasein volatility. 4.4.5.3.3 Trading Strategy The following proceduresexplain how we conduct our trading exercisesof S&P 500 straddles in this study:
st Alternatively, a long call and a short futures contracts with an appropriate hedgeratio can achievea deltazero is hedge be integer and we cannot trade fractional contracts of However, the to not necessary ratio too. position difficult deltaneutral more to obtain. Moreover, it requires a bigger investment in position futures, making the 1987). hedge (Wood at. et option margins than a pure
140
Chapter 4: An Empirical Comparison ofAPARCH Models
i) S&P 500 futures timeseries constructed in section 4.3.1 are divided into four successive nonoverlapping subperiods of four years, i. e. 19831986,19871990, 19911994and 19951998; ii) The last two years in each of the subperiods,i. e. 19851986,19891990,19931994, 19971998,are reservedfor outofsample evaluation purposes; iii)
Each volatility predictor forms a trading opinion by estimating a onestep ahead forecast on each week during outofsample periods;
iv) For each Wednesday of an outofsample period, we select the shortestmaturity straddle whose exerciseprice is closest to the current futures level; Wednesday's At trading on each week t, conditional volatility estimatesare the of end v) from processingthe most recent returns data up to and including week t 1; obtained vi)
The coefficient estimates are then applied to the information available on week t to form forecasts of the volatility change for week t+1;
is increase (decrease) from week t to week t+1, a straddle is If to predicted volatility vii) (sold); purchased viii)
Once a straddleposition is obtained at week t, the trade will be reversedat week t+1;
ix) We assume that options can be sold and purchased at daily settlement prices, and CME's futures options are used to compute the profit and on prices actual settlement loss; increased by is data Sample become one size as most recent available and each x) model's parameters are reestimated on every successive Wednesday over the the outofsample periods, therefore successive weekly estimates of of remaining be before information t can calculated week recursively on using only returns volatility week t. In addition, we assume that there is no margining requirement and agents are free to sell short. Each agent invests $100 and trades the nearesttothemoney contract.
When a straddle is
invest $100 is to the in Two proceeds plus allowed the general a agent riskfree asset. sold, 1) 2) transactions without costs; with transactions costs; we assume that considered: cases are basis ($250*0.25=$62.5) 25 both The for legs in point of trade costs commissions. a straddle is follows: buying computed straddles as rate of return on
100 (C, +P, C, P, )+100*I*rfRR, = _, , (Ct1+ pt1)
141
100 (C,  P, l 1)
* TC
Chapter 4: An Empirical Comparison of APARCH Models
where C, and P, are call and put prices at time t, respectively.
I is either 0 when the trade is
a buy or equal to 1 when the trade is a sell to allow agents accumulate interest in their accounts. TC is the transactions costs that can take on either 0 (no transactions costs) or 0.25 is This (with the transactions rf costs); riskfree rate. method of calculating the rate of point by identical Noh et al. (1994). indeed is the to one used returns
4.4.5.3.4 Why Not Other Trading Strategy? Whilst it may be argued that one can buy/sell straddle if forecasted volatility is above/below implied volatility, we must point out that this trading strategy assumes implicitly
that implied
However, less in is is forecastable. trading this our main assumption study stringent volatility forecastability directional implied but the the of change volatility, of not the only requires and level. In addition, multiperiod ahead forecast must be formed to match the maturity of the Noh, Among in the to trading aforementioned make strategy others, order workable. straddle Engle and Kane (1994) used this approach to trade straddle and found that the GARCH model was able to return profits.
Whilst this trading strategy is definitely rational, trading for the
directional change of volatility
is a more flexible strategy and we feel that that there is no
unique way to devise trading signals.
4.4.5.4 Trading Database The datasetcomprises of weekly settlement prices of S&P 500 futures options for the period from 1983 through 1998. The same options and futures databasesconstructedin Chapters3 3.3 for We trading the to 4 the outofsample sections experiment. reader refer and are used for 4.3 their contract specifications. and 4.4.5.4.1 Weekly Straddles S&P 500 index futures options are American and expire on the same day as the underlying futures contracts. The futures and option price data are Wednesday'sS2 settlement prices from CME. When a holiday occurs on Wednesday, Tuesday's observation is used in its place. The $250 is by is index level the index futures multiplied worth contract point and each size of one $250. The minimum move in the futures price is 0.1 point or $25. A onepoint change in
142
Chapter 4: An Empirical Comparison ofAPARCH Models
S&P 500 futures option premium represents the same dollar value of a onepoint change in the S&P 500 futures. As a proxy for the riskfree interest rate, we use daily middle rates on U. S. Treasury bills from Datastream matching maturity closest to the expiration date of the options.
4.4.5.4.2 Weekly TimeSeries Statistics The S&P 500 futures timeseries constructed in section 4.3.1 are divided into four successive four i. of years, subperiods e. 19831986,19871990,19911994 nonoverlapping
and 1995
1998. Results from the DickeyFuller test rejects the null hypothesis that there is a unit root in each of the four subperiods.
Skewness is negative for all subperiods53, suggesting that
be likely Excess is kurtosis for 19831986,19871990,1991to negative. more weekly return 1994 and 19951998 are 0.850,7.919,1.411 r
up to 10 lags are insignificant
autocorrelated.
and 2.081, respectively. LjungBox statistics for
for all subperiods, meaning that returns are not
In addition, correlograms for
r
also confirm
that serial correlation is
insignificant in any subperiods. These preliminary statistics posit that weekly returns are less leptokurtic and autocorrelated (closer to normally distributed) than daily returns amid weekly returns are more negatively is finding This consistent with the consensus that the longer the interval over which skewed. lesser is the the autocorrelation. returns are calculated,
Consequently, it is not necessary to
from firstorder the return series. autocorrelation remove any
4.4.5.5 Results of Trading AttheMoney Straddles 4.4.5.5.1 Preliminary Statistics for Directional Trading Signals This section aims to demonstrate that our four predictors produce very different buy/sell signals at times. Table 25 shows the correlations for the outofsample directional trading signals54generated from different volatility prediction models in each subperiod. Table 26 also exhibits some basic statistics for the forecastsof volatility changes. The mix of low values of positive and
52Wednesdaysare chosenbecausefew holidays fall on Wednesdays. 53 They are 0.242, respectively.
and 0.842 0.019 1.627,
for
19831986,19871990,19911994
143
and 19951998,
Chapter 4: An Empirical Comparison of APARCH Models
negative correlations coefficients
on outofsample
buy and sell signals in 19851986
confirms that at the inception of the S&P 500 options market volatility
predictors produce
very mixed opinions on their onestep ahead forecasts. After 19851986, however, all correlations coefficients are large and positive, indicating that buy likely issuing become have the to same or agree on more with each other our predictors sell signal. One plausible explanation for this dramatic change of forecasting behaviour is that it is caused by the increase of returns autocorrelations after the 1987 crash. It suggests that our conditional models are capable of picking up "volatility
clustering" or "memory", thus
making different volatility predictors to produce similar forecasts. Table 26 also shows that standard deviations of volatility
changes have become significantly
larger since 1987. The
ARCH is likely indicate high the that to model estimates and more produce min/max statistics therefore overpredict volatility changes. In contrary, the GARCH model is likely to have smaller estimates and underpredict volatility.
Tables 2730 presentthe beforetransactionscostsstatistics for each volatility predictor for all Since in 19851986 0.198 (ten The is straddles maturity of average weeks). year subperiods. in 1987, it has been reduced to around 0.12 year (six introduction contracts the of serial Although by increments divisible 19971998. integers in strike price are generally weeks) five, futures level raises from 139 to 1245.15 during the entire sample period. Therefore, deltaneutral towards the end of the sample period and standard to straddles are closer deviations of their delta also have decreasedsteadily from 0.107 in 19851986 to 0.021 in 19971998. In addition, the descriptive statistics show that call prices have increased from 5.403 in 19851986 to 27.349 points in 19971998. Correspondingly, put prices have also is Furthermore, 27.267 5.502 ARCH to from the very that points. model results show raised keen to produce buy signals, issuing the highest number of buys in three of out four subperiods. In contrast, both the EGARCH and GARCH models prefer selling than buying but GARCH issuing 19851986. in 99 buys to 5 willing short, more be even as sells versus perceived can buy likely that is descriptive the to twostage show statistics Finally, our as regression model
54The signal is 1 when it is a buy and 1 when it is a sell.
144
Chapter 4: An Empirical Comparison of APARCH Models
four Our in the that to suggests predictors under study are analysis subperiods. all as sell indeed quite different at times.
4.4.5.5.2 Profit and Loss: Trading AttheMoney Straddles Before Transaction Costs and No Delta Filter Without transactions costs, the EGARCH model has the highest rates of return per trade in 19851986 and 19891990, respectively.
In 19931994, the EGARCH model ranks second
in 19971998. The EGARCH is ARCH GARCH to the the model second model model. after Before transactions costs, profits can be made in 19851986 and 19931994, although only EGARCH and GARCH models can produce statistically significant returns at tratios of 1.66 and 2.48 in 1993199455, respectively.
Trading results also indicate that no predictor can
ARCH is in is in 19891990, the that and earning successful only model profit make any profit in 19971998.
Before Transactions Costs and 3% Delta Filter In the results discussed thus far, data are unfiltered.
A more rational trading approach is to
deltaneutral. to when nearestthemoney only straddles are close strategy exercise our Consequently, a filtering rule is applied to remove trades that do not satisfy putcallfutures delta 3134 less Tables 3%. by than to report with absolute trading straddles or equal parityS6 in 3% delta filter for beforetransactionscosts statistics with a± the each volatility predictor 19851986,19891990,19931994
and 19971998, respectively. Under this filter, the number
19851986,19891990,19931994 in transactions of 30.1%, 19.4% and 54.4%, respectively.
16.3%, in 19971998 traded only and are
Tables 3134 also show that standard deviations of
dramatically. been have delta reduced straddles'
After applying this filter, the EGARCH
i. in four highest trade three e. per has of return periods, rates the out of outofsample model 19851986,19891990 and 19931994. Although all predictors succeed in making profits in 19931994, only the EGARCH
and GARCH
models can produce statistically significant
in It 2.16. is 2.22 GARCH fails that also to and noted profits make any returns at tratios of Finally, in 19931994. both ARCH the and twostage regression except all subperiods
ss Since the tratio of return from trading straddle are assumedto be independent, the tratio is computed as a by divided deviation the the square root of to number of observations. standard mean ratio of 56From practical point of view it meansthe Europeanputcallfutures parity. SeeFung and Fung (1997).
145
Chapter 4: An Empirical Comparison of APARCH Models
four i. 19851986,19891990 losses in three out of outofsample periods, e. and models make 19971998although ARCH still remains first in 19971998. Before Transaction
Costs and 3% Delta Filter (excluding one spurious point)
Curiously, the performance of the EGARCH model is second to a simple ARCH model in 19971998. A careful scrutiny of our options data from 19971998 reveals that a "spurious" trade made between 22/10/1997 and 29/10/1997 is indeed very erratic.
The price of this
40.3 59.85 from During increased has to the same time period, points within a week. straddle its delta has decreased from 0.0097 to 0.6535. The timings of this "spurious" trade coincide height Asian 4.4.3.2. Financial Crisis identified in the at of the volatility of section surge with During 2023 of October 1997, the Hong Kong stock market suffers its heaviest losses ever, four its in days. A later Asian 27 October 1997, of value quarter a week nearly on of shedding jitters spill over on to world stock markets. The Dow Jones index plunges 554 points, its largest singleday point loss ever57. Therefore, it is not unreasonable to assume that a prudent trader would exercise extreme caution in such a chaotic trading environment. After removing this questionable data point, we find that the EGARCH model is first in terms of rate of Table 35 beforetransactionscosts for 19971998 19971998. the in exhibits statistics returns filter delta data the ±3% this after removal of the questionable point. These results also with is EGARCH the model the only profitable predictor in 19971998. that show
After Transaction Costs and 3% Delta Filter (excluding one spurious point) None of the profits reported in our trading strategiesthus far have attemptedto accountfor the With transactions 25 basis for legs, both the profits costs. transaction costs of points of effects between in dramatically although rankings the same orders. reduced predictors remain are The summary statistics for aftertransactionscostswith a± 3% delta filter are given in tables 3639 for each volatility predictor for the periods 19851986,19891990,19931994 and 19971998, respectively. No predictors can earn any profits in 19851986,19891990 and 19971998. In addition, EGARCH and GARCH have the first and second highest rate of from After in respectively. transactions subperiod, trade each costs, all predictors returns per but have of return rates only EGARCH and GARCH can generatereturns 19931994 positive This transactions tratios 1.49 1.45, costs at exceed of and respectively. that significantly
57Source from Tudor, G. (2000)
146
Chapter 4: An Empirical Comparison ofAPARCH Models
figure by 36, which shows the cumulative rate of return from straddle is argument supported trading of agents using EGARCH, GARCH, ARCH and a twostage regression model with transactionscosts and a± 3% delta filter in 19931994. 4.4.5.5.3 Trading Summary We report that EGARCH produces the highest rate of returns per trade in every subperiod. In addition, EGARCH
and GARCH
can generate statistical significant
exante profit
after
transactions costs. Therefore, we cannot deny that there are certain degrees of inefficiency and predictability
in the S&P 500 market.
Finally, our trading experiments also reveal the
deltaneutral trade to create a riskfree portfolio is not practical in the presumption of using event of large index movements. A new derivatives instrument is needed to allow traders and investors speculate on volatility more directly and efficiently.
4.5 Summary This chapter compares the performance of a group of welltheorised conditional volatility for 500 biases in S&P the termstructure account the that potentially can observed models futures options market. Sixteen years of daily S&P 500 futures series are used to examine the APARCH the models that use asymmetric parameterisation and power of performance transformation on conditional volatility and its absolute residual to account for the slow decay in returns autocorrelations. Our results are: i) No evidence can be found supporting the relatively complex APARCH models. Loglikelihood tests confirm that asymmetric parametersiation and power transformation in S&P 500 dynamics the characterising effective returns not are within the context of APARCH specifications; ii) Results from the 3state volatility regimeswitching model supported the notion that the performance of conditional volatility models is prone to the state of volatility of the returns series. Furthermore, loglikelihood based statistics stipulate that the EGARCH model is best in "noisy" periods whilst GARCH is the top performer in "quiet" periods; iii)
Overall, aggregaterankings for the AIC criterion show that the EGARCH model is the best performer;
147
Chapter 4: An Empirical Comparison of APARCH Models
iv) Insample results show that it is not sensible to evaluate forecasting performancewith only a single statistical loss function; v) Outofsample results demonstratethat the EGARCH model outperforms GARCH, and both of them can generatestatistically significant exante returns in one out of four sample periods; deltaneutral Trading that the trade to also reveal presumption of experiments using vi) is in large index the portfolio not practical event of a riskfree movements. create Our findings are not only significant to discretetime finance but also potentially meaningful for
continuoustime
volatility
stochastic volatility
literature
because continuoustime
stochastic
be limits the thought of as can of ARCHtype process. Nelson (1991), for models
instance, showed that EGARCH(1,1) limit. time continuous
converged to a specific bivarate diffusion model in
Moreover, Duan (1997) also proved that most of the existing bivariate
diffusion models that had been used to model asset returns volatility
could be represented as
limits of a family of GARCH models. When considering a stochastic volatility model, there look beyond incentive for little be to to simple a model volatility which allows seems Heston (1993). leverage such effect as a clustering and
148
Chapter 4: An Empirical Comparison ofAPARCH Models
Table 9: Descriptive Statistics for r 19831998
19831986
19871990
65.65442 [.000]
32.64347 [0.000]
32.16886 [.000]
32.61263 [.000]
34.76218 [.000]
Maximum
0.177493
0.037518
0.177493
0.042612
0.056547
Minimum
0.337004 0.000573
0.056886 0.000530
0.337004 0.000308
0.036987 0.000329
0.077621 0.000981
0.011845
0.009347
0.017467
0.007111
0.010881
DF stat.
Mean Std. Dev. Skewness Kurtosis3 Q(10) JarqueBera stat.
#. Obs.
19911994
19951998
5.279559 179.218
0.077911 2.470201
0.6452279 148.7873
0.211765 3.068323
0.532956 8.15034
79.085
10.925
59.226
8.9945
19.038
[0.363]
[.000]
[.532]
[.040]
5432196
257.5546
939563.7
404.9461
2848.995
[.000]
[.000]
[. 000]
[.000]
[.000]
1.0001
4045
1009
1011
1013
1012
Table 10: Descriptive Statistics for r2 19831998
19831986
19871990
19911994
19951998
58.38758 [.000]
31.20798 [.000]
29.287 [. 000]
31.23613 [.000]
23.41297 [.000]
Maximum
0.113572
0.003236
0.113572
0.001816
0.006025
Minimum
0.000000
0.000000
0.000000
0.000000
0.000000
Mean
0.000141
8.76E05
0.000305
5.06E05
0.000119
Std. Dev.
0.001883
0.000185
0.003739
0.000114
0.000374
Skewness
55.24303
7.173269
28.14637
7.123234
10.05873
Kurtosis3
3269.947
92.15109
836.6669
80.24314
132.6307
342.93
7.7053
83.672
26.486
225.80
[.0001
[.720]
[. 000]
[0.003]
[.000]
1.80E+09
365663.5
29621473
280344.5
758814.9
[. 000]
[.000]
[. 000]
[.000]
[.000]
4045
1009
1011
1013
1012
DF stat.
Q(10) JarqueBera stat. #. Obs.
149
Chapter 4: An Empirical Comparison of APARCH Models
Table 11: Descriptive Statistics for
Irl
19831998
19831986
19871990
19911994
19951998
DF scat
49.4674 [.000]
32.33748 [.000]
23.33977 [.000]
30.55499 [.000]
25.51959 [. 000]
Maximum
0.337004
0.056886
0.337004
0.042612
0.077621
Minimum
0.000000
0.000000
0.000000
0.000000
0.000000
Mean
0.007115
0.006877
0.009016
0.005122
0.007449
Std. Dev.
0.009484
0.006349
0.01496
0.004940
0.007988
Skewness
13.43799
1.903316
12.66372
2.074516
3.159958
Kurtosis3
394.0034
6.027266
245.3357
7.04718
17.11600
1332.2
6.5947
425.59
42.258
319.56
[.000]
[.763]
[.000]
[.000]
[. 000]
26285933
2136.489
2562509
2822.777
14037.23
[.000]
[.000]
[.000]
[.000]
[.000]
4045
1009
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Q(10) JarqueBera stat. #. Obs.
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W
Chapter 4: An Empirical
Comparison of APARCH Models
L2
Table 17: 12thorder LjungBox statistics for
and
`2 t
19831998
19831986
14.2359 0.2859
10.3005 0.5896
Q12 Qý 2
12.8218
ARCH
APARCH Q12
Q12
Qý
19911994
19951998
8.6629 0.7314
16.9122 0.1529
12.6628 0.394
5.0733
3.3533
7.7935
6.3095
0.3821
0.9555
0.9925
0.801
0.8997
29.2377
13.9869
29.1073
14.987
17.3921
0.0036
0.3015
0.0038
0.2421
0.1354
99.3827
23.9331
85.17
418.1956
9.864
19871990
0.000
0.6279
0.000
0.0208
0.000
16.4104
9.8438
10.973
19.0669
15.2964
Qiz
0.1732
0.6297
0.5312
0.0869
0.2256
Qz iz
8.3566 0.7567 15.9829 0.192
5.3111 0.9468 10.1537 0.6025
2.6837 0.9974 10.3789 0.5828
8.0623 0.7802 18.3103 0.1066
7.6012 0.8155 14.704 0.258
22.3838
5.4356
4.4615
9.3794
11.1127
0.0334
0.9418
0.9736
0.6702
0.5193N
14.2362
10.1624
8.864
16.9914
12.6711
Qlz
0.2859
0.6017
0.7145
0.1499
0.3934
Qz
12.7818 0.3851 15.549 0.2128X
5.4226 0.9424 10.2391 0.595
3.9355 0.9846 9.6854 0.6435
7.4777 0.8245 18.6708 0.0968
6.1887 0.9063 12.9384 0.3735
5.7647 0.9275 28.1106 0.0053
5.066 0.9557 13.5663 0.3293
3.1856 0.9941 34.7099 0.0005
7.0684 0.8531 14.6532 0.261
7.0385 0.8551 18.0688 0.1136
GARCH
TSGARCHI Ql2
Q
lz
TSGARCH1I
12
GJR Qiz Q2 'z 1
TARCH Qlz
550.9643
9.675
140.4172
24.0214
83.6644
0.000
0.6444
0.000
0.0202
0.000
10.1609
8.8706
17.226
12.5406
Qlz
14.2566 0.2846
0.6018
0.7139
0.1413
0.4033
Q2
8.2834
5.4073 0.943
2.7761 0.9969
7.1652
5.7963
0.8465
0.926
Q
12
EGARCH
12 1
0.7626
The pvalues are reported in italic.
156
Chapter 4: An Empirical Comparison of APARCH Models
Table 18: Model Rankings for the AIC Metric (Excluding EGARCH) 19831998 19831986 19871990 19911994 19951998 AIC
AIC
AIC
AIC
AIC
APARCH
1
4
1
4
2
ARCH
6
6
6
7
7
GARCH
5
1
5
1
5
TSGARCHI
3
3
4
5
4
TSGARCHH
1
5
2
2
1
GJR
4
2
3
3
3
TARCH
7
7
7
6
6
Table 19: Model Rankings for AIC Statistics (Including EGARCH)
19831998 19831986198719901991199419951998 AIC
AIC
AIC
AIC
AIC
APARCH
2
4
2
5
3
ARCH
7
7
7
8
8
GARCH
6
1
6
1
6
TSGARCHI
4
3
5
6
5
TSGARCHII
2
6
3
2
2
GJR
5
2
4
3
4
TARCH
8
8
8
7
7
EGARCH
1
5
1
4
1
157
Chapter 4: An Empirical Comparison of APARCH Models
Table 20: Aggregated Rankings for AIC Statistics (Including EGARCH) Score
Rank
APARCH
14
4
ARCH
30
7
GARCH
14
4
TSGARCHI
19
6
TSGARCHII
13
2
GJR
13
2
TARCH
30
7
EGARCH
11
1
Note: Score is the sum of the rank for each model in each subperiod.
Table 21: Model Rankings for MMEU and MMEO Criteria 19831998
19831986
19871990
19911994
19951998
MMEU MMEO MMEU MMEO MMEU MMEO MMEU MMEO MMEU MMEO APARCH
8
2
3
6
7
2
8
1
6
3
ARCH
2
8
1
8
2
7
2
7
1
8
GARCH
4
5
4
4
3
6
3
6
3
6
TSGARCHI
5
6
7
1
4
4
4
5
5
5
TSGARCHII
7
3
8
2
8
1
7
2
8
1
GJR
3
4
5
5
5
5
5
4
4
4
TARCH
1
7
2
7
1
8
1
8
2
7
EGARCH
6
1
6
3
6
3
6
3
7
2
158
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Chapter 4: An Empirical Comparison ofAPARCH Models
Table 24: Aggregated Rankings for Statistical Loss Functions
APARCH ARCH GARCH TSGARCHI TSGARCHII GJR TARCH EGARCH
MSE
MAE
MAPE
MMEU
MMEO
LL
HMSE
GMLE
Rank
Rank
Rank
Rank
Rank
Rank
Rank
Rank
(Score)
(Score)
(Score)
(Score)
(Score)
(Score)
(Score)
(Score)
3
3
3
6
3
4
6
1
(12)
(14)
(11)
(24)
(12)
(15)
(21)
(7)
7
8
5
1
7
7
1
7
(29)
(31)
(21)
(6)
(30)
(30)
(11)
(29)
4
6
2
3
6
3
8
5
(14)
(21)
(10)
(13)
(22)
(14)
(27)
(21)
5
4
5
5
4
6
5
6
(18)
(15)
(21)
(20)
(15)
(19)
(19)
(22)
1
2
4
8
1
1
2
2
(7)
(13)
(15)
(31)
(6)
(6)
(13)
(10)
2
5
1
4
5
5
7
3
(11)
(19)
(7)
(19)
(18)
(17)
(23)
(13)
6
7
7
1
7
7
2
7
(22)
(27)
(27)
(6)
(30)
(30)
(13)
(29)
8
1
8
7
2
2
4
3
(31)
(4)
(32)
(25)
(11)
(13)
(17)
(13)
Note: Score is the sum of the rank for each model in each subperiod.
161
Chapter 4: An Empirical Comparison of APARCH Models
Table 25: Correlations Between OutofSample Buy and Sell Signals 19851986 EGARCH
EGARCH
1.000
GARCH
GARCH
19891990
ARCH
2STAGE
167
070 .
064 .
1.000
058 .
.054
1.000
017 .
.
ARCH
EGARCH
1.000
GARCH
ARCH
1.000
595 .
468 .
480 .
1.000
702 .
677 .
1.000
942 .
ARCH
2STAGE
304 .
264 .
1.000
758 .
7578 .
1.000
84453 .
EGARCH
GARCH
ARCH
1.000
622 .
154
394 .
151
490 .
1.000
454 .
.
1.000
.
1.000
2STAGE
Table 26: Statistics for Forecasts of Volatility
Changes 19891990
EGARCH
GARCH
104
104
104
31.643
3.862
29.058
#.samples
2STAGE
1.000
19851986
Min
406 .
19971998
EGARCH
GARCH
2STAGE
1.000
19931994
Max
ARCH
1.000
2STAGE
EGARCH
GARCH
ARCH
2STAGE
EGARCH
GARCH
104
103
103
103
103
25.187
8.521
87.7453
60.325
112.367
209.584
1.675
455 .
7.213
30.678
17.235
48.265
61.292
ARCH
2STAGE
Mean
.
806
.008
353 .
.098
2.263
1.058
2.044
9.551
Std. dev
579 .
3.228
9.275
1274
21.747
13.126
20.898
48.297
.
19931994
EGARCH
GARCH
19971998
ARCH
2STAGE
EGARCH
GARCH
ARCH
2STAGE
103
103
103
103
103
103
103
103
Max
107.833
81.722
166.999
125.45
80.161
64.820
62.662
34.029
Min
8.565
26.559
60.505
50.472
17.288
7.476
24.824
23.583
1.079
1.258
2.524
2.160
1.269
1.020
192
286 .
14.265
12.685
24.325
21.268
14.799
10.430
10.158
9.933
#.samples
Mean FStd. dev
162
.
Chapter 4: An Empirical Comparison of APARCH Models
Table 27: Beforetransactionscosts Statistics for 19851986 without Filter EGARCH Rate of Returns
0.606881
Std. R. of Returns
GARCH
ARCH
2STAGE
0.85174
0.497811
1.30702
13.32615
13.3071
13.51409
13.26873
104
104
101
104
Ave. Delta
0.013777
0.013777
0.01441
0.013777
Std. Delta
0.10788
0.10788
0.108375
0.10788
Ave. Maturity
0.198419
0.198419
0.199023
0.198419
Ave. Call Price
5.402885
5.402885
5.442574
5.402885
Std. Calls
2.309928
2.309928
2.331023
2.309928
Ave. Put Price
5.501923
5.501923
5.527228
5.501923
Std. Puts
2.149022
2.149022
2.163903
2.149022
# of Buys
45
5
52
54
# of Sells
59
99
49
50
#. of Trades
Table 28: Beforetransactionscosts Statistics for 19891990 without Filter EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
0.12772
2.45341
2.5796
3.34284
Std. R. of Returns
14.71959
14.49996
14.47757
14.31479
103
103
103
103
Ave. Delta
0.025737
0.025737
0.025737
0.025737
Std. Delta
0.077366
0.077366
0.077366
0.077366
0.11443
0.11443
0.11443
0.11443
Ave. Call Price
7.540777
7.540777
7.540777
7.540777
Std. Calls
2.016813
2.016813
2.016813
2.016813
Ave. Put Price
7.399515
7.399515
7.399515
7.399515
Std. Puts
1.937485
1.937485
1.937485
1.937485
# of Buys
38
41
50
50
# of Sells
65
62
53
53
#. of Trades
Ave. Maturity
163
Chapter 4: An Empirical Comparison ofAPARCH Models
Table 29: Beforetransactionscosts Statistics for 19931994 without Filter EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
2.279404
3.350164
1.714102
2.004451
Std. R. of Returns
13.94236
13.72455
14.0219
13.98359
103
103
103
103
Ave. Delta
0.011027
0.011027
0.011027
0.011027
Std. Delta
0.075565
0.075565
0.075565
0.075565
0.11443
0.11443
0.11443
0.11443
6.85
6.85
6.85
6.85
Std. Calls
1.560276
1.560276
1.560276
1.560276
Ave. Put Price
6.892233
6.892233
6.892233
6.892233
Std. Puts
1.337941
1.337941
1.337941
1.337941
# of Buys
31
38
53
52
65
50
51
#. of Trades
Ave. Maturity Ave. Call Price
# of Sells
72
L
Table 30: Beforetransactionscosts Statistics for 19971998 without Filter EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
0.02342
1.03321
0.753204
0.96787
Std. R. of Returns
11.60207
11.55187
11.58106
11.55768
103
103
103
103
Ave. Delta
0.026539
0.026539
0.026539
0.026539
Std. Delta
0.021121
0.021121
0.021121
0.021121
Ave. Maturity
0.121851
0.121851
0.121851
0.121851
Ave. Call Price
27.27524
27.27524
27.27524
27.27524
Std. Calls
7.941751
7.941751
7.941751
7.941751
Ave. Put Price
27.53689
27.53689
27.53689
27.53689
Std. Puts
8.314578
8.314578
8.314578
8.314578
# of Buys
39
31
48
48
# of Sells
64
72
55
55
of Trades
164
Chapter 4: An Empirical Comparison of APARCH Models
Table 31: Beforetransactionscosts Statistics for 19851986 with ± 3% Delta Filter EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
0.342188
0.32483
2.41412
4.01786
Std. R. of Returns
13.90462
13.90696
13.67675
13.26126
17
17
17
17
0.0012
0.0012
0.0012
0.0012
Std. Delta
0.016493
0.016493
0.016493
0.016493
Ave. Maturity
0.227881
0.227881
0.227881
0.227881
Ave. Call Price
6.123529
6.123529
6.123529
6.123529
Std. Calls
1.947283
1.947283
1.947283
1.947283
Ave. Put Price
6.732353
6.732353
6.732353
6.732353
Std. Puts
2.304136
2.304136
2.304136
2.304136
# of Buys
10
1
11
7
# of Sells
7
16
6
10
#. of Trades Ave. Delta
Table 32: Beforetransactionscosts Statistics for 19891990 with ± 3% Delta Filter EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
1.170446
0.62096
2.03609
2.21947
Std. R. of Returns
13.79986
13.82599
13.6767
13.64618
31
31
31
31
0.0042
0.0042
0.0042
0.0042
0.016332
0.016332
0.016332
0.016332
0.11445
0.11445
0.11445
0.11445
Ave. Call Price
7.320968
7.320968
7.320968
7.320968
Std. Calls
1.778987
1.778987
1.778987
1.778987
Ave. Put Price
8.006452
8.006452
8.006452
8.006452
Std. Puts
2.087851
2.087851
2.087851
2.087851
# of Buys
17
16
19
20
# of Sells
14
15
12
11
#. of Trades Ave. Delta Std. Delta Ave. Maturity
165
Chapter 4: An Empirical
Comparison ofAPARCH
Models
Table 33: BeforetransactionscostsStatistics for 19931994with ± 3% Delta Filter EGARCH
GARCH
ARCH
5.28826
5.189391
3.114228
3.825862
10.68355
10.73501
11.54099
11.31548
20
20
20
20
Ave. Delta
0.00638
0.00638
0.00638
0.00638
Std. Delta
0.018011
0.018011
0.018011
0.018011
Ave. Maturity
0.123425
0.123425
0.123425
0.123425
7.05
7.05
7.05
7.05
1.208087
1.208087
1.208087
1.208087
7.5975
7.5975
7.5975
7.5975
1.462377
1.462377
1.462377
1.462377
# of Buys
6
6
9
8
# of Sells
14
14
11
12
Rate of Returns Std. R. of Returns #. of Trades
Ave. Call Price Std. Calls Ave. Put Price Std. Puts
2STAGE
Table 34: Beforetransactionscosts Statistics for 19971998 with ± 3% Delta Filter EGARCH Rate of Returns
0.061182
Std. R. of Returns
GARCH
ARCH
2STAGE
1.07436
0.483998
1.72362
12.42948
12.3787
12.42263
12.30109
56
56
56
56
Ave. Delta
0.012262
0.012262
0.012262
0.012262
Std. Delta
0.01277
0.01277
0.01277
0.01277
Ave. Maturity
0.122945
0.122945
0.122945
0.122945
Ave. Call Price
26.02054
26.02054
26.02054
26.02054
Std. Calls
6.334112
6.334112
6.334112
6.334112
Ave. Put Price
27.26696
27.26696
27.26696
27.26696
Std. Puts
6.569587
6.569587
6.569587
6.569587
# of Buys
19
16
30
24
# of Sells
37
40
26
32
#. of Trades
166
Chapter 4: An Empirical Comparison ofAPARCH Models
Table 35: Beforetransactionscosts
Statistics
for
19971998 with
± 3%
Delta Filter
(Excluding One Data Point) EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
0.942586
0.2136
0.38922
0.87467
Std. R. of Returns
10.63213
10.66822
10.66245
10.63067
55
55
55
55
Ave. Delta
0.012309
0.012309
0.012309
0.012309
Std. Delta
0.012883
0.012883
0.012883
0.012883
Ave. Maturity
0.123686
0.123686
0.123686
0.123686
Ave. Call Price
26.13364
26.13364
26.13364
26.13364
Std. Calls
6.335168
6.335168
6.335168
6.335168
27.39
27.39
27.39
27.39
Std. Puts
6.564702
6.564702
6.564702
6.564702
# of Buys
19
16
29
24
# of Sells
36
39
26
31
#. of Trades
Ave. Put Price
Table 36: Aftertransactionscosts EGARCH
Statistics for 19851986 with ± 3% Delta Filter GARCH
ARCH
2STAGE
Rate of Returns
1.91925
2.58628
4.67556
6.2793
Std. R. of Returns
13.95439
13.69692
13.72358
13.20722
17
17
17
17
Ave. Delta
0.0012
0.0012
0.0012
0.0012
Std.Delta
0.016493
0.016493
0.016493
0.016493
Ave. Maturity
0.227881
0.227881
0.227881
0.227881
Ave. Call Price
6.123529
6.123529
6.123529
6.123529
Std. Calls
1.947283
1.947283
1.947283
1.947283
Ave. Put Price
6.732353
6.732353
6.732353
6.732353
Std. Puts
2.304136
2.304136
2.304136
2.304136
# of Buys
10
1
11
7
# of Sells
7
16
6
10
#. of Trades
167
Chapter 4: An Empirical
Comparison ofAPARCH
Models
Table 37: Aftertransactionscosts Statistics for 19891990 with ± 3% Delta Filter EGARCH Rate of Returns
GARCH
ARCH
2STAGE
0.57337
2.36478
3.7799
3.96328
13.7521
13.9163
13.75325
13.72006
31
31
31
31
0.0042
0.0042
0.0042
0.0042
0.016332
0.016332
0.016332
0.016332
0.11445
0.11445
0.11445
0.11445
Ave. Call Price
7.320968
7.320968
7.320968
7.320968
Std. Calls
1.778987
1.778987
1.778987
1.778987
Ave. Put Price
8.006452
8.006452
8.006452
8.006452
Std. Puts
2.087851
2.087851
2.087851
2.087851
# of Buys
17
16
19
20
# of Sells
14
15
12
11
Std. R. of Returns #. of Trades Ave. Delta Std. Delta Ave. Maturity
Table 38: Aftertransactionscosts EGARCH
Statistics for 19931994 with ± 3% Delta Filter GARCH
ARCH
2STAGE
Rate of Returns
3.530293
3.431423
1.35626
2.067894
Std. R. of Returns
10.60153
10.69246
11.54567
11.30632
20
20
20
20
Ave. Delta
0.00638
0.00638
0.00638
0.00638
Std. Delta
0.018011
0.018011
0.018011
0.018011
Ave. Maturity
0.123425
0.123425
0.123425
0.123425
7.05
7.05
7.05
7.05
1.208087
1.208087
1.208087
1.208087
7.5975
7.5975
7.5975
7.5975
1.462377
1.462377
1.462377
1.462377
#. of Trades
Ave. Call Price Std. Calls Ave. Put Price Std. Puts # of Buys
6
6
9
8
# of Sells
14
14
11
12
168
Chapter 4: An Empirical Comparison ofAPARCH Models
Table
39: Aftertransactionscosts
Statistics
for
19971998 with
(Excluding One Data Point) EGARCH
GARCH
ARCH
2STAGE
Rate of Returns
0.447472
0.70872
0.88434
1.36978
Std. R. of Returns
10.63932
10.65304
10.66224
10.63684
55
55
55
55
Ave. Delta
0.012309
0.012309
0.012309
0.012309
Std. Delta
0.012883
0.012883
0.012883
0.012883
Ave. Maturity
0.123686
0.123686
0.123686
0.123686
Ave. Call Price
26.13364
26.13364
26.13364
26.13364
Std. Calls
6.335168
6.335168
6.335168
6.335168
27.39
27.39
27.39
27.39
Std. Puts
6.564702
6.564702
6.564702
6.564702
# of Buys
19
16
29
24
# of Sells
36
39
26
31
#. of Trades
Ave. Put Price
169
±3%
Delta
Filter
Chapter 4: An Empirical Comparison ofAPARCH Models
Figure 19: Autocorrelations
for r (19831998)
0.30 1 0.25 = 0.20 ý 0.15
0.10, 0.05 : 0.000.05 0.100.15 0.20
1
51
101
151
201
251
301
351
401
301
351
401
Figure 20: Autocorretations for T (19831998) 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.05 0.10 ! 0.15 0.20 1
51
101
Figure 21: Autocorrelations
151
1
251
for Irl (19831998)
0.30 0.25 ý 0.20 0.15 0.101 0.05 ý 0.00 0.05 0.10 0.15 0.20
201
W"ktl~
51
101
151
201
251
170
10 LAAW&AVWUlZWuw
301
351
401
Chapter 4: An Empirical Comparison ofAPARCHModels
Figure 22: Autocorrelations
for r (19831986)
0.15 0.10 0.05 1
wows
A"
0.00 0.05 0.10 ý
0.15
1
51
101
Figure 23: Autocorrelations
151
201
251
301
351
401
301
351
401
301
351
401
for r2(19831986)
0.15 , 0.10 0.05 0*
mw
0.00 0.05 0.10 0.15 1
51
101
Figure 24: Autocorrelations
151
201
251
for Irl (19831986)
0.15 ,
0.10 1 0.05
ýýý
0.00 0.000 0.05
0.10
;
0.15
1
51
101
151
201
251
171
Chapter 4: An Empirical
Comparison ofAPARCH
Figure 25: Autocorrelations
Models
for r (19871990)
0.40
ý 0.30 ý 0.20 0.10 0.00 0.10
I
0.20 1
51
101
Figure 26: Autocorrelations
151
201
251
301
351
401
301
351
401
301
351
401
for r2 (19871990)
0.40
0.30 ý 0.20 0.10 0.00 0.10 0.20
1
51
101
Figure 27: Autocorrelations
151
201
251
for Irl (19871990)
0.40 ý 0.30 0.20 0.1011 0.00 1 0.10
0.201
51
101
151
201
251
172
Chapter 4: An Empirical
Comparison ofAPARCH
Figure 28: Autocorrelations
Models
for r (19911994)
0.15 1 0.10
51
1
101
Figure 29: Autocorrelations
151
201
251
301
351
401
for rz (19911994)
0.15 0.10 0.05
kA
0.00 0.05 0.10
51
1
101
Figure 30: Autocorrelations
151
201
251
301
351
401
301
351
401
for Irk (19911994)
0.15 0.10 1
0.05
in
0.00 0.05 = 0.10I
51
101
151
201
251
173
Chapter 4: An Empirical Comparison ofAPARCH Models
Figure 31: Autocorrelations 0.40
for r (19951998)
ý
0.30 '
0.20 0.10 ý 0.00 0.10 I
0.20 1
ýrý 51
101
Figure 32: Autocorrelations
151
201
251
301
351
401
301
351
401
301
351
401
for  (19951998)
0.40 0.30 0.20 0.10 0.00 0.10 0.20 1
51
101
Figure 33: Autocorrelations
151
201
251
for Irl (19951998)
0.40 0.30 0.20 ý 0.10 0.00 0.10 0.20 ý 1
51
101
151
201
251
174
Chapter 4: An Empirical Comparison of APARCH Models
Figure 34: 3State SWARCHL(3,2):
High Volatility Regions 1.0
ý'ý
ý. 0.0
.
, ý 1, II 1
ý..
I,
.
0.0
N Co Of O Co N
0.5
0 Co N
Figure 35: 3State SWARCHL(3,2):
00 NNN
Low Volatility
175
Regions
Co
00
Chapter 4: An Empirical Comparison ofAPARCH Models
Figure 36: Cumulative
Rate of Return From Straddles Trading (19931994) With 25
bps Transactions Costs and ± 3% Delta Filter 70 50
EGARCH
30
GARCH

ARCH
10

10 30
176
2Stage
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
CHAPTER 5
Empirical Performance of Alternative Variance Swap Valuation Models
Abstract As a continuation of our study of modelling volatility, this chapter adopts a financial engineering forecasting different the to volatility performance of specifications of timeseries evaluate approach S&P 500 index. Pricing the valuation swap models on variance a variance swap can and optionsbased be viewed as an exercise in computing the weighted average of the implied volatility of the options influence It be interpreted the of skew. volatility can also as the market consensus required even under has The Demeterfi (1999) future et al. variance swap valuation variance. methodology of expected been widely accepted by practitioners but little tested and scrutinised. After the terrorist attacks on September 11,2001, the longertermed forward variance has become more volatile than the shortertermed forward variance. This research presents the first of any known attempts to use market data to Demeterfi by framework. It literature the to this of et al. the contributes nascent effectiveness evaluate 2001 from June 2001 November to ninemonth three, and variance the sixswap contracts analysing implied is Our design timeseries different of and models. research rich enough to specifications using including: 1) hoc BlackScholes 2) models prominent ad stochastic of model; number a admit GARCH 4) local 6) jumpdiffusion 5) EGARCH; 3) model; volatility model; volatility model; find We to out whether using more complex option pricing models to aim model. variance swap is improve forecastability. to anomalies an effective strategy market variance observed accommodate Based on results from six wellselected contract days, we illustrate that the optionsbased framework, incorporating future facts, be forecaster of many stylised of capable may a poor although more interest forward futures the Just rates are not rates, necessarily as good predictors of variance. framework is not necessarily an effective predictor of future Demeterfi based et al. arbitragefree data show that implied models tend to overpredict future variance and from Results our variance. The reasons could be: 1) implied strategy was originally developed models. timeseries underperform for hedging; 2) implied volatility is predominantly a monotonically decreasing function of maturity 3) strategy cannot termstructure produce patterns; enough variance and therefore optionsbased distributional dynamics implied by option parameters is not consistent with its timeseries data as likelihood to Future by the estimation need the of maximum squareroot research process. stipulated findings. in to to establish a more order statistically our larger set significant result clarify sample use a Until then we have a strong reservation about the use of Demeterfi et al. methodology for variance forecasting.
5.1 Introduction Study the Background of 5.1.1 been increased has late interest in the fact there an that since Despite the volatility products directed has been derivatives. development towards to the little 1990's, of volatility research derivatives (1996). is by Grünbichler to volatility value paper al. et first theoretical The but technically framework simple that used the a presented complicated al. Grünbichler et 177 '
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
equilibrium
approach within which specific closedform solutions for volatility
framework. derived meanreversion within a option prices were
futures and
Later, Gupta (1997) and
Engle et al. (1998) discussed the issues related to the hedging of volatility.
Subsequently,
Andersen and Andreasen (1999), Rolfes and Henn (1999), Chriss and Morokoff
(1999),
Demeterfi et al. (1999), Brenner et al. (2000), Brockhaus and Long (2000), Heston and Nandi (2000b), Howison et al. (2001), Little
and Pant (2001), Carr and Madan (1999,2002),
Javaheri et al. (2002) and Theoret et al. (2002) also researched volatility in invested volatility research amount of
derivatives, but the
products still pales in comparison with other well
barrier derivatives such products as and Asian options. studied exotic
Until now the conventional instruments for implementing a volatility hedge remain rather is The way accepted of speculating widely on volatility most usually achievedthrough crude. the purchase of European call and put options. Traditional techniques such as delta hedging deltarisk. focus In Chapter 4 we have demonstratedthe the on reduction of strategy always insufficiency of a deltaneutral hedge in the event of large market moves. Once the however, deltaneutral delta. become long index trade a moves, can short or underlying Rehedging becomes necessary to maintain a deltaneutral position as the market moves. Since transaction and operational costs generally prohibit continuous rehedging, residual from is It the ultimately arises underlying optionsbased strategies. of volatility exposure have the though options effect of adjusting the volatility profile of a portfolio, that even clear it also induces additional exposureto the underlying and other market factors. Thus volatility dealt investors be that directly traders has so to with and can expresstheir views on yet risk future volatility.
5.1.1.1 New Way of Trading: Variance Swap The arrival of variance swaps offers an opportunity for traders to take synthetic positions in They first introduced hedge in 1998 in the risk. were volatility the of aftermath and volatility (LTCM) Management Capital Term melt down when implied stock index volatility Long levels. These variance swap contracts are mostly basedon equity levels rose to unprecedented designed be to originally they a replacement for traditional optionsbased were indices and hedged few Over or straddle as such call/put the years, options. past volatility strategies
178
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
into have a sizeable market". grown variance swaps
Despite its name, a variance swap is
forward based is contract whose payoff on the realised volatility an overthecounter actually index. Their is payoff at expiration equal to: equity of a stated (Q2 R
Kvol2 )Nýk
in N is the the of swap amount some currency units per annualised variance notional where point, o
and K, are the realised stock volatility over the life of the contract (n days) quoted
in annual term, i. e.
F
n1
n rýo
S'+'  S' Sr
4 2
and the fixed
annualised
volatility
delivery
price,
is factor. F the appropriate annualisation respectively.
5.1.1.2 Usage of Variance Swap Since a variance swap provides pure exposure on future volatility levels, it is considered a It bet than an optionsbased strategy. volatility allows counterpartiesto exchange on cleaner for fixed Counterparties floating to variance. variance swap can variance use cashflows future between (floating) implied (fixed) volatility, or to the realised spread and speculate hedge the volatility exposure of other positions or businesses. According to Curnutt (2000), some of the possible strategiesusing variance swaps are: i) Speculating a directional view that implied volatility is too high or too low relative to because 1) follows volatility realised volatility a meanreverting process. anticipated In this model, high volatility decreasesand low volatility increases; 2) there is a between level. index The volatility stays volatility correlation and stock or negative high after large downward moves in the market; 3) volatility increaseswith the risk and uncertainty; implied that the Implementing view ii) a volatility in one equity index is mispriced in implied index; the volatility to another equity relative iii)
Trading volatility on a forward basis by purchasing a variance swap of one expiration and a variance swap of anotherexpiration.
58Capital Markets News,Federal Bank of Chicago, March 2001.
179
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
5.1.1.3 Variance Swap Example The following
example illustrates to the reader how variance swap really works: using the
S&P 500 as the underlying index, a volatility level of Kv01= 23% is fixed for one year. This Aa 5.29%. Counterparty Counterparty B to to of agrees variance pay a nominal corresponds for US$5,000,000 each percentage point of realised variance point above of notional amount 5.29% and Counterpary A agrees to pay Counterparty B US$5,000,000 per variance point below this value. US$26,450,000.
In this case, the notional value of the contract, or fixed leg payment, is Suppose realised volatility
turned out to be 43% (18.49%).
(variance) of S&P 500 during this time period
The payoff
to the party that receives variance is
US$5,000,000 x (18.49%  5.29%), or US$660,000. If realised volatility were 3%, the payoff to the party that pays volatility would be US$5,000,000 x (0.09%  5.29%), or a loss of only US$260,000. Figure 37 illustrates the payoff of a long variance swap under different levels of realised volatility.
Its payoff is nonlinear in volatility.
percent deviation of realised volatility
This means, for instance, that a one
above the price has a different (larger) payoff than a
below delivery deviation the of volatility price. one percent
Figure 37:Volatility vs. Variance Swap Payoffs  Long $5,000,000 $4,000,000 $3,000,000
$2,000,000 $1,000,000
0
0
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
a1 ,ooo, ooo
The maturity of variance swap contracts can run from three months to five or even seven The trades the occupy around cost most onespectrumS9. primary year although years, is bid/ask the double is the swaps spread, variance which approximately with associated bid/ask Their from 500 S&P in market. spreads on the a swap range straddle variance spread for longermaturity to two for contract a oneyear variance point points a one variance
59SeeMehta (1999) for further details.
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Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
contract.
Institutional users such as hedge funds are attracted to own variance swap,
especially when their portfolios are naturally short vega, as an alternative to using options to take on or hedge volatility exposure.
5.1.2 The Problem Statementand Hypotheses The model developed by Demeterfi et al. (1999) is the most popular tool to price variance has but no research ever considered using market data to test for its surprisingly, swaps, different This S&P the examines chapter variance swap models' performance on usefulness. 500 index from June 2001 to November 2001. After the terrorist attacks on September 11, 2001, the longertermed forward variance has become more volatile than the shortertermed forward variance.
We analyse the three, six and ninemonth variance swap contracts by
implied in different different timeseries of specifications and models at points evaluating time.
The underlying hypotheses of this project are that if optionsbased Demeterfi et al.
(1999) framework is mathematically correct then:
i) Each generalisationof the benchmark BlackScholes model should be able to improve the volatility forecastability of the optionsbasedpricing model; ii) If option prices are indeed representativeof their underlying timeseries and forwardlooking then the forecastability of optionsbased variance swap models should be superior to their timeseriescounterparts. In this study our goals are: i) To present a complete picture of how each generalisation of the benchmark BlackScholes model can really improve the variance forecastability of variance swaps and is between inconsistent generalisation each and outofsample results; whether ii)
To investigate whether there may be any systematic difference in variance forecasting between timeseries and optionsbased variance swap valuation models. performance It is intended to explore whether optionsbased models, which are forwardlooking, are discretetime outperforming capable of future in variance. information, predicting
181
processes, which
use only
historical
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
5.1.3 The Significance of the Study The Demeterfi et al. (1999) variance swap pricing methodology has been widely accepted by little have Regrettably, but tested and scrutinised. no empirical studies ever used practitioners investigate data the pricing performance of variance swap valuation models. to any market This research presents the first of any known attempts to use market data to shed light on the variance forecastability of variance swap valuation models under alternative timeseries and Since implied models. pricing volatility option competing
can be regarded as the market's
implication future forecastability by the volatility, realised of any poor variance of expectation is for look that to such practitioners and a models academicians alike may need optionsbased information historical in integrate and market to a composite option pricing model. way
5.1.4 Organisation The remainder of this chapter is organised as follows.
In section 5.2 we review the
Section 5.3 introduces 5.4 discusses dataset. Section the the models. and methodology findings. Section 5.5 the and analyses empirical summariesthe procedures/results calibration results.
5.2 Methodology This section discussesthe approachesand models used for volatility forecasting. We first judging in forecastability the different timeseries and used variance the criteria of review We for implied framework then the models. swap variance outline optionsbased variance (1999). Demeterfi This by developed traded al. et methodology options exclusively uses swap discuss Subsequently, different we forecast option pricing models that can to variance. in S&P illustrate 500 Finally the for anomalies market we options market. observed account forecasting in variance. the timeseries approach
5.2.1 Performance Criteria different forecastability variance swap models are evaluated in the following of The variance
ways:
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Chapter S: Empirical Performance ofAlternative Variance Swap Valuation Models
i) Insample analysis. In view of optionpricing, it refers to the ability of each generalisation of BlackScholes option model to fit the call option data and produce the least pricing error. Sum of price square error (SPSE) is used to judge whether one is better than the other on each contract day; model option
ii) Outofsample analysis. It comparesthe variance forecastability of all six timeseries and optionsbased models. The criteria used in selecting the best model is mean square error (MSE), and aggregateMSE ranking60is applied to evaluate the overall performance of each timeseries and optionsbasedvariance swap model for each of the three maturity months, i. e. three, six and ninemonth contracts; iii)
Consistency of optionsimplied
distributional
dynamics and timeseries properties. Maximum likelihood estimation of a squareroot process is used in order to identify potential inconsistency between optionsimplied dynamics and timeseries data by looking into the estimated structural parameters.
It should be noted that our results are based on the use of eighteen welldesigned variance 2001 December June 2001. between Although our sample is small and sample and contracts periods are overlapping, we point out that the price fitting and variance forecastability of the insensitive to the choice of sample periods because options are are models optionsbased forwardlooking do be to and not use historical data. supposed
5.2.2 The Optionsbased Variance Swap Framework The original BlackScholes model assumesthat volatility is constant or deterministic, but have developed researchers option pricing models that recognisethe stochastic many recently Hull White (1987), Heston (1993a). New financial engineering e. g. and volatility, of nature it have made possible to explore volatility trading in a more sophisticated also techniques innovations behind is idea having The these be hedged that to volatility can without manner. level. Whaley future (1993) its volatility was among first to advocatethe use of about worry its CBOE. Consequently, futures level VIX, on indicates options and the the which volatility implied S&P 100, Whaley volatility in CBOE61. on 1993 for was the created atthemoney of
60Aggregate rank is defined as the sum of the rank for each model in each subperiod. 61 The MONEP created the VX1 and VX6 indexes in October 1997. On January 19,1998, the Deutsche first in became the the world to list volatility futures based on an underlying (DTB) exchange Terminborse it launched the VOLAX futures. Readers implied when volatility Werner index to of and are referred equity VOLAX details contracts. for on (1998) Roth
183
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
free be hedge to that and could used volatility. were of price risk products such pointed out Trading volumes in thesecontracts,however, have been low. According to Neuberger(1994), As Neuberger futures to subject a addressed potentially manipulation. result, was volatility this concern by designing the logcontract to provide an accurateand flexible volatility hedge. Since then logcontract has becomean indispensablecomponent for volatility research. 5.2.2.1 LogContract Neuberger (1994) demonstratedthat by dynamically hedging the logcontract against a static futures position it was possible to engineerthe future profit or loss as an exact linear function This dependent is result variation. not on any assumptionthat returns the realised quadratic of diffusion Brownian is The "fair by that process, or a volatility price" constant. are generated be time logcontract can shown as: at any the of Lý=1og(F,)
2Q;
(T  t)
T future futures the time is t, the F, the Q, and at constant price realised volatility where is T 4= time the The at contract of value maturity.
log(FT).
The "fair price" is the direct result of dynamically hedging the logcontract with appropriate 1/F, The delta for logcontract futures is to and until maturity. contracts equal amounts of independentof volatility. If traders' view on volatility is Kv01# QR, the value of logcontract hedging In loss "fair this the this be the case, present of the price". value of profit and will not be life the contract can the shown as: of strategy over
Iz2* 2iK"°ýQx)
T
life implied KY01 the is the over the volatility of the contract and volatility realised where QR 0. logcontract time at the in the price of logcontract, it is hedging the clear that one can replicate the cashflows of By dynamically But is logcontract volatility. though on a powerful gamble even and swap variance hedge volatility, it is only a hypothetical tool. In addition to the to tool mathematical Neubeger's feasibility of logcontract, results the are also conditional on of availability hedges. dynamical Nevertheless, has Neuberger's discrete and greatly work forming
184
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
facilitated the introduction of volatility derivatives, such as volatility futures, options and swaps. In order to understandthe mechanicsof logcontract, one can take a Taylorseries expansion derivatives logarithm to the secondorder the price up which gives: of of S,  Si S,  Si +1 log Si.,  log S, = +1 1 2 Si Si Summing both sides of the aboveequation over the total number of days n in the contract and rearranging terms, one obtains: ^' S.+i Sr r=o
St
z21ogS° +2ý S"
SM s
&0
r
S.
The LHS is the floating leg of variance swap, which can be replicated by holding a derivative first RHS logcontract, forward term the the to the of position and a equal with payoff a Thus in RHS. delivery the the the sole concern setting term price of variance swap of second is to engineerthe cashflows on the RHS, in particular the log payoff. 5.2.2.2 Demeterfi et al. Framework Since logcontract is nontraded and requires dynamic hedging in order to replicate the cashflows of variance swap, it is not a "direct" bet on variance/volatility. direct and forward exposure on volatility,
In order to provide a
Demeterfi et al. (1999) developed a formal and
for framework This that the the pricing variance of showed swaps. theoretical study rigorous be inferred from the prices of traded options of the underlying level future could of volatility be focused derivatives Demeterfi the initially on on volatility valued. et al. thereby asset and with deterministic
replication
delivery BlackScholes the under the price of
volatility.
Since variance swap is a forward contract on variance, the delivery price must
framework
initially. Under interest the value and zero assumptions rates of of zero the swap make dividend yields, Demeterfi et al. proved that a constant vega, v, could be obtained by owning K2 inversely by infinite put options weighted and their call of strikes, square a portfolio of . BS how the v's vary with stock price S for portfolios consisting of 3841 Figures show inversely by K2. weighted options call different number of
185
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
Figure 38: Vega of Individual Strikes: 80,100,120
20
40
80
80
100
120
140
16o
180
200
Figure 39: Sum of the Vega contributions of Individual Strikes: 80,100,120
20
40
6o
80
100
120
140
160
180
2 oo
Figure 40: Vega of Individual Strikes: 60 to 140 spaced 10 apart
20
40
60
80
100
120
140
Figure 41: Sum of the Vega contributions of Individual
z'0
40
60
80
1 oo
120
140
160
180200
Strikes: 60 to 140 spaced 10 apart
160
180
2 oo
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models 5.2.2.2.1
Derivation
of Demterfi
et at. Framework
To obtain an initial exposure of a unit of currency per volatility point squared, this portfolio at time 0 can be constructed as follows:
IIo =T
So Sý S,
o +Q, log(sS .)
forward is S* the atthemoney stock or spot level and o usually where
is the view or
from future traders. variance realised estimate of The hedging of the above portfolio is similar to that of logcontract: if the realised variance turns out to have been QR the net payoff on the dynamically hedged position until expiration The (o inside be terms the squaredparenthesisare values of the "fair" to C2). equal will first inside The brackets is term the swaps. variance of price

forward contract with
delivery price S" which can be statically replicated. The second term describesa short log , So It is log be to that the term clear to only reference with rehedged needs position . dynamically. Demeterfi et al. also relaxed some BlackScholes assumptions and derived the diffusive delivery jump. by: for The is the price conditional on no asset price given evolution solutions d5,
_ ,u(t, ")dt+Q(t, ")dWt st
) ) is Brownian functions W, p(t,... motion, and C(t,... are arbitrary of time and other where delivery The is for theoretical this price process general stochastic variables, respectively. given by:
187
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
V_TfToQz K=ý
E[V]
K4_ý
fT E[
Q2(t....)dt] Q2 dt + dZr
d (log S,) _UI S`
S, ) d(1og =2Qzdt t22T LS' ST Jo log gyý _TE S St0
:.
ST ST dS, S. rr log lo E +gS rT, =log = S S' o st ro0 and S
logs;SSKK
sT+
2ý K,, :. _? rT TS
S. 12
fo
s°
Max(KST, ))dK+ý,
12
Max(ST K, ))dK
s""
e'T 1
s1 f log P(K)dK + e'T So 0 K2
+ e'T
1
C(K)dK
S K2
denote fair P(K) European the European C(K) current value and of put of a call and a where T K time that at mature with riskfree interest rate r and some arbitrary at strike struck boundary S' separatingactively traded outofthe money call and put options. On the basis linear Ki, for finite to this approximation payoff strikes, a set put of call and of a piecewise c by: the K,, appropriate option portfolio weights are respectively, given and P,
2S
g(Sr) =TS. )=I w(K,, c
logS;
g(K; +,.c)g(K;. K,
c)
K;. c +i.c 
g(K; +i.r)g(K;, p) )_w(K,, P K;.  K; +t. P P
;ý
w(Kj, c )
for calls
w(Kj, P)
for puts
J=O rt j=o
is: the strikes where the order of KrI. P < < K3,P < K2.P < KI.P < S* = Ko < Kl. < K2, < K3, < < Kri,c c c c ... ...
188
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Appendix B. 7 summarises the procedures to calculate the "fair" delivery price. The above "adjusted" weights guarantee that option payoffs will always exceed or match the value of logcontract.
Clearly the essence of this derivation is that log payoff can be decomposed into
a portfolio consisting of a forward contract and outofthemoney call and put options62. This approach to the fair value of future variance is the most rigorous from a theoretical point of view and makes fewer assumptions than the initial intuitive
treatment.
From a hedging
perspective, it makes precise the intuitive notion that implied volatility can be regarded as the future of realised volatility. expectation market's
Most importantly,
it provides a direct
connection between the market cost of options and the strategy for capturing future realised is implied there an when volatility skew and the simple BlackScholes formula even volatility, is invalid.
From a practical perspective, traders may express views on volatility
using
having delta hedge. to without variance swaps
5.2.2.2.2 Implementation Issues with Demeterfi et al. Framework Few issues merit our attention in pricing variance swap using Demeterfi et al. framework. First, since log() payoffs are not traded in the marketplace, one will have to approximate them in limited European Because options a these strikes cannot exactly traded strike range. with duplicate such cashflows, they will capture less than the true realised variance. According to Little and Pant (2001), this reduction is greater for the longermaturity diffusive. fail to remain asset price may
swaps. Second, the
When asset price displays jumps, the impact of
jumps on the pricing and hedging of volatility
derivatives is significant and it can cause the
is that the true realised variance. to a quantity not capture strategy
To fully implement a
for variance swaps, one needs price continuity and a consistent stochastic replication strategy volatility discretely
for options. model
Finally the above analysis is based upon approximating the
in the contract terms of most variance swaps by a variance used sampled
Whilst this be variance. approximation sampled can expected to provide very continuously for frequent, is shortdated variance the they may swaps estimates when sampling reasonable less frequent for longer Chriss sampling We with to well the reader periods. refer perform not for practical risk management issues in regard to variance swaps. (1999) Morokoff and
62See Carr and Madan (2002) for its derivation.
189
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
Despite Demeterfi et al. framework is not perfect, it remains an essential component for the in this research. exercises valuation variance swap
5.2.3 Option Modelsfor Variance Swaps In a study of finding an arbitragefree framework for pricing of volatility
derivatives, Carr et
long found (2002) that as as the movement of the underlying asset is continuous63, the al. is independent hedging contracts of variance completely of the choice of the pricing and Carr et al. showed that modelindependent prices of variance swaps could process. volatility be inferred from the market prices of Europeanstyle vanilla options.
Therefore, pricing a
in be implied the the an as exercise viewed computing can weighted of average variance swap volatility
of the options required to replicate the swap even under the influence of volatility
delivery is is, in That to the the terms of the price set so as reflect cost aggregate skew. implied volatility of the hedge portfolio.
However, results in Chapters3 and 4 demonstratethat the termstructure of implied volatility is pronounced in the S&P 500 marketplace. In addition to many studies, Rubinstein (1985, 1994) also documentedevidence that implied volatility tended to rise for deep inthemoney The degree, lesser termto options. presence of skews, smiles and, a outofthemoney and basic it BlackScholes the the assumptions most of model and makes structures violates hedging In to the concept of pricing to order and revisit of vanilla options. necessary it is in BlackScholes to a the reality, necessary model market extend accommodate In fashion. distributions lepotokurtic to via a particular, one needs generate meaningful for The hidden the spot and main possible some additional variables. stochastic process difficulty is that there are many models and processesthat can be used for this purpose and drawbacks depend hand. partly and on a specific merits problem at their relative The 1990's witnessed several important developments in order to describe smile effects. For instance, Dupire (1994), Derman & Kani (1994) and Rubinstein
(1994) developed the
deterministic smile models. An alternative approach would be to consider the volatility
as
is Merton there hypothesis. and growing to variable, this evidence support another stochastic European first derived the option pricing solution for the jumpdiffusion model; (1973)
63There is not an equivalent framework for assetthat follows a jumpdiffusion process.
190
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
(1985) Bates (1991) Torous jump Ball that and component could confirmed and subsequently, in Bates (1996), Bakshi the observed mispricing empirically options market. of explain some (1999) & Andreassen invented jumpAnderson (1997), stochastic and many others also et al. diffusion models. More general stochastic volatility models were developed by Hull and White (1987), Johnson and Shanno (1987), Scott (1987), Wiggins (1987), Stein et al. (1991), Ball and Roma (1994) and Schöbel and Zhu (1999). This list is by no meansexhaustive. The models developed by most of the above research papers require either the use of MonteCarlo simulation or numerical solution of a twodimensional is computationally which equation,
intensive to implement.
parabolic partial differential Too often, option models are
instance, for hoc, the on grounds of their tractability and solvability. ad chosen
Finding a
framework implementing it in practice remains a major challenge theoretical and meaningful to practitioners and academicians alike.
In the following
subsections we will explain what
types of option pricing models are selected for the pricing of variance swaps.
5.2.3.1 Stochastic Volatility Models 5.2.3.1.1 Justification for the Stochastic Volatility Approach Diffusion models assume that volatility is, like the underlying asset, a continuous random is timestatedependent There This the socalled approach. are many reasonswhy variable. diffusive For it a as process. volatility example, could simply represent model we should it it friction from could arise transaction or as could a costs, or estimation uncertainty, (heavytailed) distributions, leverage it returns effect or could simulate simulate nonGaussian (2000) Bakshi as a stationary, meanreverting suggested volatility process. et al. and capture diffusion inadequate inconsistency to models were explain pricing that onedimensional After for 500 S&P in timedecay controlling options. and market microstructure observed factors, Bakshi et al. stipulated that if one had to introduce another state variable that affected (2002) be Shu Zhang this process stochastic would also second volatility. and option prices, BlackScholes that volatility stochastic models the model outperformed evidence provided is in In moneynessmaturity all groups. almost volatility other words, stochastic significantly BlackScholes' lognormal the describing of farreaching extension a much more model, a complex market.
191
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
However, parameter estimation and stability of the estimates in time presents the major in challenge practical and using the stochastic volatility mathematical formula for option prices under a particular stochastic volatility
model.
Without a
model, estimating the risk
intensive. is Many questionable models are often chosen computationally neutral parameters so that there is a closedform solution, and this usually means taking the volatility
to be
independent of the Brownian motion driving the underlying asset price, whereas common for instance, between stock index and that a negative correlation exists, suggests experience Furthermore, the relatively poor performance of some of these models in capturing
volatility.
the observed implied volatility surface (see Das and Sundaram, 1999), as well as their difficult calibrations
and inherent
market
incompleteness,
them
make
unattractive
to
both
Consequently, and practitioners. pricing of options in the presence of stochastic academicians be done difficult is and seldom can analytically. volatility
5.2.3.1.2 Heston Model Recent research has shown that allowing for correlation as a free parameter can explain many Rubinstein (1994) discovered that the local volatility anomalies. market observed index was negatively correlated with the level of the index.
of stock
In a pure diffusive model, this
be can only achieved through a negative correlation between returns and negative skewness volatility. volatility
In addition, Nandi (1998) found that accounting for correlation between returns and in the stochastic volatility
model substantially improved the mispricing of outof
both to the zero correlation version of the stochastic when compared themoney options volatility
BlackScholes the widely used and model model.
Since Heston (1993a) invented
the Fourier approach to option pricing under stochastic volatility, volatility
the study of stochastic
become for has This much easier. approach permits a closedform solution models
European options and at the same time allows a nonzero risk premium for volatility as well between One asset correlation returns and volatility. can also use the arbitrary as an long in timeseries or the options market to calibrate model a information contained insample in thereafter context and compute outofsample option prices. an parameters The most important feature of Heston model is that it can account for correlation between Correlation between is returns. asset volatility and and necessary to asset returns volatility in distribution the it skewness and of asset skewness returns and affects the pricing of generate inthemoney
options relative
to outofthemoney
192
options.
Without
this correlation,
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
increasing the volatility of volatility of stochastic volatility only increases the kurtosis of asset far fromthemoney in the turn versus only affects pricing nearthemoney of returns, which options.
Since options are usually traded nearthemoney and the BlackScholes formula
for identical to the stochastic models atthemoney virtually volatility produces option prices for the use of stochastic volatility model. the this empirical support explains some of options, The stochastic volatility
model used in our variance swap pricing exercises is Heston's
in is pricing model, option which correlated with the underlying volatility volatility stochastic is The The modelled as a process squareroot process with meanreversion. variance asset. Heston model is nested within Bakshi et al. (1997) framework.
It is given as follows:
dS(t) = rdt + V,dW, dV, = (6,  x, V, )dt + Q, V,dW.
interest is V, diffusion is the the spot rate; constant component of returns variance where r jump Ws Wv Brownian and occurring; are each a motion with no standard on conditional 9, / ky dWy) Cov[dW3 K, pdt the ; and a, are of adjustment, = respectively speed correlation , longrun mean, and variation coefficient of the diffusion process V,. The solution for the above set of formulas is basedon the idea that whilst the probability that is greater (less) than the strike price cannot be expressed price the underlying asset function indeed be described the characteristic corresponding analytically. can analytically, For a European call option written on the stock with strike price K and maturity T, its time t by: is price given C(t, T)=S, *1I, (t, T; S, r, Vt)K*B(t,
T)IIz(t, T; S, r, V,)
bond The is in that Tt T) (t, the B1 price pays zerocoupon a unit periods. of currency where from be Ramaswamy Since the obtained European and can put putcall parity. of a Bakshi (1993) Scott (1997) found that the stochasticinterest rate (1985), and et al. Sundaresan improve the BlackScholes did performance the significantly of model, we will not not model interest in is Therefore, (t, T) to this B, rate model reduced study. the stochastic consider e'(T') .
Given the characteristic functions fj"'s,
193
the conditional probability density
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
functions Ii, and II2 can be recoveredfrom inverting the respective characteristic functions Bates (1996) Pan (2002): Heston (1993), in and as IIj(t,
et4m(K)f"
11 T; S,, r, V)=2+ý0Re
(t, T, Sf, r, V,; ý)
vo
io
for j=1,2. The characteristicfunctions are given in appendix C. 1. 5.2.3.2 JumpDiffusion Models 5.2.3.2.1 Justification for the JumpDiffusion Approach The explanation that volatility
smile is the sole consequence of timestatedependent or
diffusive local volatility is far from common intuition, and it has become increasingly clear that the assumptions underlying the pure diffusive approach are not particularly realistic. It is diffusion fact the that pure model overprices longterm options and cannot take a wellknown by the effects smile exhibited strong shortterm options. In addition, many studies account of have showed that modelling jump component can improve option pricing performance. For (1988) discovered Jorion that there was evidence of jump component in equities and example, foreign
exchange
even
explicit
allowance
was
made
for
possible
conditional
heteroskedasticity. The importance of introducing a jump component in modelling stock price dynamics had also been noted in Bates (1996,2000) and Bakshi et al. (1997) who stated that had difficulties in in diffusionbased models explaining shortsmile effects, particularly pure Bakshi et al. concluded that the Poissontype jump components in jumpterm option prices. diffusion models could be used to address these concerns. In addition, Madan et al. (1998) introduced a pure jump process with a random time change for European options and found be BlackScholes could model rejected in favour of the variancegamma model. that the Furthermore, empirical investigations of timeseries conducted by Carr et al. (2000) indicated devoid dynamics diffusion Carr index was essentially of a et al. stated component. that stock for indices infinite be jump tended and stocks to processes of pure processes that riskneutral Moreover, Lipton (2001) finite that took variation. the advocated models of use and activity features jumps local stochastic and of the volatility dynamics for pricing and risk into account foreign exchange options. Finally, using Bates's (2000) model with timeof management (2002) found Pan that dominated the jump jumprisk premia, stochastic pure model varying
194
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
diffusion models. Pan concluded that introducing volatilityrisk
premia in addition to jump
risk premia would not result in any significant improvement in the goodness of fit. Qualitatively,
jumpdiffusion
models produce distributions of returns that are mixtures of
do leptokurtic have features, at least for short maturities. distributions and attractive normal The jump model can capture some types of crash phenomena, e.g. stock market crashes, 9/11type events, currency devaluation etc. The jumpdiffusion
asset dynamics can be modelled as
the resultant of two components:
i) The continuous part which is a reflection of new information that has a marginal impact on the underlying asset; ii)
The jump part which is a reflection of important news that has an instantaneous, nonimpact on the underlying asset. marginal
The jump parameterallows better tracking of volatility by accounting for sudden changesin downward in It the that upward or movements accompanies asset. gives the model volatility flexibility in different Such dimension valuing of options across models also strikes. an extra imply an inverse relationship between option maturity and the magnitude of skewness,with little skewnessfor longmaturity options. However, the use of Demeterfi et al. framework is based on the approximation of 1o jumps. do not prices stock payoff when longer capture realised volatility.
Sr So
When stock prices do jump, logcontract can no
This is because to
ST
can be replicated by an infinite
0 number of weighted market call and put options only when the sample path of the underlying is continuous. process
Given the shortcomings of pure diffusion models, the extension to
is in jumps well motivated. options include pricing
Although the use of Demeterfi et al.
framework requires the underlying process to be pure diffusive,
it would be pedantic to
its validity simply because its sample path may not be strictly continuous. ignore completely To highlight the "impact" of noncontinuous asset dynamics on variance swap pricing, we Demeterfi jumpdiffusion to the framework model the et al. whilst maintaining all will apply by We the original analysis. this made that though strategy emphasise even assumptions other
195
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
is not perfectly consistent on a scientific basis, it may demonstrate any possible pricing improvements over the classic timestatedependentapproach. 5.2.3.2.2 Bakshi et al. Model We adopt the closedform jumpdiffusion the jumpanalysis.
(1997) for developed by Bakshi al. option model et
Following Baskshi et al., this riskneutral jumpdiffusion
known to variations6S of the BlackScholes admit many enough including:
1) BlackScholes model: A=0
The jumpdiffusion
setup is rich
model as special cases
and 0v = KY = 6y = 0; 2) Heston model: A=0.
model is given by:
dS(t) = (r  A,u )dt + V,dW, + J, dg, j dV, = (8v  KvV,)dt + Qv V,dWv A interest is frequency is V, jumps is the the the spot rate, constant of per year; where r diffusion component of return variance conditional on no jump occurring; W, and W, are each a standardBrownian motion with correlation Cov[dW,, dWvI= pdt ; J, is the percentage jump size conditional on a jump occurring that is lognormally, identically, and independently distributed over time with unconditional mean u.,. The standard deviation of ln(1+ J, ) is Adt jump A Poisson intensity is P(dq, counter that with and a so a., ; q, =1) = Adt 0) P(dq, = =1; Kv,9y 1kv and u, are respectively the speed of adjustment, longrun diffusion, V, the coefficient of mean, and variation . The advantageof modelling volatility as a squarerootprocessis that volatility never becomes T, European K For the call option written on maturity a stock and with strike price negative. its time t price is given by: C(t, T)=S, *II, (t, T; S, r, V,)K*B(t,
T)II2(t, T; S, r, V,)
64See Demeterfi et al. (1999) for details. 65Note that we simplify the Bakshi et al. model by eliminating the stochasticinterest part.
196
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
Given the characteristic functions fj''s,
conditional probability density functions fI, and
I12 can be recovered from inverting the respective characteristic functions as in Heston (1993), Bates (1996) and Pan (2002): 1 IIj (t T, Sr, r, V) __f +1, 2z for j=1,2,
e'4'n(")fjýJ(t, T, Sý,rVt, 'o) vo Re io
with the characteristic functions 17'.
The characteristic functions are given in
European be from 2. The The C. total the of a put can obtained price putcall parity. appendix into be decomposed two components: can return variance
SSý 1 Vý Var = dt t where Vi., _t
VarI(J, dq, ) = 2(,uß + (e' 1)(1 +, uý) Z) is the instantaneousvariance of the
jump component.
5.2.3.3 Local Volatility Models 5.2.3.3.1 Justification for the Local Volatility Approach The local volatility model, also known as deterministic volatility function, is the most natural in BlackScholes be formulated the term the to model as a which volatility can extension function of assetlevel and time. The local volatility model assumesthat assetlevel and time In implied BlackScholes to theory, dominant the smile effects. contribution constant are the local be formulated time for t of at can as weighted average an option maturing volatility Consequently, for before t. time the t) one can extract market's consensus volatility Q(S, future local volatility from a spectrum of available market options as quoted by the implied BlackScholes volatility. how derive first to to the local volatility (1994) show uniquely was Dupire
function given
Dupire's strikes and maturities all with are available. prices continuoustime market option by discretetime For been a number has of numerical supplemented methods. example, result (1994), Denman Kani (1994), Rubinstein Derman, Kani and Chriss (1990), and Longstaff fit (1996a) the volatility Chriss (1996) and
smiles through careful manipulation of the local
197
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
branching probabilities in implied binomial or trinomial tree framework.
These "implied"
European the of a complete set of call option prices, existence spanning assume methods which, in practice, requires the use of extrapolation and interpolation of the available market option prices. They offer a relatively straightforward approach fitting the volatility smile, but suffers from a number of setbacks: 1) tree methodology needs extensive "engineering" treatment to infer probabilities because negative transition probabilities are not allowed; 2) trees such as DermanKani
use options at each time interval.
frequently and lead to extremely erratic convergence behaviour.
Bad probabilities
occur
The reader is referred to
"implied" detailed for 2.4 the of survey methodology. a section Whereas the impliedtree is primarily based on a discretisation of the asset price process, the finitedifference
focuses on developing a discretetime model by discretising the approach
fundamental noarbitrage partial differential scheme to the volatility Osher
(1997),
equation.
The application of finitedifference
smile problem has been studied by many authors, e.g. Lagnado and
Andersen
and
BrothertonRatcliffe
(1998),
Coleman
et
al.
(1999),
Chryssanthakopoulos (2001) and Little and Pant (2001). Whilst somewhat more complicated however, finitedifference calibrate, to evaluate and
scheme is shown to exhibit much better
impliedtrees finitedifference because than properties convergence and stability not involve
explicit
adjustments of branching probabilities
timepartitioning. the and stockof prescription be shown to be similar to a trinomial implicit
or CrankNicolson
method does
and allows for independent
The explicit finitedifference scheme can also
tree, however, it is commonly acknowledged that
schemes is unconditionally
stable whilst explicit schemes are
66 not .
5.2.3.3.2 OneFactorModel The inspiring research by Breeden et al. (1978) stated that the riskneutral probability distributions could be recoveredfrom Europeanstyle options by pricing butterfly spreads,and derivative the second as of the call option price with respect to the therefore expressed Breeden Based the Dupire how (1994) et upon al's results, one could price. showed exercise European derivatives The function. local standard of to options the partial volatility relate is implied to distribution and construct the whole Dupire's method extract behind idea
66Zvan et at. (1998) deal with the necessaryconditions to avoid spurious oscillations.
198
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
diffusion processthat is consistentwith the market observedprices. In the riskneutral world, Dupire's local volatility model is assumedto evolve according to the following onefactor diffusion model: continuoustime
dS= ((r(t)
s
 q(t))dt + Q(s,t)dW
drift, is the q(t) is the dividend yield, and dW is a Wiener process. riskneutral where r(t) Given a continuum of traded European calls with different strikes and maturities, Breeden et al. found that: pcS, t, K, T>=e"/r
a2Cxr
ax
is K, T) density function C. the t; conditional probability and where p(S,
is the current
K is level T S strike time with t; an option price and of maturity and at rf asset marketvalue the constantriskfreerate. In the continuoustimelimit whenriskfree rate and dividendare a2cý.
is 0, determined from * t) Q(K, the volatility completely and constant, a2K2
smile.
At
time t and strike K, Dupire relates option prices to a(K, t) as follows67:
( öCT Q(K, T)=ý
v[
acrP (rr + gK T+ 11
 g)Ck7
a2cKr K2
1 1
;)V2
The major advantage of the above onefactor continuous model, as compared to the jumpdiffusion or stochastic model, is that no nontraded source of risk such as the jump or In is introduced. first derivative European the addition, the call or put of stochastic volatility is the to tail strike to the price respect proportional with riskneutral relevant option price derivative is its density. to second proportional the whilst conditional probability probability Given there are enough strike prices, the patterns of implied volatility across different strike identify the distribution. the density shape of uniquely riskneutral and prices can Consequently,the completenessof this onefactor diffusion model allows for arbitrage pricing hedging. and
199
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
5.2.3.3.3 Coleman et al. Approach Dupire's continuoustime results have been supplemented by a number of finitedifference methods. For example, Zou and Derman (1997) applied the "pseudoanalytical"
method to
by local surface approximating the derivatives of options prices with respect volatility extract to the strike levels and maturity using Edgeworth expansion for the pricing of lookback options. Andersen et al. (1998) illustrated how to construct the stable finitedifference to extract local volatility consistent with the equity option volatility implicit interest using rate of
and CrankNicholson
demonstrated its application by pricing
downandout
Coleman et al. (1999) developed a CrankNicholson
lattice
smile and termstructure
lattices68; Andersen et al. also knockout
options.
scheme to "optimise"
In addition, local volatility
"smoothness" in introducing by the BlackScholes PDE discretisation process. surface
In this study we adopt the spline functional approach of Coleman et al.(1999) to directly local the volatility surface and price variance swap via finitedifference method. In construct Little (2001) Coleman to et al., et al. also approximated a variance swap by using additional the CrankNicholson method in an extended BlackScholes framework that was based on a decomposition of a twodimensional problem into the solving of a set of onecleverly dimensional BlackScholes partial differential equations. At a glance, Little et al.'s method because finitedifference be this directly to attractive model prices a variance swap seems based on a discretely sampled variance and allows for the incorporation of local volatility. Besides computationally intensive, the major deficiencies of Little et al.'s setup are: 1) one has to make an assumptionof the underlying assetprocess;2) local volatility is assumedto be incorporate therefore the requiring use to of and separate a and method extract exogenous is in This to the Demeterfi contrast assumptionfree smile. et al. model that only volatility for different implied in the volatilities maturities order to value a variance swap, and requires Little 's the consider et not al. will methodology here. therefore, we for 's the local volatility function by directly discretising the nosolves Coleman et al. method differential finitedifference the equation using arbitrage partial
method. Given Si,,i,, r, q and
cr(S, t) and under the noarbitrage condition, the option value must satisfy the BlackScholes
67 See also pp. 810 of Andersen and BrothertonRatcliffe
(1998) for a detailed derivation of this formula.
200
Chapter S: Empirical Performance ofAlternative Variance Swap Valuation Models
for differential every price of the assetlevel and for every time from starting equation partial time to the expiry given by Merton (1973): ac
+(r _ q)S
ac
asC= +l Q(S, t) ZSZ rC
aZs
at aCäS, as 2
lim !
t)
/i
tE [0, T]
=eecTt>
C(O,t) = 0,
tE [0, T]
C(S, T) = max(ST  K, 0) denotes C(S, the option value of an underlying asset with an arbitrary strike at K and t) where [O, T]. T, tE expiry at
The boundary conditions for the upper (u) and lower (1) spatial
boundaries are:
a2c a2c S1 as 2I S=O as 2I
=0
Before applying finitedifference
method to calculate option prices, Q(S, t) needs to be
lack Due to of market option price data, i. e. noncontinuum of strikes, this can approximated. be regarded as a wellknown
but illposed function approximation problem from a finite
dataset with a nonlinear observation functional.
Therefore, there are an infinite number of
solutions to the problem given a set of the market option price data. To tackle this problem, the Coleman et al. model introduces "smoothness" to facilitate accurate approximation of the local volatility
function from a finite set of data. The Coleman et al. model assumes that the
follows diffusion in incorporates bicubic a onefactor the asset model and spline underlying choice of parameterisation.
After choosing the number of spline knots and their placement,
by interpolating be fixed an represented t) spline can with a a(S, end condition.
The spline
knots uniquely construct Q(S, t) and the knots are determined by solving a constrained nonlinear optimisation
problem to match the market option prices, therefore effectively
inverse into it spline minimisation problem with respect to local volatility an turning
at the
local volatility calibration procedures are summarised as follows: The knots. spline
i) Assume there are m observedoption closing prices Ci
j =1,..., m
68Andersen et al. (1998) found that explicit finite difference method was not wellbehaved in the fitting of the volatility smile.
201
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
ii)
Choose p spine knots { s; tr }? with corresponding local volatility 1 ,
a: = 0(s1 , t; )
iii)
Define an interpolating spline c(s;, t = Q; j)
i =1,... p
iv)
Let Cj (c(S, t; Q' )) = C(c(S, t; a* ), Kj Tj ), ,
j =1,..., m
by (s,, (sp, for knots, Given the tl).... tp), cf solve minimising the pvector spline v) p objective function: 1mw
MIN f(o)=ý(Cj(c(S, d2 j=i
t; a*))_Cj)2
subject to l
In contrast to Dupire (1994), Coleman et al. do not emphasiseon the matching of the market is The data. local to the objective reconstruct as smooth volatility as possible option price function Qf (S, t) This way, the local volatility surface possessescertain properties a priori, . better The chance and of convergence. approximation of o smoothness namely,
in the
A
European the process requires evaluation of options C. above minimisation
This inverse
finitebe tree can only computed or a numerically via a method minimisation problem difference approach. Several issues merit our attention in this inverse minimsation problem. First, to construct a finitedifference in be knots the method, a via placed spline should spline efficiently is knots D69. Second, the to the assettime covering space of spline mesh number rectangular be no greater than the number of option prices (p S m) in order not to allow too many degreesof freedom in approximating cr(S,t) . Under mild assumptions,the Coleman et al. decreasing function to a values monotonically sequence of objective approach corresponds bounds i. k 1, lower o a, the e. upper and convergence, are =1,..., c*. u > and guarantees European local knots. both In the be imposed traded the volatility at on addition, that can be local to the the to used calibrate volatility may spline approximation call/put options function or*(S, t). A thorough examination of finitedifference method is beyond the scopeof Andersen We (1998) implicit for the et treatment dissertation. recommend al. of a proper this local to difference extract volatility surface consistent with smile effects. approach infinite Chriss Tsiveriotis (1998) and Little and Pant (2001) have also a (1993), and Wilmott et al.
69In general, we have no a priori knowledge of D within which the volatility values are significant for pricing available options.
202
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
in
good discussion
pricing
options
under
the onedimensional
BlackScholes
PDE
environment.
5.2.3.4 Ad hoc BlackScholesModel In light of the BlackScholes model's moneyness and maturity
biases, researchers and
find "live have tried to to always with the smile". ways especially practitioners
One of the
hoc, is to estimate and use an "implied volatility matrix". arguably ad whilst proposed ways, This formulation
is also termed as "practitioner
BlackScholes".
We adopt Dumas et al.
(1998) methodology and construct an ad hoc BlackScholes model in which each option has its own implied volatility depending on the strike price K and timetomaturity implied BlackScholes the that volatility al. observed
for S&P 500 options tended to have a
forms for that argued quadratic and volatility parabolic shape implied volatility. parameterise
T. Dumas et
function were suffice to
Specifically we use the functional form:
K+a2K2 +a, +a3z+a4z2 +asKz z) QN(K, =ao where cr.
is implied volatility using the BlackScholes formula for an option of strike K and
timetomaturity
'r.
This formulation is not only internally inconsistent with the BlackScholes assumptions but demand forces to which are subject prices not option and violates and supply also generates local noarbitrage conditions, and therefore potentially erroneous. But Dumas et al. did show deterministic binomial tree the implied or volatility the that
models of Derman and Kani
(1994), Dupire (1994) and Rubinstein (1994) underperformed
the ad hoc BlackScholes
in 500 S&P index in options valuation the terms outofsample of errors option market. model Furthermore, the ad hoc BlackScholes model is very different from the local volatility local The the approach volatility models smile effects with the spots and strikes approach. depend BlackScholes hoc This prices on the moneyness regressionbased alone. ad whereas is definitely flexible the naive, than hoc although more using challenging and strategy, ad local volatility model for pricing variance swaps. Comparing the ad hoc BlackScholes local jump stochastic and volatility volatility the to with/without strategies should strategy insights their in relative on terms of forecasting volatility and efficacies therefore yield valuing options.
203
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
5.2.4 TimeSeries Models for Forecasting Variance The models discussedso far for pricing variance swaps includes only the implied models stochasticvolatility with/without jump models and local volatility model under the Demeterfi et al. assumptionfreeframework. Although interesting, these approachesdo largely depend on a very limited number of option strikes to infer the prices of an entire continuum of options of every strike and maturity on the underlying asset. 5.2.4.1 Justification for the Conditional Volatility Approach On the other hand, timeseries models can be used to directly approximate the delivery price for a variance swap. Indeed discretetime GARCHtype processescan be linked to bivariate diffusion models, and vice versa. For example, Nelson (1991) showed that EGARCH processesconvergedweekly to a specific stochasticvolatility bivariate diffusion model. More (1997) Duan generalisedthese results and brought the largely separateGARCH and recently, bivariate diffusion literatures together. Duan showed that most of the existing bivariate diffusion models that had beenused to model assetreturns and volatility could be represented GARCH family Despite limits of the fact that the timeseries approach models. a of as ignores the direct modelling of volatility smile effect and uses only historical information, there may still be some advantagesin implementing heteroskedasticmodels using the vast literatures on numerical proceduresfor GARCHtype models.
5.2.4.2 GARCHVariance Swap An alternative way for pricing variance swaps is to use stochastic volatility models that are in historical (GARCHVS) time GARCHvariance the series, such as with swap good agreement (2002). Javaheri by The OrnsteinGARCHVS invented et al. the model uses model its for to variance in continuous time: model Uhlenbeck process dv = k(9  v)dt + yvdW 0 is the longterm mean reversion level, y is the is reversion, k the of mean speed where dW is Brownian and the motion. volatility volatility of differential a partial used equation approach to determine the first Javaheri et al. originally Expected and approximate the variance realised of expected realised volatility. two moments 204
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
by determining is the first moment of realised variance in a evaluated realised variance discrete and continuous context. It uses the discrete time process GARCH(1,1) to calculate expectedrealised variance EI=Tfo
v(t)dt . The structural parametersestimated from the
GARCH(1,1) environment were derived by Engle and Mezrich (1995) and Javaheri et al. (2002) as follows: E, = hu,,
u,  i.i.d.
ht ) N(O, et r, =e, h; =ao+a, e,,2, +ß, h;, 0=v
dt 1a, A dt
a° GARCH(1,1), V= the ß, autoregressive parameters are of a,, where (1a, ßl) is h, the conditional volatility, unconditional variance,
is the
T is time to maturity and v is the
instantaneous variance to the last observation in the GARCH(1,1)
model.
The expected
delivery price of a variance swap can be written as: k''T [O*(Týe_1)+!
E fIJoTv(t)dt =
(l e'k4T)*v
k
IT
Since the GARCHVS model has a closedform solution for variance swap valuation, one can information in history timeseries the the of asset prices to estimate model easily use (2002). Theoret in parametersas
5.2.4.3 EGARCH Simulations In Chapter 4 we have shown evidence that EGARCH GARCH
in both in and outofsample tests.
include the EGARCH
model (Nelson,
is the best model and outperforms
In addition to the GARCHVS
model, we
1991) and analyse its performance by directly
delivery prices of variance swaps from its simulated sample paths. the calculating by: is given EGARCH(1,1)
205
The
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models e, = h, v, r, = E, log h, '
where z, =
=ao +a, z,,
h'
+Y,
1'(2/9t)os)+ßl
(Izr,
log h1_12
degrees freedom. Given i. d. is the with unit studentt variance and of v, v , i.
, data 100,000 (ao, Nstep the y, a, we simulate return with parameter of structural a set , ,/ 1) , delivery for in the to calculate order prices variance swaps, where N is number of replications trading days.
5.2.5 Summary of the Methodology Sections 5.2.14 have explained the performance criteria, methods, models and underlying hypothesesused in this study for variance swap valuation. It should be emphasisedthat our S&P investigate both timeseries to the and optionsbased variance swap models study uses 500 index market. Our researchis designedto include the results from the best models shown in Chapters 3 and 4 including: 1) ad hoc BlackScholes model; 2) stochastic volatility model; 3) jumpdiffusion model; 4) local volatility model; 5) EGARCH; 6) GARCH variance swap S&P 500 data in the The this study. presents section used next model.
5.3 Data Description 5.3.1 Specifications and Filtering The dataset comprises of the daily closing prices of the S&P 500 index for the period from June 1999 through December 2001. The option prices used in this study are S&P 500 call We for CBOE. CBOE bid/ask the traded obtain closing on traded on option prices options70 from before three Friday's the to after September 11,2001. These options months the third for index 100 between difference to cash a amount times European equal the settled and are data Similar option were formerly used by Rubinstein (1985), Bakshi et price. level and strike (1998). Nandi (1997) and al.
70The S&P 500 index is a valueweighed index. S&P 500 index options are traded on CBOE whilst S&P 500 3 4 in Chapters CME. traded and are used on futures options index
206
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
S&P 500 index options expire on Saturday immediately following the third Friday of the expiration month. There are three nearterm expiration months followed by three additional Marchquarterly i. from the cycle, e. March, June, September and December. Our months option databaseis supplied by an option specialist". Following the lead of Bakshi et al. (1997), several exclusion filters are applied to remove uninformative options records from our database: i) Options with less than six days to expiration may induce liquidityrelated
biases and
they are excluded from the sample; ii)
Price quotes lower than $0.375 are eliminated
to mitigate
the impact of price
discretenesss on option valuation;
iii)
Quotes not satisfying the arbitrage restriction: C(t, T) >max(O,SDK*
e'ýT`>)
are taken out of the sample; iv) Options with no open interest are not included becauseof liquidity problem.
5.3.1.1 Dividends S&P 500 index options are chosen becausethese are the second most active index options S. in interest U. in terms in and, of open the options, they are the largest. In contrast to market S&P100 index options, there are no wild card features that can complicate the valuation hedge S&P 500 is index to It is because easier also there a very active options process. fact, 500 futures. In it is S&P best for for European the one testing of option markets a market As in S&P is the 500 index the dividends, of stocks there many a need model72. pay valuation index level. We collect daily cash dividends for the S&P 500 spot the exdividend to obtain index from Bloomberg from June 2001 to December 200273.We arrive at the presentvalue of it from index dividendthe level in dividends subtract current the to and the order obtain is used as input into the option models. The exdividend spot index 500 that S&P exclusive index level is:
71Option data are provided by ivolatility. com in New York. 72Refer to Rubinstein (1994) for more details. 73The calculation of exdividend spot level requires the use of up to 18 months of future dividends to make level. index its adjustmentson
207
Chapter S: Empirical Performance ofAlternative Variance Swap Valuation Models Tt `S esdividend(t)

Sclose (t)
eý
i*r
Dt+l
i=1
dividend in is is the closing index price, r, is the D1+r the future, SciOSe the actual where continuously constant riskfree compounded rate corresponding to i periods to expiration from day t calculated from interpolated U. S. Treasury yields provided by the U. S. Treasury Department. Implied volatility is computed by applying the NewtonRaphson method to the BlackScholes call option formula:
(t)N(d, ) Xercrt)N(d2 ) C(t, T) = Sexeevraend (t) /X]+ (r + 0.5Q2)(T t) ln[SexdivJdene dl QT t d2 =d, Q Tt 5.3.1.2 Calibration Using Call Options Few issues merit our attention when using the call options database for option models' calibration.
First, we have demonstrated in section 3.4.3 that the implied volatility
of call
(outofthemoney) inthemoney in given a category are quite similar to the implied options in the (inthemoney) opposing options outofthemoney put of category regardless volatility termtoexpiration. or period sample of
For a fixed termtoexpiration,
call and put options
imply the same Ushaped volatility pattern across strike prices. Such similarities in pricing between due call and put to the working of the putcall options mainly existing structure link is it that this makes call and put options of the same strike price and the same and parity, levels 500 Second, S&P Bakshi (1997) of mispricing. similar exhibit used et al. expiration found the to of some that results were parameters stochastic estimate models and put options Because these two of reasons, only call options are used to calibrate the similar. qualitatively We jump local BlackScholes, volatility hoc stochastic with/without and volatility models. ad follow basing to 500 S&P from calibrations our solely that on call results obtained argue biased the After a picture present of not candidate applying the should models. options data, the the to day is 100. average number criteria of options available on each exclusionary
208
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
5.3.2 Financial and Political Events The decision to value the JuneNovember 2001 variance swap contracts is neither incidental nor arbitrary.
Many significant global macroeconomic and political events occurred during
the 20012002 period. For example, the September 11 terrorist attacks in New York, U. S. led U. S. Tyco, Afghanistan, in global economic scandals weak growth, such as corporate war investment banks' scandals, the collusion between Enron and its auditor Arthur & Andersen, bankruptcies, e.g. United Airlines,
US Airways
and WorldCom,
worldwide
bursting of
technology, media and telecoms bubble, E. U. enlargement, circulation of Euros, surging oil Iraq IsraelPalestine have in East Middle the the and against all war conflicts possible price, conspired to spook markets. On the equity side, the global market was extremely volatile and depressing during the 20002002 period. In the U. S. there were more than 186 bankruptcies recorded with $368 billion in Tokyo finished 2002. 2002 in 225 decline 19 in Nikki the with a percent collapsed assets 19year low in November losses 2002 to The sank a over nine market and suffered average. in December days 2002, its longest losing In 2002 for 11 the trading streak years. consecutive European bourses suffered their worst year since 1974 with a fall of 22.1 percent in the MSCI Europe index.
Germany had also lost almost 35 percent as hopes for a recovery were
frustrated in 2002. On Wall Street the Dow Jones index had plummeted 17 percent during 2002, its worst performance for 28 years. The technology weighted NASDAQ composite had done even worse with a fall of 32 percent. London's FTSE 100 plunged 25 percent in 2002. In December 2002 the FTSE 100 index extended a losing streak into eight consecutive falls since its inception in 1984. Cumulative losses for the longest its of sequence sessions, FTSE World index since the start of 2000, after the bursting of the technology, media and 43 bubble, totalled percent. telecoms
The 20002002 period was the worst threeyear
19291931 fell 58.8 By when world markets since percent. comparison, world performance 1973 1974 in had height 39 Investors the lost and at the percent of world oil shock. markets 20002002. turbulent over ride a indeed endured
for Call Options Statistics Descriptive 5.3.3 and S&P 500 Index in 19992002 index 500 data S&P for option are shown in table 40. Table 40 reveals that Basic statistics is higher in postSeptember 11 period. It is also evident notably implied volatility average
209
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
from figure 42 that returns cluster in time. Appendices B. 1B.6 exhibit the specifications and for daily Causal inspection input parameters our option contracts. of various option 1) lower have higher implied 2) B1B6 that: strikes a volatility; reveals volatility appendices for is nearterm options. These results agree with the "stylised" fact smile more pronounced 3. in Chapter presented
Table 40: Basic Statistics for S&P 500 Index Options 6/15/2001
7/20/2001
8/17/2001
9/21/2001
10/19/2001
131
89
78
117
83
106
72.457 (87.751) 8001900
58.135 (62.02) 10251900
68.471 (81.118) 8001900
28.852 (42.526) 8001900
53.868 (66.467) 8001700
89.913 (98.478) 7001700
40
33
35
41
32
50
0.61 (0.4389) 0.2094 (0.0481)
0.5531 (0.4466) 0.1853 (0.0213)
0.6085 (0.4455) 0.1981 (0.051)
0.4983 (0.4056) 0.2856 (0.06707)
0.5633 (0.386) 0.2287 (0.04624)
0.4444 (0.3597) 0.2363 (0.08588)
#. of Options Mean Call Price Strike Range #. of strikes Mean Maturity Mean Imp. Vol.
Table 41: Descriptive Statistics for r Full Period 16/06/199931/1212002 DF stat.
Maximum Minimum Mean Std. Dev. Skewness Kurtosis3 Q(10) JarqueGerastat. #. Obs.
Pre9/11 15/06/199910/09/2001
Post 9/11 17/09/200131/12/2002
29.85260 [. 000)
23.5922p [.000]
17.82220 [.000]
055732 . 060052 .
048884 . 060052 .
055732 . 050468 .
0.0004325 014137 . 16579 . 1.22370
0.000299 013112 . 0.073602
0.00066424 015778 . 26741 . 0.83187
5.42500 [. 861)
10.18850 [.424)
5.93000 [. 821)
59.74210 [.0001
47.06940
13.28510
[.000]
[.0013]
892
505
326
1.40510
210
11/16/2001
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Figure 42: S&P 500 index and Returns: 19992002 1600 ,T
10
1400
1200
ö LL 6
4ä Q V)
ö c ý ) N
ö °D = Q U,
ö " Ö Il)
ý U N a I()
N
2 m LL U7
NN 44 CL Qý
ýuI)
N 4
c
N 0 v 0 ý
N R U
y U7
5.3.4 Contract Specifications Since variance swaps are not traded on organised markets, contract terms such as maturity, factor Investment banks etc are negotiable. quote daily delivery prices for their annualisation for from maturities three months to two year. Figure 43 plots various running counterparties the future realised74 three, six and ninemonth variance75 from September 1999 to March 2002. During these periods, average returns are close to zero. Table 41 shows that LjungBox statistics up to the 10`horder are not significant, which suggest that returns are not serial kurtosis Both and skewness excess are slightly positive, but JarqueBera test correlated. hypothesis that returns are normal in all intervals. the null statistics reject Descriptive statistics in table 41 indicate that returns in the pre and post9/11 periods are statistically similar.
But a close inspection of realised forward variance in figure 43 reveals
displays index's 500 S&P process variance a meanreverting property. In addition, the that between the 3 and 9month contracts has been forward spread variance widening realised is in It figure 9/11 43 also evident have forward that attacks. the realised variances since inverted at different maturities after September 11,2001, i. e. the longertermed forward
74 Readers should not be confused it with the smoothing average approach. Our results represent what the if had been have the obtained we entered day. variance trades that would swap on a particular variances 75 Variances are calculated by summing the arithmetic returns and the mean of returns is assumedto be zero. frequency is daily. is 252 factor and observation Annualisation
211
Chapter 5: Empirical Performance ofAliernative Variance Swap Valuation Models
variance has become more volatile than the shortertermed forward variance. September 11, 2001 has indeed served as a reflection point where investors have clearly changed their risk investment horizons. different appetites at
5.3.4.1 Design of VarianceSwap Contracts Evidence in figure 43 clearly shows that the JuneNovember 2001 period is an interesting time to value variance swaps. An accurate variance swap valuation model should be able to price into the inverted volatility termstructure relationship correctly during this period.
In
different how to variance swap models can predict the changing termstructure of assess order included have three, six and ninemonth variance swap contracts which are variance, we Money International Market (IMM) the compatible with
rulebook76. The specifications for
the three, six and ninemonth variance swap contracts are shown in table 42. It is noted that begin always contracts on the third Fridays and end on the Thursdays prior swap our variance to the third Fridays of the maturity month. For example, the start and end dates for the three2001 June variance swap contract correspond to the inception of the June 2002 S&P month 500 futures contract and the last trading day of the September 2001 S&P 500 futures contract on CBOE, respectively.
Figure 43: Realised Forward Variances 0.09 1 3M
00
°ö
212
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
Table 42: Contract Specifications for Variance Swaps Pre9/11 Maturity
June 2001 Start
End
August 2001
July 2001 Start
Start
End
End
3Month
15/06/01
20/09/01
20/7/01
18/10/01
17/08/01
15111101
6Month
15/06/01
20/12101
20/7/01
17/01/02
17/08/01
14/02/02
9Month
15/06/01
14/03/02
20/7/01
18/04/02
17/08/01
16/05/02
Post9/11 Maturity
September2001 Start
End
November 2001
October 2001 Start
End
Start
End
3Month
21/09/01
20/12/01
19/10/01
17/01/02
16/11/01
14/02/02
6Month
21/09/01
14/03/02
19/10/01
18/04/02
16/11/01
16/05/02
9Month
21/09/01
20/06/02
19/10/01
18/07/02
16/11/01
15/08/02
5.4 Results & Analysis Six variance swap models are investigated to determine the quality of variance forecastability following In to the deliver. this compare carry section we out analytic procedures the models the variance forecasting performance of various timeseries and optionsbasedvariance swap models: i) OutofSample Analysis. The outofsample error criterion is judged by MSE tests. Each model's performance is based on the aggregate ranking for each of the three 6M 3M, 9M; i. and e. contract months, ii) InSample Analysis. Insample test, which relies on the sum of price square error how is to good an option model can fit a given set of call option (SPSE), used evaluate data for each contract day. Insample analysis is primarily used to investigate whether data; are misspecified and overfit options models pricing option iii)
future be in All the to calibrated order calculate expected Calibrations. models must data Optionsbased by timeseries models are calibrated whilst call option variance. data Calibration historical to their estimate on structural parameters. rely models both inby and outofsample analysis; results are shared
Estimation Likelihood of the squareroot process. We apply this procedure Maximum iv) dynamics implied by the that underlying illustrate options are not consistent with to data. timeseries
76We thank Philipp Jokisch for contributing to this idea.
213
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
We explain our calibration procedures in section 5.4.1.
In and outofsample results are
5.4.2.2, likelihood in and maximum sections estimation of the squarereported and analysed 5.4.2.3. in is section conducted root process
5.4.1 Calibration Procedures Sections 5.4.1.14 discuss the econometric
and numerical
BlackScholes, hoc the ad stochastic volatility of calibrations volatility
1997), local (Bakshi jump et al., volatility with
methods that are used for (Heston, 1993), stochastic
(Coleman et al., 1999), EGARCH
(Nelson, 1991) and GARCHVS (Javaheri et al., 2002) models.
5.4.1.1 Calibrations for Stochastic Volatility withlwithout jump Estimation of stochastic processes on discretetime data is difficult.
Since volatility
is not
directly observable, many parameter estimation methods have relied either on timeseries analysis of volatility
proxies such as conditional volatility
or on cumbersome econometric
(1987) (1987) Scott Wiggins and using moment matching procedures". as techniques such Instead of estimating parameters from the underlying asset return data, we imply out the from the the models crosssection of observed option prices using all stochastic of parameters in Bakshi (1997). A disadvantage the prices as option et al. with traded call major actively "implied"
is it lack is formal that of a statistical theory. methodology
This approach is to
data and information to determine the structural the a wide range of uses market that assume the riskneutral of parameters
underlying
asset and variance processes.
The primary
it is for however, that option prices market parameter estimations, advantage of using by from information inferred the crossthe the marketplace using of "gauges" the sentiments section of the market option
prices, information
that essentially
is forward
looking.
into translate unique values for the volatility of volatility and smiles Consequently, volatility in a stochastic volatility correlation assetvolatility jumpdiffusion model. in a parameters
model, and into unique jump distribution
for are available solutions our selected stochastic models, a natural Since closedform the riskneutral parameters, which enter the pricing and for of the estimation candidate
77Both Scott and Wiggins found that the parameterestimateswere sensitive to the momentswhich they fitted.
214
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
hedging formula, is a nonlinear least squares(NLS) procedure involving minimisation of the between full For the the models and market prices. stochastic errors squared sum of is jump the (D model, set of stochastic volatility and parameters: volatility/jumpdiffusion Au o. }. The first four are the parameters of the stochastic volatility 4) ={p, Kv Qv j, , ,O, following jump The three the steps summarise our are parameters. remaining model whilst calibration procedures: i)
Collect N call options on the S&P 500 index on the same day, for N greater than or be the to to number of parameters one plus estimated. equal
For n=1,..., N and
A
let C,, (t, T,,, K,, ) be observed price and C,, (t, T,, K,, ) its define: For each n, model price. A
8(Vt,
ii) Choose «
(D) = Cn(t, TT, K,, )C.
(t, T,,, Kn)
and instantaneous volatility
V, to minimise the following objective
function: N
ý MIN SSE(t) = Eý(VI, (P) n=1
An alternative objective function, the percentage error, which can be obtained by dividing dollar errors by the underlying index price, may be used to estimate implied parameters. This is a sensible metric becauseoption prices are theoretically nonstationary but optionasset hypothesised However, under most this metric would stationary are processes. price ratios lead to a more favourable treatment of cheaper options, e.g. outofthemoney options at the longterm have Based inthemoney the options. and we on above considerations, expenseof The SSE is MATLAB to the approach. computer employed program to adopt chosen formulas and minimisation routines. Among others, Bates implement the option pricing (1997) have Bakshi al. et also applied this technique for similar purposes. (1995,1996) and
215
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
5.4.1.2 Calibrations for Local Volatility Model We apply finitedifference
method in MATLAB
using a trustregion optimisation algorithm
(Coleman et al., 1999) with a partial differential equation (PDE) approach's to directly solve for local volatility
Q(S, t).
The BlackScholes partial differential
logspacing. Sdimension the with along
CrankNicholson
discretised is equation
finitedifference
method is used
for solving the BlackScholes partial differential equation because it improves the stability finitedifference the of and convergence c(s, t; c')
algorithm.
Given any a*,
the bicubic spline
is functions the the end condition79 computed evaluated variational and using with SPLINE TOOLBOX.
in the MATLAB
We use a uniformly
spaced mesh with NxM
grid
[0, f*S; ]x is [0, the in rectangular region z] r the where maximum predetermined points , n;t data f is is for local the in the option and the range parameter volatility which market maturity discretisation The is by: for scheme given pricing. significant
S, = (11f) * Smit+i* AS,
i=0,..., M 1
z tt = jM1,
j0,...,
AS =[f
*S;
(1/f)*S;,, tt
u]l(M1)
We use backwards difference to approximate a2C
ac
N1
ät
and central difference to approximate
The resulting system is tridiagonal and can be solved by MATLAB at each time .
as2' as
inversion, i. than LU decomposition the rather matrix method. reduction e. row step using Starting from j=M 1 for which time the terminal condition is known and progressing backwards through time, we successivelysolve for the j 1 option values until j=1,
which
Sdimension. In boundary the the along conditions values addition, option timezero gives a zC Isu into the finite difference scheme by setting their incorporated 0 are =az S=L= 21 as as to zero. Further descriptions of finitedifference method go difference approximation central
78We sincerely thank Demetri Chryssanthakopoulosfor making printed copies of his codes available. 79This is a MATLAB option to ensurethat secondderivatives are zero.
216
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
beyond the scopeof this study but the readeris referred to Chapter 2 of Presset at. (1992) and Andersen et al. (1998) for a more thorough investigation of the implementation issues. 5.4.1.2.1 TrustRegion Reflective QuasiNewton Method Bicubic
important in implementation is the the element most spline
of finitedifference
determined is by Its solving a constrained nonlinear optimisation parameterisation method. problem to match the market option prices as closely as possible.
Andersen et al. (1998)
bicubic from drawback the that that smoothness was only splines might suffer suggested guaranteed in the Sdirection.
The reader is referred to Dierckx (1995) for discussions of
interpolation spline schemes that are smooth in both T and Sdirections. more sophisticated The "csape" and "fnval" functions available within the MATLAB
Spline Toolbox are used for
bicubic natural splines to ensure that a(K, T) and its partial derivatives of the construction
au au a2Q are well behaved. aT' aK' a2K The
builtin
minimisation "optimset"
MATLAB
Optimisation
is "lsgnonlin".
Through
Toolbox
function
the MATLAB
Large Scale Algorithm the options: select we
for
nonlinear
Optimisation
least squares
Toolbox
function
ON, Jacobian OFF, and Function
Tolerance lx 103. Preconditioned Conjugate Gradient is left to the default value of zero. These settings refer to, respectively, the "trustregion
reflective
quasiNewton"
method
(1999). Coleman by al. et proposed
5.4.1.2.2 Calibrations for Absolute Diffusion Process In order to demonstratethe effectiveness of the Coleman et al. method in reconstructing the local volatility surface,we consider the casewhere volatility is inversely proportional to index is follow the diffusion to In underlying assumed this process: example, an absolute price. SS,
=, u(S,, t)dt+`
dWW
t European for formula options of the absolute diffusion processis available (seeCox Analytic local volatility surface is known a priori, we have chosento set the Since the 1976). Ross, and to by finitedifference prices equal call values the European option provided routines. market index be initial let S;,,,,= 100, riskfree interest rate r= 4% and 25 the stock and We set a= 217
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
dividend rate q =1 %.
We consider twentyfour European call options on the underlying
following the above absolute diffusion process. Call options are equally spaced with strike T=[0.2: 0.2: 0.8]. discretisation for The 125] 10: K=[75: and maturities parameters asset prices M=200 N=50, time and set as respectively. are steps steps and
The lower and upper bounds
knots are 1. = 1 and u; =1 for i=1,2,3... 24. We let the
for the local volatility at the KxT
knots knots 24 to the the number equal of options p and calibrate m= spline spline of number equidistantly on the grid
f:
S*f
where the range parameter f=2.
The initial volatility values at the spline knots are specified as 0.2. The optimisation method function is iterations five 7.877 10 With the 6 and computed optimal objective an x requires . 2.712 l0' of x error average pricing
index point, the CrankNicholson
method excellently
full demonstrates the Figure 44 pricing across option call range of strikes. actual reproduces the accuracy of this local volatility excellent.
reconstruction.
The local volatility
reconstruction is
Indeed our methodology can reliably reconstruct the local volatility surface in the
[0.2,0.8]. [75,125] x region Figure 44: Calibrated Local Volatility Surfaces for Absolute Diffusion Process
Theoretical
0.1
Surface
.,. ..:... ............................................................. .............. ...::.............. ._.". ............. ......... . ýý i'ý120
110
100
90
Calibrated
80
Surface
. ............... 0.4 0.3 0.2 0.1
218
.................... ...........
0.2
0.6 0.4 
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
5.4.1.2.3.. FiniteDifference Settings Having verified that our algorithm accurately reproduces the volatility the pricing of variance swaps. We choose the number of knots p
smile, we now turn to where p= 72 < m.
The spline knots are placed uniformly between the endpoints of these intervals, with twelve knots along the Sdimension and six knots along the Tdimension.
Cubic splines are fit to all
T columns of the (S  7)space and then a second cubic spline is fit along the S direction. The ]x [0, r] of the (S local volatility surface has been calibrated over a set R: [(1/ f) * Si,,,,,f*S; ;, We is 200 50 M= N= the 7) maturity. r maximum choose asset steps and space, where time steps for the PDE discretisation and the range parameter f is set to 2 in order to level The dividend is to to the strikes. of constant set maximum yield equal accommodate 1.46%, which is the average yield over 20012002 obtained from Bloomberg80. We have be 2.31% interest U. Treasury S. to rate using average oneyear yields the constant proxied during the period studied. A summary of the parameters and settings for the problem is D. 1 in appendix provided
5.4.1.3 Calibrations for Ad Hoc BlackScholes Model Following function
Dumas et al. (1998) and Heston and Nandi (2000), we estimate the volatility 5.2.3.4 by fitting in section Q(X, r)
the deterministic
volatility
function to the
implied volatility at time z. The coefficients of the ad hoc model are BlackScholes reported in least by the the squares each of variance swap minimising contracts ordinary via estimated BlackScholes between implied different the strikes errors volatility across squared of sum functional form implied the of model's volatility. and maturities and
for TimeSeries Models Calibrations 5.4.1.4 GARCHVS by EGARCH in 5.2.4 and presented estimated of section are The parameters from the history of S&P 500 spot levels. First differences in logs of filtered using volatility levels to At calculate index are employed returns. 500 each time, we use the timeseries S&P two years (504 trading days) to filter the variance for the from the previous of returns We have GARCH(1,1) longer filtered models. also and experimented with EGARCH(1,1)
219
Chapter 5: Empirical
Performance ofAlternative
Variance Swap Valuation Models
intervals such as three or four years for estimations.
The results, however, are similar,
due in to the mean reversion strong variance. perhaps
Given a set of structural parameter
we simulate the Nstep return data using the EGARCH model.
The Monte
Carlo values reported are basedon simulating 100,000 paths with a time step of At = 252 . The diffusion parameters k, V, 8 and the expected delivery price of the GARCHVS
are
formulas described in section 5.2.4.1. based the on calculated
5.4
.2
Empirical Results
Figure 45 confirms the changing nature of the termstructure of realised variances in the S&P index: 500 longterm realised variance have gone up the on contracts variance swap September 11,2001 whilst realised variances for shorter swaps have tended after significantly to decline.
A convenient way to examine the deviation between the BlackScholes model
is to plot the BlackScholes implied volatility price price and market exercise price.
as a function of the
Figure 46 validates the usual findings in numerous studies that implied
volatility tends to vary across exercise prices, with implied volatility higher for inthemoney flattens out monotonically options and
as maturity increases.
The substantially smaller
magnitudes of the pre9/11 smiles relative to the post9/11 smiles is also evident in figure 46. In view of forecasting variance, an accurate variance swap forecasting model should not only but the effects also the changing termstructure of variance correctly. smile of take account Figure 45: Future Realised Variances for 3M, 6M & 9M Variance Swap
0.06
t3M f6M 9M ýý
0.05
0.04
0.03
0.02
r
Jun
Jul
Aug
Sep 220
Oct
Nov
!
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
Figure 46: TermStructure
of Implied Volatility
June15,2001
September21,2001
ö
O 7
7
500
Ö 7
1un
rrvI
11Afl1
July20,2001
6000ctober1000 19,2001
1500
2000
ýovember 1062001
1500
2000
Ö >
I=
I
2
1ý 5OOAugUSt 17,2001
ä >
Ö >
5.4.2.1 Calibration Results for Optionsbased Models Calibrated parametersof the stochastic volatility and jumpdiffusion models are shown in hoc BlackScholes parameters are summarised in table 45. Regressed 4344. ad tables Estimated structural parametersof the EGARCH and GARCHVS models are given in tables 46 and 47, respectively. The bracketedvalues are standarderrors. 5.4.2.1.1 Calibration The implied with/without
Results for Stochastic Volatility
structural parameters are generally different jump models. The stochastic volatility
Models with/without
Jumps
across the stochastic volatility
model controls skewness and kurtosis
diffusion jump the by whereas model is supposed to be able to internalise levels p and cr
221
Chapter S: Empirical Performance ofAlternative Variance Swap Valuation Models
kurtosis. higher A be drawn from and of number observations can skewness negative more tables 4344: i)
V,, V. and A are larger in the secondhalf of the sample, reflecting the more volatile immediate 9/11 the after attacks; conditions market
ii)
K is higher for the jumpdiffusion model;
iii)
Ov I longterm the xv are significantly lower for the jumpdiffusion variance Qv and model;
iv) The magnitude of p is lower for the jumpdiffusion model; be jumpdiffusion to The model appears able to explain negative skewnessand excess v) kurtosis via the jump parameters 2, u, and a. without making other parameters , "unreasonable". In addition, the averagejump frequency A is 0.69 time per year; the averagejump size 6l, 6.23 deviation The its cr, are and percent, respectively. above results standard and 14.35 (1997). Bakshi full in al. et agreementwith are
Stochastic for Parameters Volatility Calibrated 43: Table 9,,
1'{'v
Qv
June2001
0.0989
1.9194
0.4219
July2001
0.0757
1.9360
0.3104
August2001
0.0818
2.2232
0.3271
September2001
0.2136
3.3672
1.3677
October2001
0.1547
3.5877
0.5816
November2001
0.1209
3.0570
0.5246
222
Model
9v I K,
p
V,
0.7011
0.0482
0.0515
0.6485
0.0378
0.0391
0.7135
0.0467
0.0368
0.6388
0.1770
0.0634
0.6505
0.0845
0.0431
0.0565
0.0396
0.6358
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Table 44: Calibrated Parameters for Stochastic Volatility with Jump Model
Ov
p
Qv
Ky
VV
Q,
m,
11
V,
Ov/ K
June2001
0.0711 4.2926 0.1812 0.5333
0.0366 0.4589 0.1836
0.1439 0.0219 0.0166
July2001
0.0684
1.9683 0.2850 0.7293
0.0347 0.4884 0.0191
0.0827 0.0034 0.0348
August2001
0.0421
5.9795
0.0354
0.0261
September2001 0.1166 3.1058
0.0231
0.5747
0.1892
1.6002 0.6294 0.1643 0.6808 0.1578
October2001
0.0749 5.5933 0.7492 0.4159 0.0722
November2001
0.0245
4.5700
0.6491
0.3216
0.1037
0.0359
0.0235
0.0070
7.8e7 0.0170 0.0375
1.0116 0.1438
0.0659 0.0242 0.0134
0.8581
0.0553
0.1679
0.0260
0.0054
5.4.2.1.2 Calibration Results for ad hoc BlackScholes Model Next we focus on the ad hoc BlackScholes model. Table 45 shows that the regressed is This hoc BlackScholes the contracts. model are variable very across ad parameters of Durbinimplied is 0.85, Average R2 due the to nature of volatility. and changing probably Watson's statistics cannot reject the null hypothesis that residuals are not autocorrelated.
Table 45: Estimated Parameters for Ad Hoc BlackScholes Model
ao 9.044E01 June2001
at 7.775E04
a2
a3
1.895E07
1.948E01
a4
1.689E02
as
1.217E04
R2
DW Stat
0.7975
p=0.376
0.9266
p=911
0.7404
p=0.85
(5.789E02) (9.343E05) (3.960E08) (3.608E02) (1.114E02) (3.032E05) 5.476E01 July2001
4.099E04
1.021E07
6.174E02
1.470E02
2.931E05
(3.133E02) (5.070E05) (2.080E08) (1.358E02) (3.911E03) (1.138E05) 8.026E01 August2001
6.223E04
1.206E07
3.013E01
3.178E02
1.845E04
(8.163E02) (1.373E04) (6.080E08) (5.197E02) (2.068E02) (4.228E05) 1.123E+00 September2001 (6.978E02)
8.287E01
1.113E03 (1.260E04)
7.017E04
4.011E07 (5.880E08)
2.619E01 (4.7145E02)
1.862E07
October2001
2.553E01
1.250E01 (2.004E02)
4.645E02
7.369E07 0.8524 p=0.436 (4.343E05)
1.264E04
0.9438
p=0.619
0.848
p=0.520
(4.201E02) (7.327E05) (3.290E08) (2.362E02) (1.119E02) (2.193E05) 1.387E+00 November2001
(9.328E02)
1.469E03
4.207E07
(1.670E04)
(7.700E08)
5.254E01 (6.293E02)
223
4.402E02
3.471E04
(3.703E02) (5.663E05)
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
5.4.2.1.3 Calibration Results for EGARCH and GARCH Variance Swap Models On the other hand, it appears that the evolutions of the estimated parameters of the EGARCH and GARCHVS
models are more stationary as compared to the implied parameters of the
stochastic models. Based on the statistical results of the estimated parameters in tables 46 and 47, EGARCH dynamics.
seems to be more capable than GARCH to describe the underlying returns
Negative and significant
al's
also indicate that EGARCH
in returns. asymmetry
Table 46: Estimated Parameters for EGARCH ao June2001
July2001
August2001
September2001
October2001
November2001
0.4255 (0.1901)
0.4562 (0.2369) 0.4499 (0.2409) 0.4397 (0.2332) 0.5201 (0.2561) 0.3742 (0.2744)
al
Y,
0.1839 (0.0358)
0.1734 (0.0379) 0.1789 (0.0388) 0.1823 (0.0382) 0.1831 (0.0413) 0.1869 (0.0407)
224
iß,
0.05622
0.9518
(0.0381)
(0.022)
0.07421
0.9485
(0.0406)
(0.0274)
0.07508
0.9493
(0.0401)
(0.0278)
0.0732
0.9504
(0.0398)
(0.027)
0.0823
0.9412
(0.0424)
(0.0297)
0.0669
0.9583
(0.0488)
(0.0318)
is able to capture
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Table 47: Estimated Parameters for GARCHVariance
a,
A
1.23e05
0.0921
0.8411
(1.68e05)
(0.0681)
(0.1480)
1.27e05
0.0899
0.8415
(1.85e05)
(0.0705)
(0.1594)
1.24e05
0.0941
0.839
(1.98e05)
(0.0812)
(0.1764)
1.58e05
0.1175
0.8041
(1.62e05)
(0.0723)
(0.1365)
1.804e05
0.1234
0.7845
(1.58e05)
(0.0697)
(0.1294)
1.903e05
0.1277
0.7755
(1.50e05)
(0.0651)
(0.1196)
ao June2001
July2001 August2001 September2001 October2001 November2001
Swap
5.4.2.1.4 Calibration Results for Local Volatility Model Last, we turn our attention to the local volatility
model.
Figure 47 displays the calibrated
Notably the variation in surfaces from each of the six sets of option prices. . local volatility is greater than the variation in implied volatility that produced it in figure 46. local volatility
For skewed option markets, this behaviour is consistent with the Zou et al. 's (1997) heuristic index level local implied the twice varies that volatility with about volatility as as rapidly rule leads This believe to that our calibration procedures can reliably result us strike. varies with local the volatility reconstruct volatility
surfaces.
However, the highly variable shape of the local
is because it implies that future local volatility potentially problematic surfaces
different from is, The be downwardsloping today's. typical very smile volatility smiles will by fear driven downward the large of rapid to a extent, movements of the underlying index. The local volatility approach typically predicts that future volatility will tend to flatten out and disappear over time. This prediction, however, is clearly at odds with market reality that the be to tends quite stationary over time. volatility smile
225
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Figure 47: Calibrated Local Volatility Surfaces
September 21,2001
ý1 July
[100
1800 1i771 11LAJ i Ann
1800
20,2001
1800 1600 1200 1400
1ý
1800 1600 120(] 1400
BOB 1000 November 16,2001
August17,2001 ..............
time
800 1000
1800 1600 1200 1400
800 1000
1800 1600 1200 1400
indexlevel 5.4.2.2
Variance Swap Forecasting
5.4.2.2.1 Implementation
Results
Issues for Optionsbased Variance Swap Model
by: is The variance estimator given
F
n ;=o
2 5;  S; +ý
Si
days is S 252; factor, is trading is F, the the to the of and number annualisation set n where index. Moreover, S&P 500 be is the the to sample zero. of mean assumed closing price
Demeterfi for the using et al. framework is the "wing effect", which refers One major concern high low for The hedged implementable and strike the prices portfolio. replicating to the be this to by focusing strategy the can attributable chosen central region on strikes of range
226
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
liquidity. Clearly, judgement is in determining is there the range of sufficient required where strikes81. Table 48 shows variations of the S&P 500 index during the lifespan of the corresponding variance swap contracts. It has a range between 29.94% and 21.4% in 6/20018/2002. Consequently,we price variance swap for a 30% variation of the index with 1,000 discrete strikes.
Table 48: Variation of S&P 500 Index June 2001
Pre9/11 Maturity 3Month 6Month 9Month Post9/1I Maturity
Min
Max
18.93%
1.87%
20.47%
1.87% 1.87%
20.47% September2001 Min
Max
3Month
0%
21.18%
6Month
0%
21.4%
9Month
0%
21.4%
July 2001 Min
Max
20.24%
0.82%
20.24%
0.82% 0.82%
20.24% October 2001 Min
Max
1.28%
9.23%
1.28%
9.23%
17.88%
9.23%
August 2001 Min
Max
16.88%
1.98%
16.88%
1.98% 1.98%
16.88% November 2001 Min
Max
5.14%
2.97%
7.83%
2.97%
29.94%
2.97%
5.4.2.2.2 OutofSample Test: Variance Forecastability Estimated delivery prices for the three, six and ninemonth variance swap contractsfor each in 49; future table swap models are given the variance realised variance are also shown six of in the second column of the same table. Table 50 reports the aggregatemeansquareprice for for three, the (MSPE) sixand ninemonth rankings contracts; aggregate model errors bracketed displayed in the sametable. contracts are ninemonth and three, six and be from can eighteen variance Based on results swap contracts, a number of observations 50: 49 from tables and drawn i)
ii)
Aggregate MSPE ranking of the models from table 50 is robust across maturities;
Conditional heteroskedasticmodels outperform optionsbased models in predicting GARCHVS first in find More that ranked with all maturities. strikingly, we variance
81We thank Tom Ley for this invaluable comment.
227
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
even a naive EGARCH simulation can deliver less forecasting errors than the highly sophisticatedoptionsbasedpricing models; model is similar to the stochastic volatility model in producing variance forecasts. Adding a jump component to a stochastic volatility model serves The jumpdiffusion
iii)
to increase variance in short maturity
but does not seem to enhance variance
forecastability; The local volatility variance forecasts;
iv)
model underperforms ad hoc BlackScholes model in making
v) All models predominately overpredict variance. On average there is a 81% chance that any variance swap model will overprice future variance; vi) The amount of overpricing is more manifest in the aftermath of the 9/11 attacks; Last and most importantly, we observe that the optionsbased variance swap pricing
vii)
models cannot produce enough variance termstructure patterns.
5.4.2.2.3 Comments on OutofSample Results Although our sample is small, it is still puzzling to see there is such a large discrepancy between optionsbased and timeseries models in terms of variance forecastability. One for disappointing the explanation performance of the optionsbased pricing plausible framework concerns with the fact that the Demeterfi et al. methodology was originally developed for hedging. The timeseries methods use historical information to price variance from be different future the the could expectations about evolution of the asset and swaps in Theoretically option prices. that embedded are option prices should summarise all price future information expected regarding volatility whereasthe timeseriesapproachcan relevant index information inferrable history from that the the stock of subset of past exploit only prices.
228
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
Table 49: Delivery Prices for 3M, 6M and 9M Variance Swap Contracts Realised Variance
EGARCH
GARCHVS
Ad Hoc BS
Jun2001 3M
0.04093
0.05052
0.03573
0.05704
0.07840
0.04860
0.05400
Jun2001 6M
0.03939
0.04761
0.04114
0.05269
0.07036
0.04761
0.04937
Jun2001 9M
0.03599
0.04671
0.04282
0.04873
0.06616
0.04633
0.04660
Jul2001 3M
0.05225
0.04212
0.03532
0.03917
0.05498
0.03802
0.03817
Jul20016M
0.03819
0.04224
0.04100
0.03788
0.05604
0.03770
0.03769
Jul2001 9M
0.03568
0.04244
0.04289
0.03669
0.05398
0.03701
0.03693
Aug2001 3M
0.05583
0.04580
0.03503
0.05344
0.06946
0.04431
0.04857
Aug2001 6M
0.04030
0.04419
0.04076
0.04589
0.06019
0.04196
0.04237
Aug2001 9M
0.03878
0.04355
0.04274
0.03976
0.05910
0.03985
0.03905
Sep2001 3M
0.03702
0.07539
0.04076
0.13005
0.13874
0.12703
0.12880
Sep2001 6M
0.03291
0.06063
0.04550
0.10088
0.10017
0.09670
0.09656
Sep20019M
0.03432
0.05425
0.04747
0.08109
0.08297
0.07843
0.07761
Oct20013M
0.02514
0.04324
0.04063
0.07396
0.08366
0.06959
0.07152
Oct2001 6M
0.02798
0.04324
0.04501
0.06220
0.06390
0.05914
0.05810
Oct2001 9M
0.03743
0.04309
0.04648
0.05340
0.06411
0.05260
0.05131
Nov2001 3M
0.02566
0.03114
0.04106
0.06467
0.06323
0.05107
0.05302
Nov20016M
0.03089
0.03548
0.04539
0.05182
0.05292
0.04643
0.04612
Nov20019M
0.05703
0.03709
0.04680
0.04151
0.05570
0.04308
0.04215
Contracts
Local Volatility
Stochastic Volatility.
JumpDiffusion
Table 50: Aggregate MeanSquare Price Errors and Model Rankings for 3M, 6M and 9M Variance Swap Contracts
3M 6M 9M
EGARCH
GARCHVS
MSE
0.00213
Rank
Ad Hoc BS
Local Volatility
Stochastic Volatility
JumpDiffusion
0.00124
0.01300
0.01678
0.01112
0.01174
(2)
(1)
(5)
(6)
(3)
(4)
MSE
0.00112
0.00067
0.00644
0.00797
0.00535
0.00530
Rank RS E
(2)
(1)
(5)
(6)
(4)
(3)
0.00101
0.00047
0.00285
0.00474
0.00248
0.00240
Rank
(2)
(1)
(5)
(6)
(4)
(3)
229
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
5.4.2.2.4 InSample Fit for Option Pricing Models
Given the implied framework is supposedto provide a forwardlooking means to "gauge" Demeterfi is important it to the et al. optionsbased understand why market sentiment, framework has such a poor variance forecasting performance. Table 51 reports the sum of day. A few for for (SPSE) the contract each each of option models price square error in order: are observations i)
The SPSE is successively lower as we extend from the ad hoc BlackScholes to the jump local models with/without and volatility model; volatility stochastic
ii) The local volatility model has the lowest SPSE in all contract days whilst allowing jumps to occur reducesthe SPSEfurther over the stochasticvolatility model; iii)
Overall, modelling for skewed and leptokurtic distributions via the relaxed BlackScholesspecificationsfurther enhancesthe model's ability to fit option prices. But the finding that the local volatility model does not improve variance forecastability over local highly is BlackScholes hoc the given model surprising, especially the ad insample excellent pricing performance. volatility model's
The above observations suggest that a flexible but theoretically inconsistent model may dominate insample fit but has much less predictive power for predicting future variance, by insample the implies that overfitting misspecified model achieves good a results which options data.
Option for Fit (SPSE) InSample 51: pricing Models Table Ad Hoc BS
Local Volatility
Stochastic Volatility
JumpDiffusion
JUNE2001
995.3221
35.0748
178.2048
81.7872
JULY2001
51.4489
2.2035
25.1238
23.8116
AUGUST2001
554.6932
5.9678
120.7058
32.8119
SEPTEMBER2001
759.4753
16.7763
170.2324
106.6719
OCTOBER2001
197.8162
0.5986
67.0353
13.2934
2079.2547
8.3802
238.1859
42.5544
NOVEMBER2001
230
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
5.4.2.3 Consistency with the Timeseries Properties of Volatility Based on our limited sample,we have demonstratedthat insample fit of daily option prices is from hoc BlackScholes to the stochastic volatility better the extend as we ad progressively local By far jump and volatility model. our evidence shows that: models with/without i)
Incorporating stochastic volatility and jumps to the option model does not lead to a GARCHtype in forecasting future to the terms models of performance superior variance but it does contribute to a better insample fitting;
ii)
The meansquare error based ranking of the local volatility model is in sharp contrast based the on the insample fit of option prices; hence there may obtained ranking with be an issue of overfitting.
A possible interpretation of these results is that the local volatility
model does not properly
dynamics its the to pathdependent surface of volatility. volatility relate
Since the sole source
local is index, the the under volatility model underlying options of option prices, of variations regardless of maturity and moneyness, must perfectly covary with each other and with the imposes This dynamics. In potentially a stringent on restriction option price asset. underlying the next section our goal is to investigate whether option prices are consistent with its dynamics. underlying
5.4.2.3.1 CIR SquareRoot Process Basing the results upon Cox et al's (1985) stochastic interest rate model, Bates (1996) developed an econometric method for testing the consistency of the distribution implied in its Bakshi S&P (1997) timeseries the test to this properties. with et al. applied option prices 500 index and found that the stochastic volatility with/without jump models were misspecified because the volatility of volatility
cr,
too high. But Bakshi et al. 's study only tested for
implied Their its the structural parameters of with prices. evolution of option the consistency because implied by the stochastic option prices problematic volatility potentially were results for Consequently, known true little the volatility. surrogate a whether as was used were implausible
structural parameters were caused by misspecification
of the models, or by
The distribution the the procedure. estimation question remains open whether problems with is directly the that by same as prices observed from market asset price. implied option
investigate directly the consistency of implied distributional assumption In this section we index Following Bates (1996), underlying of price. the evolution when volatility riskwith 231
Chapter 5: Empirical Performance of Alternative Variance Swap Valuation Models
density Vt 2cV is the transition to the conditional variance of y= proportional premium , 4,, /a is (40, for 2cVe'r°°t), where V, X2 process noncentral a squareroot conditional on , v, I The density ln(VV,, ) is by: / V, (1transition 0.50 K,, given of a"") c' = . P(ln(V %... 1.11+&.
a
sv ý s(e=+n) (eZ)o. eo. /V 1= 'l1 SV
2U.
(0.25ezA)J
tor(o.sv+j)j!
2cV, 2cV, ! A= 4G and Qy e'r"°' e' v= = where , +,.
5.4.2.3.2 Results of Maximum Likelihood Estimation Maximum likelihood estimatesof the parameters 9,,, K, and a, using historical timeseries Average implied for 52. in the the stochastic table values of structural parameters are shown in 43 44 jumpdiffusion in tables the same table models and are also presented and volatility for easeof comparison.
Table 52: Estimated & Implied Structural Parameters ev 30Day Historical Volatility Weighted 30Day BS Implied Volatility
0.12117
K,
Dry
1.35228
P
0.32318
(3.22l.e3)
(0.63628)
(3.486e4)
0.26997
3.22071
0.41141
(7.038e3)
(1.53316)
(5.715e4)
Implied Stochastic Volatility Model
0.12427
2.68185
0.58888
Implied Jump
0.06627
4.25158
0.52672
Diffusion Model
MLE Value
0.21600
602.0167
0.76250
557.5923
0.66470
0.30615
N. A.
N.A.
for CIR process are: Estimation procedures i) ii) iii)
by index between is returns; calculating correlation volatility changes and estimated p We use a weekly 30day historical volatility series as a proxy for the true volatility 92; Since Bakshi et al. (1997) reported that implied
instantaneous volatility
was on
0.5 less BlackScholes the than percent the apart stochastic among and average
82The timeseries comprisesof the weekly observations for the period from June 1999 through December2001.
232
Chapter S: Empirical Performance ofAlternative Variance Swap Valuation Models
have for the also we estimated models, parameters a vegaweighted Blackvolatility Scholes implied volatility series. This series is constructed by averaging the vega30day implied BlackScholes call and put volatility. weighted Since options are priced off the riskneutral process but not the true process, parameters distributions be different. Because from true riskneutral can and of volatility and estimated jump risk premiums, only 0, cr, and p are directly comparable in table 5283. Four observations are in order: i)
Optionimplied
a, 's are significantly higher than its historical estimate, although it is
it level high as a was suggested by Bakshi et al. (1997), who found a 300% not as difference between the "true" and "implied" parameters;
ii) Estimated historical Kv, which is 1.352, is consistent with Das et al. (1999) assertion that plausible values for Kv are in a neighbourhood of unity. Furthermore, higher estimates for optionimplied K,, confirm that volatility riskpremium is significantly positive; iii)
Estimated correlation between index returns and historical volatility is significantly different from the correlations implied by the 30day BlackScholes implied volatility and the stochastic option models. Based on an EGARCH specification for equityNelson (1991) dynamics, gave an estimate of 0.12 for the correlation between return in the true volatility, which is closer to our historical timeseries changes and returns estimates;
iv) The results for 0, 's are mixed and we cannot draw any consistent observation to finding. our explain Nevertheless, it appearsindisputable that the distributional dynamics implied by option prices index its are not consistent. and underlying
5.5 Summary This chapter has emphasised the empirical conditional
implications
of forecasting variance using
heteroskedastic approaches and an arbitragefree optionsbased variance swap
framework in the period from three months before to after the 9/11 attacks. The exercises are
83See Bates (1996) for a detailed explanation of riskneutral versus true distributions.
233
Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models
latest in finance literature by timeseries the and option pricing models employing carried out to generate skewness and kurtosis in returns distributions.
The Demeterfi
et al. (1999)
been has from framework have forecasting examined a perspective practical and we variance limitations. its and properties of understood some
During the six contract days from three
framework Demterfi 9/11 that the before terrorist the to et al. show attack, we after months forecasting have In MSPE. future that timeseries models a smaller and variance overpredicts directional illustrate the that optionsbased models cannot predict changes of the addition, we 3M, 6M and 9M future variance.
Our results are in direct violation to the underlying hypothesesthat: i) Each generalisationof the benchmark BlackScholes model should be able to improve the volatility forecastability of the optionsbasedpricing model; ii) If option prices are indeed representativeof their underlying timeseries and forwardlooking then the forecastability of optionsbased variance swap models should be superior to their timeseriescounterparts. In particular, we cast doubt on the usefulnessof the local volatility model as a forecasting tool becauseit has the best insample fitting result but worst volatility forecastability. We observe improve insample flexible the and sophisticated option pricing model may more a that using fitting of option prices but not necessarilyforecastability of future variance. In our view, it is important that an accurate variance forecasting model should not only take account of the (crossovers) but the termstructure changing correctly. also of variance smile effects Therefore we have a strong reservation about the effectiveness of forecasting future variance through logcontract replications in inconsistency is in that there small sample In summary, we provide some evidence between optionsbased models and timeseries models from forecasting performance volatility Demeterfi 9/11 1) be: The before the the terrorist to after attacks. could reasons three months developed for hedging and its strategy can only guarantee that framework originally was et al. have the same payoff at maturity as the variance swap regardless will portfolio the replicating index. Bakshi by Our finding the taken with and consistent arguments are of the actual path (2003) that volatility where a negative Kapadia risk an equilibrium premium suggests and downside hedge higher the to investors therefore to a a risk, pay as making index options act hold in their portfolio than its price when volatility is not to options (implied volatilty) price
234
Chapter S: Empirical Performance of Alternative Variance Swap Valuation Models
future delivery Rather the than predicting variance, price probably only reflects the priced. implied 2) volatility of replication; costs
is largely a monotonically
decreasing function of
maturity and therefore the optionsbased strategies cannot produce enough variance termstructure patterns; 3) distributional dynamics implied by option parameters is not consistent by likelihood data its the timeseries estimation of the squareroot as stipulated maximum with likelihood Results of maximum estimation of a squareroot process also suggest that process. implausible levels lesser to on may rely of correlation and, a extent, volatility models option variation to rationalise the observed option prices.
In particular, the high magnitude of
in levels option prices, which generates excessive of negative skewness, correlation negative for biases in be the the equityindex market. the observed strike culprit price could Finally, although the forecast periods are overlapping, we must point out that this will only forecasting the performance of timeseries models. affect be forwardlooking to supposed models are periods.
Optionsbased variance swap
and therefore insensitive to the choice of sample
A larger sample group is indeed required in order to draw a more consistent and
forecasting the timeseries conclusion about superiority of variance significant statistically models.
Until
then we have a strong reservation about the use of Demeterfi
forecasting for volatility methodology
235
et al.
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
CHAPTER
6
Discussion Summary, Further Research
and
Suggestions
for
6.1 Introduction As an aid to the reader, this final chapter of the dissertation restates the research problems in this study. The major sections of this chapter summarise and discuss the results. The final for future research. section makes recommendation
6.2 Statement of the Problem Throughout the first project (Chapter 3) we examine the empirical behaviour of S&P 500 futures option's implied volatility using daily data from 1983 through 1998. The primary implied is the the termstructure to of patterns of observe, characteriseand analyse objective investigate 500 The is S&P in to the marketplace. objective second whether option volatility hypothesis line in the with rational expectations under a meanreverting volatility prices are final in is The identify to this objective work what types of option models would assumption. be consistent with the observedmoneynessbiasesin the S&P 500 options market. In the second project (Chapter 4) we investigate the performance of APARCH models that 500 for S&P daily decay in the slow returns autocorrelations using can potentially account futures seriesfrom 1983 through 1998. The objectives are: i) To investigate the effectiveness of asymmetric parameterisation and power transformation within the context of APARCH specifications; ii)
To study the impact of structural change of volatility on the performance of asymmetrical and symmetrical conditional volatility models;
iii)
To compare the performance of EGARCH (Nelson, 1991) with APARCH models;
iv) To explore the ability of different symmetrical and asymmetrical statistical loss functions to track the insample forecasting performance of conditional volatility models; different by the forecasts To of quality conditional assess conducting exante volatility v) straddle trading exercises. 236
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
In continuation of our study of modelling volatility, the third project (Chapter 5) evaluates the volatility
forecasting performance of timeseries and optionsbased variance swap valuation
models on the S&P 500 index. The primary goal is to present a complete picture of how each generalisation
of the benchmark BlackScholes
model can really improve the volatility
forecasting performance of variance swaps and whether each generalisation is consistent between in and outofsample results. The second goal is to investigate whether there is any systematic difference in performance between timeseries and optionsbased variance swap valuation models. It is intended to explore whether optionsbased models, which are forwardlooking,
are capable of outperforming
discretetime processes, which use only historical
information, in predicting future variance.
6.3 Summary of the Results In this research we have examined many empirical issues relating to the modelling of volatility from both the options market and timeseries perspectives. The results are in summarised this section. The first project (Chapter 3), entitled "A Report on the Properties of the TermStructure of S&P 500 Implied Volatility", analyses the termstructure of implied volatility using S&P 500 futures and its options data from 1983 to 1998. BlackScholes
formula, implied volatility
Contrary to the basic assumption of the
exhibited both smile effects and termstructure
Termstructure analysis revealed that: 1) implied volatility patterns.
tended towards a long
term mean of about 16%; 2) put options had higher premiums and a larger range of fluctuation 3) shortmaturity options were more volatile than longmaturity options. than call options; Results from harmonic analysis showed that put options were more "responsive" to a change for In found be than to options. call sentiment addition, strongest smile effects were of market indicating by that the options, shortterm options were most severely mispriced shortterm BlackScholes formula and therefore presented the greatest challenge to any alternative option Furthermore, evidence suggested that option prices were not consistent with models. pricing the rational expectations under a meanreverting volatility
assumption. We also conducted a
test to find out whether observed moneyness biases were consistent with the distribution derived the from any specific distributional hypothesis. riskneutral of skewness distributional
The 4% skewness premiums results agreed with the termstructure analysis that the degrees of
237
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
had been 500 S&P in the market options gradually worsening since around 1987. anomalies As correlation might be responsiblefor skewness,our diagnostics suggestedthat leverage and jumpdiffusion (with negativemeanjumps) models were more appropriate for capturing the futures 500 S&P in biases the options market. observed Based upon the skewness premiums results in Chapter 3 that a leverage model was suitable to (Chapter 4), "An Empirical the the second anomalies, project market entitled observed model Comparison of APARCH Models", compares a group of welltheorised conditional volatility for biases in S&P 500 the termstructure the observed that account can potentially models futures options market. Sixteen years of daily S&P 500 futures series were used to examine the performance of the APARCH models that used asymmetric parameterisation and power its volatility absolute residual to account for the slow decay and conditional transformation on in returns autocorrelations.
No evidence could be found supporting the relatively complex
APARCH models. Loglikelihood ratio tests also confirmed that asymmetric parameterisation in S&P 500 dynamics the characterising transformation were not effective returns and power within
the context of APARCH
specifications.  In addition, a 3state volatility
regime
detect "quiet" "noisy" the to and was used periods and provide evidence that switching model the performance of conditional volatility
models was prone to the state of volatility
of the
EGARCH best in "noisy" AIC The that showed metric was periods whilst return series. GARCH was the top performer in "quiet" periods. In an effort to rate the performance of different conditional volatility models, aggregated rankings were used to determine the best for best AIC EGARCH Aggregated the the that was rankings metric showed overall model. for We to that apply attempted additional allowed also statistical criteria model. in the loss functions of investors to select the best volatility forecasting symmetry/asymmetry but results were mixed. model, superior.
Insample results showed that no single model was clearly
Since it was not sensible to evaluate forecasting performance with only a single
function, we evaluated the performance of volatility predictors based on their loss statistical from trading changes and generate nearestthevolatility to exante profits predict ability 500 straddles in four twoyear outofsample periods. Outofsample results S&P money EGARCH the that model outperformed GARCH and both of them could demonstrated Therefore, in four exante significant returns periods. one out of sample generate statistically inefficiency in degrees S&P Finally, 500 futures the of our market. options there were certain that the deltaneutral to trades revealed also presumption create of using experiments trading
238
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
in large index the of movements. We concluded practical event was not portfolios riskfree that a new derivatives instrument was needed to allow traders and investors speculate on directly and efficiently. more volatility by the findings in Chapter 4 that traditional
Motivated
optionsbased volatility
trading
5), (Chapter large third the to project entitled market moves, strategies were vulnerable Perfonnance of Alternative
"Empirical volatility
Variance Swap Valuation Models", evaluated the
forecasting performance of different specifications of timeseries and optionsbased
S&P 500 index from in the the three months before on models valuation period variance swap to after the 9/11 attacks. Our exercises were carried out by employing the latest timeseries in finance literature to generate skewness and kurtosis in returns models and option pricing distribution.
Based on results from six wellchosen contract days from threemonths before to
Demterfi framework 9/11 that the terrorist showed attack, we et al. the overpredicted after future variance and that timeseries forecasting models have a smaller MSE. In addition, we illustrated optionsbased models could not predict the directional changes of 3M, 6M and 9M future variance.
We observed that using a more flexible and sophisticated option pricing
insample fitting but forecastability future improve the of option prices not of might model from likelihood Finally, maximum results estimation of the square root process variance. high in the that of magnitude negative correlation option prices, which generated suggested be for levels the observed strike price negative skewness, might of responsible excessive biases in the S&P 500 index market.
6.4 Discussions of the Results This dissertation is a quantitative
study whose primary
investigate is the to objective
forecasting different timeseries specifications of and optionbased volatility performance of influence based is biases. Our the the of observed primarily market research under models data for 500 but S&P 19822002. It the three period selfcontained contains upon the use of This (less discusses implications the section projects. anticipated and seemingly related in this study as well as the relationship of the current research to findings anticipated) previous research.
239
Chapter 6: Summary, Discussions and Recommendations for Future Research
6.4.1 TermStrucutre of Implied Volatility The study started with a graphical inspection of the termstructure of implied volatility. Termstructure
analysis revealed that implied
systematically
volatility
followed
implied that models such as stochastic volatility which patterns, predictable
market.
models and
heteroskedastic models could be used to account for the inefficiency
conditional
some
in the
The anticipated results included the finding of moneyness and maturity biases.
Observed irregularities in relative implied volatility
constituted strong evidence against the
hypothesis that the BlackScholes' implied volatility was the market's fully rational volatility forecast.
The Ushape could be the result of: 1) illiquid
distribution.
Bidask spread in illiquid
options and this could artificially forming the basis for "volatility
market; 2) nonnormality
markets was typically
introduce high volatility
returns
huge for outofthe money to outofthemoney
options,
skew". But perhaps the more credible reason responsible for
data. The "volatility in U the returns was nonnormality shape the observed
skew" could also
be a result of active use of portfolio insurance policies to protect investors' portfolios, thus driving for demand their prices and outofthe up money put and options surging a creating volatility.
Our termstructure evidence also showed that the convexity of relative implied
Thus insensitive longerterm time. to was options relatively of calendar evolution volatility of for indicating the that were shortterm options options, shortterm strongest were smile effects by BlackScholes formula the the greatest perhaps mispriced and presented most severely in Das the et al. to results any alternative option pricing models, which agreed with challenge (1999). been had implied the getting that volatility The termstructure evidence also supported notion importance increasing of using the time thus stressing evolved, calendar as skewed more degrees Moreover, of the relative leverage models to characterise skewness properly. lengthened. Once again this evidence suggested that decreased termtomaturity as anomalies formula In severely mispriced BlackScholes addition, evidence also shortterm options. the implied volatility of call options in a given inthemoney (outofthemoney) the that revealed implied in to the outofthesimilar volatility quite opposing of put options was category (inthemoney) or true period category, which sample was generally of regardless money in Such between similarities pricing structure call and put options existed termtomaturity. the due the to of working putcall parity. mainly
240
Other more regular results in Chapter 3
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
included:
1) put options commanded a higher premium than call options in each maturity
group, which was consistent with Black's leverage effect.
A possible explanation of these
results was that purchase of S&P 500 futures was a convenient and inexpensive form of portfolio
insurance.
Thus excess buying pressure of frontmonth put options might cause
prices to increase, resulting in higher puts' implied volatilities. put implied
volatilities
meanreverted to their
Furthermore, average call and
longterm
mean of
16% and 16.8%,
implied That to that was say when volatility was above its longterm mean level, respectively. the implied volatility of an option would have to be decreasing in the time to expiration, and vice versa; 2) implied volatility of shorter maturity options were more variable than longer 3) implied variation of put options' options; volatility maturity
was higher than call options.
The latter result could be viewed as evidence that put options were more "responsive" to the arrival of new information.
Less anticiplated, though, were the findings that option prices
might not be consistent with the rational expectations hypothesis under an AR(1) process. In addition to Stein (1989), Bates (1996) and Bakshi et al. (1997), the results from elasticity requirements
questioned whether the volatility
process implied
consistent with the properties implied in its timeseries. assumption of constant volatility
by traded options was
Given that the BlackScholes'
was so poorly violated, it was not surprising that the 4%
skewness premiums results recommended the use of a leverage model or a jumpdiffusion biases. the to observed market capturing model
6.4.2 Conditional HeteroskedasticModels In order to find a timeseries model that could take account of the observed termstructure biases in Chapter 3, the performance of the APARCH and EGARCH models were compared by using different types of insample criteria.
Likelihoodbased
statistics questioned the
APARCH the complex more using models. In particular, the use of asymmetric rationale of transformation APARCH be ineffective power and to the were shown within parameterisation specifications.
According to the AIC metric, EGARCH
"noisy" and "quiet" periods, respectively.
and GARCH were top models in
Overall, EGARCH was the best model based on
Since EGARCH and GARCH converged to some specific AIC rankings. aggregated diffusion in limit, models continuous these results indicated that there was volatility stochastic little incentive to look beyond a simple stochastic model which allowed for volatility
241
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
leverage Heston (1993). These findings effect such as a and are in full agreement clustering based (2002), Christoffersen their analysis upon evaluating the in and outal. which et with determine best MSE to the prices on option model specifications. ofsample
Christoffersen et
al. also pointed out that more might be gained from changing the specification of other fundamental building blocks of the stochastic models, such as jump.
Additional
statistical
for in loss functions investors to select the the symmetry/asymmetry of allowed which criteria, best volatility forecasting model, were also used to determine the best model, but results were mixed.
Given the conflicting
ranking results from different statistical loss functions, we
proceeded to use an economic criterion  trading nearestthemoney straddles, to measure the outofsample
performance of the EGARCH
and GARCH
models.
We evaluated the
forecasting performance of different volatility forecasting models by assessing whether profits from trading weekly nearestthemoney straddles on S&P 500 futures with be generated can based forecasts to times maturity on outofsample of volatility changes. remaining shortest As exante volatility predictions a priori did not take account of future unexpected events, it be difference in there that the outofsample performance of would much anticipated was not different volatility predictors. We found EGARCH and GARCH were able to make exante four highest In EGARCH had in twoyear the outofsample of periods. addition, one profits in best in S&P trade therefore the the subperiods all and per value economic returns rate of 500 futures options market. predictable
GARCHtype and
Our findings reinforced the idea that volatility
changes were
models might be able to make adjustments for market
imperfections that could not be explained by the BlackScholes formula.
Finally, our trading
deltaneutral that the trades to create riskfree revealed also presumption of using experiments in large large index the practical event changes needed not of and was movements, portfolios dates be at critical predicted when there were potential profit available. to correctly
Many
(e. demonstrated deltaneutral have the trading strategies g. problems associated with studies 1985; Leland, Figlewski et al., 1994). In general, the longer it takes to 1980; Boyle et al., But deltaneutral fluctuations. has trade, the it to more exposure volatility a reverse dynamically hedging and rebalancing the position once a day until expiration would be so prohibitively 1989b).
it is impractical even for an option market marker (see Figlewisk, that expensive
Rebalancing less frequently can reduce costs, but risk increases. Therefore, we
derivatives instrument would be needed to allow traders that a new the verdict reached speculate on volatility.
242
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
6.4.3 Timeseries and Optionsbased Variance Forecasting Models The skewness premiums analysis conducted in Chapter 3 indicated that jumpdiffusion
and
leverage models were best for capturing observed termstructure biases in the S&P 500 In addition, results in Chapter 4 suggested that EGARCH
market.
and GARCH
were
behaviour 500 S&P timeseries the to of market whilst a continuous model model adequate for ideal (1993) Heston option pricing. was such as
Furthermore,
volatility
deltaneutrality that the presumption showed of was unrealistic. experiments these findings,
Chapter 5 evaluated the volatility
trading
Motivated by
forecasting performance of different
S&P timeseries the and optionsbased variance of swap valuation models on specifications 500 index.
Our research was designed to include the results from the best models shown in
Chapters 3 and 4 including:
1) EGARCH;
volatility
model; 3) jumpdiffusion
model.
Based on our limited
2) GARCH variance swap model; 3) stochastic
model; 5) local volatility model; 6) ad hoc BlackScholes sample, we showed that the Demterfi
et al. framework
forecasting MSE. have In future that timeseries and variance models a smaller overpredicts had local least the the the model, which was specification, volatility parsimonious particular, best insample fitting performance but worst variance forecasting performance. We illustrated that the use of more flexible and sophisticated option pricing models within the context of the Demeterfi et al. framework might not be able to improve the performance of variance swap pricing.
These findings had brought two major questions to our attention.
First, since
future by huge overpriced models variance a margin, we asked whether option optionsbased prices were consistent with timeseries properties?
Maximum likelihood estimation of the
distributional that the confirmed process squareroot
dynamics implied by option prices not
index. its The implication finding this underlying of was that academicians with consistent have historical look for integrate to to market and alike would a way practitioners and information in a composite option pricing model. The second question was why did local volatility
in forecasting insample future its poorly so excellent variance given perform
One inconsistent flexible but that theoretically explanation model was a performance? pricing fit but had much less predictive power for predicting future insample dominate might implied by insample that a misspecified results model achieves good variance, which data. As Bakshi et al. (2002) pointed out, the poor performance of onethe options overfitting factor models, such as the local volatility
model, could also be a result of the monotonicity
how imposed that correlation property perfect option on and a stringent constraint property
243
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
Therefore be the taken asset extreme caution with underlying price. must change could prices for forecasting. the volatility options when using
6.4.4 Final Comment Finally, a note has to be made in regard to the use of S&P 500 data in this dissertation. Although the S&P 500 market data were employed throughout our analysis, we must stress that our findings are not likely
to be market specific.
It is important for investors to
because 500 S&P 500 S&P liquid the the market products are most one of understand financial However, in dissertation in this the world. we expect obtained can results contracts be generalised to other markets as well.
6.5 Recommendations for Further Research This dissertation has provided new insights into modelling volatility but also raised many new following for The areas are recommended additional research. questions. Firstly, there is an urgent need to establish a consensus on whether option prices, which are forwardlooking,
should be used for forecasting purposes. Recent studies such as Dumas et
Gemmill (1998) and at.
et al. (1999) cast doubt on the usefulness of option prices as
forecasting tools. As the latter points out, options only react to crucial events but they do not predict them.
According to Flamouris (2001), the criterion for the goodness of a implied
distribution more often was the fit it provides to the observed option prices and less frequently its ability to forecast the statistical properties of future data. Given the fact that the main is the noarbitrage pricing of exotic and using optionsbased methodologies of advantage lines further first the of research should consider work along vanilla products, perhaps hedging performance of optionsbased variance swap models. Secondly, one might want to repeat our variance forecasting experiments using a larger infer This to to will allow one a statistically conclude whether significant result set. sample In in forecasting addition, outperform optionbased models models volatility. timeseries
244
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
forecast periods should be nonoverlapping so the sample does not consist of dependent observations84. Thirdly, it would be of interest to extend the work on the overreaction hypothesis in Chapter 3. Since we restricted our investigation to testing the rational expectation hypothesis of Stein (1989) using aggregated data, further analysis can be performed on daily data using fixed maturity
series to check whether option
prices are really
consistent with
the AR(1)
specification. Fourthly, an interesting extension of our work on model rankings in Chapter 4 would be to compare the optionsbased results using alternative economic criteria. (1995) presented the idea of transforming
volatility
For instance, Lopez
forecasts into probability
forecasts.
However, one should be cautious when using these metrics as it is not clear whose utility function they reflect (Orakcioglu, 2000). Alternative trading approaches are also possible, but with certain caveats.
For example, although it does not seem practicable, one can trade
data basis. One daily the the window when can also experiment with size of options on forecasting horizon the to the remaining the structural parameters, or match estimating All lead different these the to straddle. amendments may results. of maturity
Fifthly, special attention should be given to the incorporation of jumps into the delivery prices dominant Currently, framework. theoretically the within a swap consistent of variance Demeterfi et al. framework requires price continuity and a consistent stochastic volatility As longer for to price variance swaps. options one would expect more maturity gets model jumps to occur. Since the evidence in Chapter 5 shows that the replication of logcontracts is forecast future variance, another possible to traded options an not effective way through be MonteCarlo logcontracts. to conduct simulations on would extension Next, a reasonable improvement in estimating structural parameters of stochastic volatility be 5 Chapter by in techniques such as can achieved using more advanced econometric models by Scott (1987) and Wiggins procedures employed matching moment
(1987).
Unlike
likelihood in Bakshi (1997) estimation method used et al. and this study, moment maximum do distribution. assume not a priori procedures a matching
Thus it can offer an alternative
84We thank Roy Batchelor for pointing out the problems associatedwith Chapter 5 of this dissertation.
245
Chapter 6: Summary, Discussions and Recommendationsfor Future Research
view of finding out whether option prices are consistent with the timeseries properties of the underlying asset. Finally, and most importantly, we must change our view that volatility is not a tradable asset. For example, the MONEP created the VX1 and VX6 indexes in October 1997; on January 19, 1998, the Deutsche Terminborse (DTB) became the first exchange in the world to list volatility
futures based on an underlying equity index of implied volatility
the VOLAX
when it launched
futures. Recent advances in financial engineering have also developed a number
(see Howison to trade contracts volatility et al., 2001). Yet it is difficult to conduct ways of research on these volatility
contracts because their existence is largely at the development
liquid is Therefore, investigation to test there the market any potential no models. and stage of modelling more complex volatility products such as options must be postponed until more OTC data are available.
246
EPILOGUE
EPILOGUE Volatility
is a timeliness subject. It is one of the core concepts of financial theory, especially
in modem portfolio theory, risk management and option pricing.
The past two decades have
witnessed an explosion of volatility models, both in option pricing and forecasting, in order to take account of the imperfections displayed in options market and timeseries. As part of this research, I have implemented, applied and scrutinised many volatility models from a practical perspective.
I believe that recent studies have attached too much weight to theory and
financial research is frequently devoid of financial logic and argument. In this dissertation, I have tried to strike a balance between practicalities and technicalities whilst not scarifying any academic vigour.
I believe that more immediate question does not lie in the realm of more
in but in their terms of forecasting, trading checking out market performance models complex and pricing.
Finally, I stress that analytical skills are as important as mathematical skills, and
is finance as much an art as a science. studying
247
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266
Appendices
APPENDICES
A. 1 APARCH Models y, ° ao + at Y, + E, _t E, = s, e, e,  N(0,1) aa>0,6Z. 0,
> 0,i =1,...,p, i=1,..., <1, p, wl
)e+ cr,(Is, E, _ýI'Y, , wt
J. t
ßs8 J ,
>
Model 2: ARCH 6=2
Y, =0"ß, =0 , ar£,
s,: ' ao + r. r
,
Model 3: GARCH 62, Yr =0 s,2 = aa ++s,:, ý,
J. t
Model 4: Taylor & Schwert's GARCH I
6=1,Y, =0 1+±, 6,
s, =aa +±a,
s,,
r"t J=t Model 5: Taylor & Schwert's GARCH  II 61
±cr,
s, =ao + , 1
(je,, IYre,)+± , j. t
ßisrJ
Model b: GJR
62 32 =aa +a,
r. t
(lE,q
Y,E, )ý +±ß, _, J. t
stj
Model 7. TARCH
6=1, ß, =0 sr =aa +± 01
a, (IE, a
IY,E,, )
267
Appendices
A. 2 InSample Model Selection Criteria
Ir.:
QI
MSE=l
T
Mean Square
Q,
Error
., "2
IT MAE=EjQdQ; T _,
j
Mean  Absolute Error
w2
MAPE=1rYIara; T .,
I
Mean  Absolute Percent Error
Qi
1O"7ý
ý2!
21+
MME(U)=7, ý.ý
a,  a,
ýo Q 11 ýsý
Mean Mixed
Error (under prediction)
(Brailsford and Faff, 1996) U
O
wl
Q; 1+
wMME(O)=1T.
ýara'; 0=,
ý_,
'Al
11
Mean  Mixed Error (over prediction) (Brailsford and Faff, 1996)
[In(a,"222
1T
)J
LL)ln(Q, T ,.,
Logarithmic
Loss
(Paganand Schwert, 1990) 2
1'' Q HMSE=ý 1 T ,.,
Heteroskedasticity  adjusted MSE (Bollerslev and Ghysels,1994)
1Tw2
GMLE=ý
T '``
ai22
ln(Q, )+
Guassian quasi  MLE Qt at
(Bollerslevet al.,1994)
268
Appendices
B. 1 June 15,2001 Call Options
Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield
Dis.
Open
Div.
Interest
0.6479
1
1214.35
995
227.4
225.4
226.4
0.4356
07/21/01 0.0959
(%) 3.5200
1214.35
1005
217.5
215.5
216.5
0.4198
07/21/01 0.0959
3.5200
0.6479
1
1214.35
1075
150
148
149
0.3361
07/21/01 0.0959
3.5200
0.6479
22
1214.35
1100
126.5
124.5
125.5
0.3084
07/21/01 0.0959
3.5200
0.6479
277
1214.35
1125
103.7
101.7
102.7
0.2833
1214.35
1150
82
80
81
0.2611
3.5200 3.5200
0.6479 0.6479
209 257
1214.35
1175
62.2
60.2
61.2
0.2439
07/21/01 0.0959 07/21/01 0.0959 0721/01 0.0959
3.5200
0.6479
29
1214.35
1200
44.2
42.2
43.2
0.2257
07/21/01 0.0959
3.5200
0.6479
1882
1214.35
1225
29.3
27.3
28.3
0.2112
07/21/01 0.0959
3.5200
0.6479
2443
1214.35
1250
17.3
16.3
16.8
0.1986
0721/01
0.0959
3.5200
0.6479
9026
1214.35
9.9
8.9
9.4
0.1927
07/21/01 0.0959
3.5200
0.6479
11371
1214.35
1275 1280
8.4
7.4
7.9
0.1881
07/21/01 0.0959
3.5200
0.6479
618
1214.35
1285
7.6
6.6
7.1
0.1890
0721/01
0.0959
3.5200
0.6479
3283
1214.35
1300
5.1
4.4
4.75
0.1874
07/21/01 0.0959
3.5200
0.6479
6717
1214.35
1325
2.35
1.9
2.125
0.1824
07/21/01 0.0959
3.5200
0.6479
4321
1214.35
1350
1
0.9
0.95
0.1819
07/21/01 0.0959
3.5200
0.6479
10642
1214.35
1375
0.7
0.4
0.55
0.1905
07/21/01 0.0959
3.5200
0.6479
4452
1214.35
1025
202.3
200.3
201.3
0.3217
08/18/01 0.1726
3.5200
0.8969
10
1214.35
1175
73.3
71.3
72.3
0.2331
08/18/01 0.1726
3.5200
0.8969
781
1214.35
1200
56.3
54.3
55.3
0.2216
08/18/01 0.1726
3.5200
0.8969
953
1214.35
1250
29.3
27.3
28.3
0.2024
08118/01 0.1726
3.5200
0.8969
3831
1214.35 1214.35
1275
19.8
18.3
19.05
0.1965
08/18/01 0.1726
3.5200
0.8969
2083
1300
12.8
11.3
12.05
0.1905
08/18/01 0.1726
3.5200
0.8969
2993
1214.35
1325
7.5
6.8
7.15
0.1851
08/18/01 0.1726
3.5200
0.8969
614
1214.35
1350
4.5
3.8
4.15
0.1824
08/18/01 0.1726
3.5200
0.8969
1039
1214.35
1375
2.7
2
2.35
0.1810
08/18/01 0.1726
3.5200
0.8969
2534
1214.35
1400
1.5
1.05
1.275
0.1799
08/18/01 0.1726
3.5200
0.8969
1719
1214.35
1425
0.9
0.45
0.675
0.1794
08/18/01 0.1726
3.5200
0.8969
164
1214.35
800
424.2
422.2
423.2
0.4289
0922/01
0.2685
3.5222
0.9658
4715
1214.35
1050
186
184
185
0.2777
09/22/01 0.2685
3.5222
0.9658
267
1214.35
1100
142.5
1405
141.5
0,2548
0922101 0.2685
3.5222
2158
1214.35
1125
122.2
120.2
121.2
0.2454
09/22/01 0.2685
3.5222
0.9658 0.9658
1214.35
1150
102.7
100.7
101.7
0.2350
0922/01
0.2685
3.5222
0.9658
4169
1214.35
1200
68.7
66.7
67.7
0.2196
0922101 0.2685
3.5222
0.9658
9200
1214.35
1225
54.1
52.1
53.1
0.2119
0922/01
0.2685
3.5222
0.9658
3077
1214.35
1240
46.7
44.7
45.7
0.2095
0922/01
0.2685
3.5222
0.9658
434
1214.35
1250
41.5
39.5
40.5
0.2053
09/22/01 0.2685
3.5222
0.9658
17406
1214.35
1260
37.1
35.1
36.1
0.2031
09/22/01 0.2685
3.5222
0.9658
407
1214.35
1275
31.1
29.1
30.1
0.1999
09/22/01 0.2685
3.5222
0.9658
6423
1214.35
27.3 22.2
25.3 20.2
26.3
0.2685
3.5222
0922/01
0.2685
3.5222
0.9658 0.9658
1104
21.2
0.1970 0.1928
0922/01
1214.35
1285 1300
8441
1214.35
1325
15.5
14
14.75
0.1887
0922/01
0.2685
3.5222
0.9658
5339
1214.35
1350
10.5
10
0.1857
09/22/01 0.2685
3.5222
0.9658
11196
1214.35
1375
0.9658
10587
0.9658
7051
3.5222
0.9658
4733
1214.35
1450
1.7
1.25
1.475
0.1792 0.1777 0.1753
3.5222
2.85
3.7 2.2
0922101 0.2685 0922/01 0.2685 0922101 0.2685
3.5222
1400 1425
6.25 4.05 2.525
0.1804
1214.35 1214.35
6.5 4.4
9.5 6
3.5222
0.9658
5744
1214.35
1475
1.05
0.6
0.825
0.1729
0922101 0.2685 09/22/01 0.2685
3.5222
0.9658
4431
1214.35
1500
0.9
0.45
0.675
0.1806
0922/01
3.5222
0.9658
2766
269
0.2685
167
Appendices Bid
SX
Mid
Ask
BS Imp. Vol.
Exp.
Maturity
Yield M
Dis. Div.
Open Interest
1214.35
1525
0.65
0.2
0.425
0.1816
09/22/01 0.2685
3.5222
0.9658
3482
1214.35
800
430.7
428.7
429.7
0.3021
12/22/01 0.5178
3.5493
1.0005
7
1214.35
900
336.3
334.3
335.3
0.2849
12/22/01 0.5178
3.5493
1.0005
1663
1214.35
950
290.6
288.6
289.6
0.2748
12/22/01 0.5178
3.5493
1.0005
2
1214.35
995
250.7
248.7
249.7
0.2652
12/22/01 0.5178
3.5493
1.0005
1680
1214.35
1025
224.4
222.4
223.4
0.2558
12/22/01 0.5178
3.5493
1.0005
1462
1214.35
1050
203.6
201.6
202.6
0.2509
12/22/01 0.5178
3.5493
1.0005
252
1214.35
1100
163.5
161.5
162.5
0.2388
12/22/01 0.5178
3.5493
1.0005
1264
1214.35 1214.35
1150
124.8
125.8
0.2275
12/22/01 0.5178
3.5493
1.0005
4531
107.8 92.3
108.8 93.3
0.2216 0.2169
3.5493 3.5493
1225
79.5
77.5
78.5
0.2108
12/22/01 0.5178 12/22/01 0.5178 12/22/01 0.5178
3.5493
1.0005 1.0005 1.0005
1328
1214.35 1214.35
1175 1200
126.8 109.8 94.3
7006 4005
1214.35
1250
66.7
64.7
65.7
0.2069
12/22/01 0.5178
3.5493
1.0005
7606
1214.35
1275
54.9
52.9
53.9
0.2022
12/22/01 0.5178
3.5493
1.0005
3996
1214.35
1300
44.1
42.1
43.1
0.1965
12/22/01 0.5178
3.5493
1.0005
15712
1214.35
1325
35.2
33.2
34.2
0.1924
12/22/01 0.5178
3.5493
1.0005
8315
1214.35
1350
27.7
25.7
26.7
0.1887
12/22/01 0.5178
3.5493
1.0005
6407
1214.35
1375
21.3
19.8
20.55
0.1855
12/22/01 0.5178
3.5493
1.0005
1534
1214.35
1400
14.5
15.25
0.1813
12/22/01 0.5178
3.5493
1.0005
12452
1214.35
1425
16 12
10.5
11.25
0.1784
12/22/01 0.5178
3.5493
1.0005
3918
1214.35
1450
8.8
7.8
8.3
0.1766
12/22/01 0.5178
3.5493
1.0005
9054
1214.35
1475
6.5
5.5
6
0.1747
12/22/01 0.5178
3.5493
1.0005
83
1214.35
1500
4.7
4
4.35
0.1736
12/22/01 0.5178
3.5493
1.0005
12551
1214.35
1525
3.4
2.7
3.05
0.1720
12/22/01 0.5178
3.5493
1.0005
777
1214.35
1550
2.35
1.9
2.125
0.1707
12/22/01 0.5178
3.5493
1.0005
5363
1214.35
1575
1.7
1.25
1.475
0.1698
12/22/01 0.5178
3.5493
1.0005
145
1214.35
1600
1.4
0.95
1.175
0.1728
12/22/01 0.5178
3.5493
1214.35
1650
0.8
0.35
0.575
0.1723
12/22/01 0.5178
3.5493
1.0005 1.0005
10306 4125
1214.35
1675
0.6
0.15
0.375
0.1709
12/22/01 0.5178
3.5493
1.0005
525
1214.35
1025
238.5
236.5
237.5
0.2446
03/16/02 0.7479
3.5401
1.0018
28
1214.35
1050
218.5
216.5
217.5
0.2406
03/16/02 0.7479
3.5401
1.0018
1433
1214.35
1100
180.1
178.1
179.1
0.2314
03/16/02 0.7479
3.5401
1.0018
191
1214.35
1125
162.4
160.4
161.4
0.2281
03/16/02 0.7479
3.5401
1.0018
289
1214.35
1150
145.2
143.2
144.2
0.2238
03/16/02 0.7479
3.5401
1.0018
748
1214.35
128.8
126.8
127.8
0.2194
03/16/02 0.7479
3.5401
1.0018
19
1214.35
1175 1200
113.2
111.2
112.2
0.2146
03/16/02 0.7479
3.5401
1.0018
2384
1214.35
1225
98.6
96.6
97.6
0.2100
03/16/02 0.7479
3.5401
1.0018
238
1214.35
1250
85.4
83.4
84.4
0.2062
03/16/02 0.7479
3.5401
1.0018
622
1214.35
1275
73.3
71.3
72.3
0.2027
03/16/02 0.7479
3.5401
1.0018
3160
1214.35
1300
62.2
60.2
61.2
0.1989
03/16/02 0.7479
3.5401
1.0018
4015
1214.35
1325
52.2
50.2
51.2
0.1952
03/16/02 0.7479
3.5401
1.0018
9
1214.35
1350
43.6
41.6
42.6
0.1923
3.5401
1.0018
1214.35
1375
35.7
33.7
34.7
0.1884
03/16/02 0.7479 03/16/02 0.7479
3.5401
1.0018
1001 441
1214.35
1400
26.8
1.0018
7239
21.2
0.1846 0.1817
3.5401
1425
27.8 22.2
03/16/02 0.7479
1214.35
28.8 23.2
03/16/02 0.7479
3.5401
1.0018
297
1214.35
1450
18.2
16.7
17.45
0.1788
3.5401
1.0018
1703
1214.35
1475
14.2
12.7
13.45
0.1757
03/16/02 0.7479 03/16/02 0.7479
3.5401
1.0018
16
1214.35
1500
11
10
0.1741
03/16/02 0.7479
3.5401
1.0018
869
1214.35
1600
3.7
0.1670
1.0018
365
1050
237.3
235.8
0.2403
3.5379
1.0019
1
1214.35
1100
200.7
197.7
199.2
0.2331
03/16/02 0.7479 06/22/02 1.0164 06/22/02 1.0164
3.5401
1214.35
3 234.3
10.5 3.35
3.5379
1.0019
102
1214.35
1150
166.9
163.9
165.4
0.2262
3.5379
1.0019
1350
1214.35
1200
136
133
134.5
2788
1250
108.2
105.2
106.7
3.5379 3.5379
1.0019
1214.35
0.2192 0.2119
1.0019
1577
270
06/22/02 1.0164 06/22/02 1.0164 06/22/02 1.0164
Appendices Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield
Dis. Div.
Open Interest
1214.35
1300
84.2
81.2
82.7
0.2055
06/22/02 1.0164
(%) 3.5379
1214.35
1350
64.1
61.1
62.6
0.1999
06/22/02 1.0164
3.5379
1.0019
4656
1214.35
1400
47.4
44.4
45.9
0.1942
3.5379
1.0019
6940
1214.35
1450
33.9
30.9
32.4
0.1883
06/22102 1.0164 06/22/02 1.0164
3.5379
1.0019
6897
1214.35
1500
23.1
20.1
21.6
0.1814
06/22/02 1.0164
3.5379
1.0019
5974
1214.35
1550
14.7
13.2
13.95
0.1758
06/22/02 1.0164
3.5379
1.0019
851
1214.35
1600
9.4
8.4
8.9
0.1719
06/22/02 1.0164
3.5379
1.0019
6076
1214.35
1650
6.1
5.1
5.6
0.1692
06/22/02 1.0164
3.5379
1.0019
2849
1214.35
1700 1750
3.7
3
3.35
0.1660
06/22/02 1.0164
3.5379
1.0019
5639
2.2
0.1636 0.1612
1120
0.1591
06/22/02 1.0164 06/22/02 1.0164 06/22/02 1.0164
1.0019
1.35 0.85
1.975 1.125 0.625
3.5379
1800 1850
1.75 0.9 0.4
3.5379 3.5379
1.0019 1.0019
4132 1150
0.6
0.15
0.375
0.1589
06/22/02 1.0164
3.5379
1.0019
3644
1214.35
1900 1100
233.5
230.5
232
0.2290
12/21/02 1.5151
3.7772
1.0019
64
1214.35
1150
201.1
198.1
199.6
0.2238
12/21/02 1.5151
3.7772
1.0019
1785
1214.35
1200
171
168
169.5
0.2184
12/21/02 1.5151
3.7772
1.0019
4897
1214.35
1250
143.2
140.2
141.7
0.2126
12/21/02 1.5151
3.7772
1.0019
4127
1214.35
118.4
115.4
116.9
0.2073
12/21/02 1.5151
3.7772
1.0019
3116
1214.35
1300 1350
96.3
93.3
94.8
0.2022
12/21/02 1.5151
3.7772
1367
1214.35
1400
77.5
74.5
76
0.1979
12/21/02 1.5151
3.7772
1.0019 1.0019
3569
1214.35
1450
60.6
57.6
59.1
0.1923
12/21/02 1.5151
3.7772
1.0019
3783
1214.35
1500
46.4
43.4
44.9
0.1868
12/21/02 1.5151
3.7772
1.0019
3740
1214.35
1550
35.1
32.1
33.6
0.1823
12/21/02 1.5151
3.7772
1.0019
1667
1214.35
1600
26.3
23.3
24.8
0.1786
12/21/02 1.5151
3.7772
1.0019
4587
1214.35
1650
18.6
17.1
17.85
0.1748
12/21/02 1.5151
3.7772
1.0019
1060
1214.35
1700
13.2
11.7
0.1709
12/21/02 1.5151
3.7772
1.0019
2046
1214.35
1800
6.4
5.4
12.45 5.9
0.1653
12/21/02 1.5151
3.7772
1.0019
1200
1214.35
1900
3.1
2.4
2.75
0.1621
12/21/02 1.5151
3.7772
1.0019
6500
1214.35 1214.35 1214.35 1214.35
271
1.0019
4395
Appendices
B.2
Call Options July20,2001
Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1210.85
1050
164
162
163
0.2144
08/18/01 0.0767
3.5300
0.8520
362
1210.85
1075
139.9
137.9
138.9
0.2388
08/18/01 0.0767
3.5300
0.8520
3
1210.85
1100
116.1
114.1
115.1
0.2341
08/18/01 0.0767
3.5300
0.8520
660
1210.85
1125
93.2
91.2
92.2
0.2280
08/18/01 0.0767
3.5300
0.8520
326
1210.85
1150
71.4
69.4
70.4
0.2177
08/18/01 0.0767
3.5300
0.8520
1237
1210.85
1175
51.7
49.7
50.7
0.2090
08/18/01 0.0767
3.5300
0.8520
3451
1210.85
1200
34.3
32.3
33.3
0.1967
08/18/01 0.0767
3.5300
0.8520
4437
1210.85
1225
21
19.5
20.25
0.1911
08/18/01 0.0767
3.5300
0.8520
14228
1210.85
1250
11
10
10.5
0.1812
08/18/01 0.0767
3.5300
0.8520
10578
1210.85
1275
5.4
4.7
5.05
0.1782
08/18/01 0.0767
3.5300
0.8520
15186
1210.85
1300
2.7
2
2.35
0.1797
08/18/01 0.0767
3.5300
0.8520
15988
1210.85
1325
1
0.7
0.85
0.1756
08/18/01 0.0767
3.5300
0.8520
4888
1210.85
1050
170.3
168.3
169.3
0.2357
09/22/01 0.1726
3.5300
1.0870
279
1210.85
1100
124.9
122.9
123.9
0.2233
09/22/01 0.1726
3.5300
1.0870
2150
1210.85
1125
103.6
101.6
102.6
0.2166
09/22/01 0.1726
3.5300
1.0870
644
1210.85
1150
83.8
81.8
82.8
0.2108
09/22/01 0.1726
3.5300
1.0870
4212
1210.85
1190
55.4
53.4
54.4
0.1990
09/22/01 0.1726
3.5300
1.0870
483
1210.85
1200
49.6
47.6
48.6
0.1984
09/22/01 0.1726
3.5300
1.0870
14007
1210.85
1210
43.6
41.6
42.6
0.1948
09/22/01 0.1726
3.5300
1.0870
7358
1210.85
1225
35.5
33.5
34.5
0.1903
09/22/01 0.1726
3.5300
1.0870
27100
1210.85
1240
28.5
26.5
27.5
0.1867
09/22/01 0.1726
3.5300
1.0870
4860
1210.85
1250
24.7
22.7
23.7
0.1861
0922/01
0.1726
3.5300
1.0870
21295
1210.85
1275
15.8
14.3
15.05
0.1798
09/22/01 0.1726
3.5300
1.0870
6893
1210.85
1285
13.3
11.8
12.55
0.1789
09/22/01 0.1726
3.5300
1.0870
1202
1210.85
1300
9.6
8.6
9.1
0.1755
09/22/01 0.1726
3.5300
1.0870
10994
1210.85
1325
5.6
4.9
5.25
0.1726
09/22/01 0.1726
3.5300
1.0870
10914
1210.85
1350
3.3
2.6
2.95
0.1715
09/22/01 0.1726
3.5300
1.0870
12910
1210.85
1375
1.8
1.35
1.575
0.1705
09/22/01 0.1726
3.5300
1.0870
9773
1210.85
1400
1.2
0.75
0.975
0.1751
09/22/01 0.1726
3.5300
1.0870
6684
1210.85
1425
0.7
0.25
0.475
0.1735
09/22/01 0.1726
3.5300
1.0870
4933
1210.85
1450
0.45
0.3
0.375
0.1838
09/22/01 0.1726
3.5300
1.0870
5829
1210.85
1025
208.8
206.8
207.8
0.2219
12/22/01 0.4219
3.5300
1.2069
1462
1210.85
1050
187.3
185.3
186.3
0.2204
12/22/01 0.4219
3.5300
1.2069
700
1210.85
1060
178.8
176.8
177.8
0.2190
12/22/01 0.4219
3.5300
1.2069
450
1210.85
1100
145.7
143.7
144.7
0.2111
12/22/01 0.4219
3.5300
1.2069
1708
1210.85
1150
108.6
106.6
107.6
0.2041
12/22/01 0.4219
3.5300
1.2069
4545
1210.85
1175
91.8
89.8
90.8
0.2001
12/22/01 0.4219
3.5300
1.2069
1411
1210.85
1200
76.4
74.4
75.4
0.1961
12/22/01 0.4219
3.5300
1.2069
7525
1210.85
1250
49.6
47.6
48.6
0.1863
12/22/01 0.4219
3.5300
1.2069
8715
1210.85
1275
38.2
36.2
37.2
0.1801
12/22/01 0.4219
3.5300
1.2069
4858
1210.85
1300
29.4
27.4
28.4
0.1770
12/22/01 0.4219
3.5300
1.2069
16318
1210.85
1325
21.5
20
20.75
0.1723
12/22/01 0.4219
3.5300
1.2069
8751
1210.85
1350
15.9
14.4
15.15
0.1700
12/22/01 0.4219
3.5300
1.2069
6892
1210.85
1375
11.6
10.1
10.85
0.1681
12/22/01 0.4219
3.5300
1.2069
2822
1210.85
1400
8.2
7.2
7.7
0.1670
12/22/01 0.4219
3.5300
1.2069
14456
1210.85
1425
5.3
4.6
4.95
0.1628
12/22/01 0.4219
3.5300
1.2069
4269
272
Appendices SX
Bid
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield
Dis. Div.
Open Interest
1210.85
1450
3.6
2.9
3.25
0.1609
12/22/01 0.4219
(%) 3.5300
1210.85
1475
2.35
1.9
2.125
0.1598
12/22/01 0.4219
3.5300
1.2069
112
1210.85
1500
1.75
1.3
1.525
0.1618
12/22/01 0.4219
3.5300
1.2069
13476
1210.85
1525
1.15
0.7
0.925
0.1597
12/22/01 0.4219
3.5300
1.2069
785
1210.85
1550
0.85
0.4
0.625
0.1606
12/22/01 0.4219
3.5300
1.2069
5218
1210.85
1575
0.6
0.15
0.375
0.1593
12/22/01 0.4219
3.5300
1.2069
155
1210.85
1025
221.6
219.6
220.6
0.2103
03/16/02 0.6521
3.5482
1.2115
28
1210.85
1050
201
199
200
0.2089
03/16/02 0.6521
3.5482
1.2115
1433
1210.85
1100
162.1
160.1
161.1
0.2050
03/16/02 0.6521
3.5482
1.2115
202
1210.85
1125
143.6
141.6
142.6
0.2016
03/16/02 0.6521
3.5482
1.2115
339
1210.85
1150
126.3
124.3
125.3
0.1988
03/16/02 0.6521
3.5482
1.2115
750
1210.85
1175
110
108
109
0.1957
03/16/02 0.6521
3.5482
1.2115
19
1210.85
1200
94.8
92.8
93.8
0.1925
03/16/02 0.6521
3.5482
1.2115
2982
1210.85
1225
80.2
78.2
79.2
0.1877
03/16/02 0.6521
3.5482
1.2115
757
1210.85
1250
67.6
65.6
66.6
0.1849
03/16/02 0.6521
3.5482
1.2115
1433
1210.85
1275
56.2
54.2
55.2
0.1819
03/16/02 0.6521
3.5482
1.2115
3782
1210.85
1300
46
44
45
0.1786
03/16/02 0.6521
3.5482
1.2115
3889
1210.85
1325
37.1
35.1
36.1
0.1753
03/16/02 0.6521
3.5482
1.2115
340
1210.85
1350
29.4
27.4
28.4
0.1719
03/16/02 0.6521
3.5482
1.2115
973
1210.85
1375
23.2
21.2
22.2
0.1695
03/16/02 0.6521
3.5482
1.2115
444
1210.85
1400
17.7
16.2
16.95
0.1667
03/16/02 0.6521
3.5482
1.2115
7715
1210.85
1425
13.5
12
12.75
0.1643
03/16/02 0.6521
3.5482
1.2115
294
1210.85
1450
10.1
9.1
9.6
0.1628
03/16/02 0.6521
3.5482
1.2115
1965
1210.85
1475
7.4
6.4
6.9
0.1601
03/16/02 0.6521
3.5482
1.2115
84
1210.85
1500
5
4.7
4.85
0.1576
03/16/02 0.6521
3.5482
1.2115
788
1210.85
1600
1.5
1.05
1.275
0.1551
03/16/02 0.6521
3.5482
1.2115
464
1210.85
1325
54.7
52.7
53.7
0.1786
06/22/02 0.9205
3.5805
1.2117
1010
1210.85
1425
25.8
23.8
24.8
0.1690
06/22/02 0.9205
3.5805
1.2117
1
1210.85
1100
212.9
209.9
211.4
0.2031
12/21/02 1.4192
3.7535
1.2117
64
1210.85
1150
180.4
177.4
178.9
0.2002
12/21/02 1.4192
3.7535
1.2117
1980
1210.85
1200
150.4
147.4
148.9
0.1963
12/21/02 1.4192
3.7535
1.2117
5218
1210.85
1250
123.1
120.1
121.6
0.1918
12/21/02 1.4192
3.7535
1.2117
4484
1210.85
1300
98
95
96.5
0.1857
12/21/02 1.4192
3.7535
1.2117
3846
1210.85
1350
77.1
74.1
75.6
0.1815
12/21/02 1.4192
3.7535
1.2117
1867
1210.85
1400
59.1
56.1
57.6
0.1767
12/21/02 1.4192
3.7535
1.2117
4417
1210.85
1450
44.1
41.1
42.6
0.1718
12/21/02 1.4192
3.7535
1.2117
4930
1210.85
1500
32.9
29.9
31.4
0.1689
12/21/02 1.4192
3.7535
1.2117
3740
1210.85
1550
24.3
21.3
22.8
0.1666
12/21/02 1.4192
3.7535
1.2117
1717
1210.85
1600
16.3
14.8
15.55
0.1623
12/21/02 1.4192
3.7535
1.2117
6711
1210.85
1650
10.9
9.9
10.4
0.1588
12/21/02 1.4192
3.7535
1.2117
1065
1210.85
1700
7.2
6.2
6.7
0.1553
12/21/02 1.4192
3.7535
1.2117
2071
1210.85
1800
3.5
2.8
3.15
0.1546
12/21/02 1.4192
3.7535
1.2117
12350
1210.85
1900
1.55
1.1
1.325
0.1525
12/21/02 1.4192
3.7535
1.2117
6540
273
1.2069
9344
Appendices
B. 3 August 17,2001 Call Options
Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1161.95
800
365.6
363.6
364.6
0.5364
09/22/01 0.0959
3.4525
0.6392
5774
1161.95
1025
145.6
143.6
144.6
0.3187
09/22/01 0.0959
3.4525
0.6392
15
1161.95
1050
121.8
119.8
120.8
0.2897
09/22/01 0.0959
3.4525
0.6392
284
1161.95
1100
78.2
76.2
77.2
0.2563
09/22/01 0.0959
3.4525
0.6392
2369
1161.95
1125
58.4
56.4
57.4
0.2389
09/22/01 0.0959
3.4525
0.6392
1633
1161.95
1150
41.1
39.1
40.1
0.2244
09/22/01 0.0959
3.4525
0.6392
5930
1161.95
1190
18.5
17.5
18
0.1980
09122/01 0.0959
3.4525
0.6392
9816
1161.95
1200
15
14
14.5
0.1962
09/22/01 0.0959
3.4525
0.6392
27223
1161.95
1210
12.4
11
11.7
0.1961
09/22/01 0.0959
3.4525
0.6392
10767
1161.95
1225
8
7.3
7.65
0.1896
09/22/01 0.0959
3.4525
0.6392
30525
1161.95
1240
5.1
4.6
4.85
0.1852
09/22/01 0.0959
3.4525
0.6392
5130
1161.95
1250
3.8
3.3
3.55
0.1835
09/22/01 0.0959
3.4525
0.6392
30757
1161.95
1275
1.95
1.5
1.725
0.1851
09/22/01 0.0959
3.4525
0.6392
9960
1161.95
1285
1.4
0.95
1.175
0.1829
09/22/01 0.0959
3.4525
0.6392
1232
1161.95
1300
0.7
0.35
0.525
0.1747
09/22/01 0.0959
3.4525
0.6392
21666
1161.95
900
275.2
273.2
274.2
0.2664
12/22/01 0.3452
3.3486
0.9878
1730
1161.95
950
229.4
227.4
228.4
0.2656
12/22/01 0.3452
3.3486
0.9878
2
1161.95
995
189.1
187.1
188.1
0.2541
12/22/01 0.3452
3.3486
0.9878
1931
1161.95
1025
163.4
161.4
162.4
0.2466
12/22/01 0.3452
3.3486
0.9878
1468
1161.95
1050
142.8
140.8
141.8
0.2400
12/22/01 0.3452
3.3486
0.9878
700
1161.95
1060
134.9
132.9
133.9
0.2378
12/22/01 0.3452
3.3486
0.9878
450
1161.95
1100
103.1
101.1
102.1
0.2204
12/22/01 0.3452
3.3486
0.9878
2257
1161.95
1150
70.2
68.2
69.2
0.2092
12/22/01 0.3452
3.3486
0.9878
4897
1161.95
1175
55.6
53.6
54.6
0.2016
12/22/01 0.3452
3.3486
0.9878
1757
1161.95
1200
43.3
41.3
42.3
0.1964
12/22/01 0.3452
3.3486
0.9878
9020
1161.95
1225
32.7
30.7
31.7
0.1907
12/22/01 0.3452
3.3486
0.9878
7039
1161.95
1250
24
22
23
0.1854
12/22/01 0.3452
3.3486
0.9878
13786
1161.95
1275
17.1
15.6
16.35
0.1816
12/22/01 0.3452
3.3486
0.9878
7011
1161.95
1300
12.2
10.7
11.45
0.1792
12/22/01 0.3452
3.3486
0.9878
17208
1161.95
1325
8.2
7.2
7.7
0.1763
12/22/01 0.3452
3.3486
0.9878
9602
1161.95
1350
5.5
4.8
5.15
0.1748
12/22/01 0.3452
3.3486
0.9878
8906
1161.95
1375
3.6
2.9
3.25
0.1722
12/22/01 0.3452
3.3486
0.9878
2816
1161.95
1400
2.1
1.65
1.875
0.1680
12/22/01 0.3452
3.3486
0.9878
15650
1161.95
1425
1.4
0.95
1.175
0.1677
12/22/01 0.3452
3.3486
0.9878
4464
1161.95
1450
0.95
0.5
0.725
0.1674
12/22/01 0.3452
3.3486
0.9878
9393
1161.95
1475
0.65
0.4
0.525
0.1711
12/22/01 0.3452
3.3486
0.9878
122
1161.95
900
283.4
281.4
282.4
0.2293
03/16/02 0.5753
3.3390
1.0046
18
1161.95
1025
177.4
175.4
176.4
0.2269
03/16/02 0.5753
3.3390
1.0046
28
1161.95
1050
157.7
155.7
156.7
0.2219
03/16/02 0.5753
3.3390
1.0046
1433
1161.95
1100
120.7
118.7
119.7
0.2111
03/16/02 0.5753
3.3390
1.0046
209
1161.95
1125
104.2
102.2
103.2
0.2071
03/16/02 0.5753
3.3390
1.0046
339
1161.95
1150
88.6
86.6
87.6
0.2023
03/16/02 0.5753
3.3390
1.0046
750
1161.95
1175
74.5
72.5
73.5
0.1980
03/16/02 0.5753
3.3390
1.0046
24
1161.95
1200
61.3
59.3
60.3
0.1927
03/16/02 0.5753
3.3390
1.0046
4023
1161.95
1225
49.8
47.8
48.8
0.1883
03/16/02 0.5753
3.3390
1.0046
1394
1161.95
1250
40.1
38.1
39.1
0.1849
03/16/02 0.5753
3.3390
1.0046
3302
274
Appendices Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1161.95
1275
30.8
28.8
29.8
0.1786
03/16/02 0.5753
3.3390
1.0046
3032
1161.95
1300
24.3
22.3
23.3
0.1769
03/16/02 0.5753
3.3390
1.0046
3853
1161.95
1325
18.5
17
17.75
0.1745
03/16/02 0.5753
3.3390
1.0046
1826
1161.95
1350
13.7
12.2
12.95
0.1707
03/16/02 0.5753
3.3390
1.0046
2874
1161.95
1375
9.9
8.9
9.4
0.1682
03/16/02 0.5753
3.3390
1.0046
459
1161.95
1400
7.1
6.1
6.6
0.1653
03/16/02 0.5753
3.3390
1.0046
8375
1161.95
1425
4.9
4.2
4.55
0.1627
03/16102 0.5753
3.3390
1.0046
294
1161.95
1450
3.4
2.7
3.05
0.1602
03/16/02 0.5753
3.3390
1.0046
1951
1161.95
1475
2.3
1.85
2.075
0.1589
03/16/02 0.5753
3.3390
1.0046
84
1161.95
1500
1.7
1.25
1.475
0.1593
03/16/02 0.5753
3.3390
1.0046
869
1161.95
1600
0.45
0.4
0.425
0.1642
03/16/02 0.5753
3.3390
1.0046
921
1161.95
850
337.8
335.8
336.8
0.1964
0622/02
0.8438
3.3713
1.0054
286
1161.95
1325
32.7
30.7
31.7
0.1777
06/22/02 0.8438
3.3713
1.0054
1010
1161.95
1425
12.1
10.6
11.35
0.1640
0622/02
0.8438
3.3713
1.0054
1
1161.95 900
316.4
313.4
314.9
0.2113
1221/02
1.3425
3.4859
1.0054
552
1161.95 950
276.9
273.9
275.4
0.2134
1221/02
1.3425
3.4859
1.0054
1
1161.95
1100
170.1
167.1
168.6
0.2035
12/21/02 1.3425
3.4859
1.0054
64
1161.95
1150
139.5
136.5
138
0.1981
12J21/02 1.3425
3.4859
1.0054
2622
1161.95
1200
112.5
109.5
111
0.1934
12/21/02 1.3425
3.4859
1.0054
5801
1161.95
1225
99.8
96.8
98.3
0.1903
1221/02
1.3425
3.4859
1.0054
251
1161.95
1250
87.9
84.9
86.4
0.1871
1221/02
1.3425
3.4859
1.0054
4534
1161.95
1300
67.7
64.7
66.2
0.1825
12/21/02 1.3425
3.4859
1.0054
4125
1161.95
1350
50.7
47.7
49.2
0.1776
12/21/02 1.3425
3.4859
1.0054
2016
1161.95
1400
36.8
33.8
35.3
0.1725
12/21/02 1.3425
3.4859
1.0054
4418
1161.95
1450
26.2
23.2
24.7
0.1682
1221/02
1.3425
3.4859
1.0054
5431
1161.95
1500
17.5
16
16.75
0.1642
12/21/02 1.3425
3.4859
1.0054
4338
1161.95
1550
11.8
10.3
11.05
0.1607
12/21/02 1.3425
3.4859
1.0054
1577
1161.95
1600
7.5
6.5
7
0.1571
12/21/02 1.3425
3.4859
1.0054
6753
1161.95
1650
4.9
4.2
4.55
0.1557
12/21/02 1.3425
3.4859
1.0054
1070
1161.95
1700
3.2
2.5
2.85
0.1538
12/21/02 1.3425
3.4859
1.0054
4214
1161.95
1800
1.15
0.7
0.925
0.1481
1221/02
1.3425
3.4859
1.0054
12361
1161.95
1900
0.6
0.15
0.375
0.1487
12/21/02 1.3425
3.4859
1.0054
6540
275
Appendices
B. 4 September 21,2001 Call Options
Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
965.8
800
179.2
175.2
177.2
0.6323
1020/01
0.0767
2.1200
0.8605
167
965.8
900
91.6
87.6
89.6
0.4811
10/20/01 0.0767
2.1200
0.8605
177
965.8
975
40.5
36.5
38.5
0.3980
10/20101 0.0767
2.1200
0.8605
75
965.8
995
30.5
27
28.75
0.3832
10/20/01 0.0767
2.1200
0.8605
2448
965.8
1010
24.4
20.4
22.4
0.3714
10/20/01 0.0767
2.1200
0.8605
4085
965.8
1020
20
17
18.5
0.3617
10/20/01 0.0767
2.1200
0.8605
2322
965.8
1025
18
15.3
16.65
0.3562
10/20/01 0.0767
2.1200
0.8605
4395
965.8
1030
16.9
13.9
15.4
0.3563
10/20/01 0.0767
2.1200
0.8605
694
965.8
1040
12.1
10.2
11.15
0.3324
1020/01
0.0767
2.1200
0.8605
761
965.8
1050
10.5
9.1
9.8
0.3391
10/20/01 0.0767
2.1200
0.8605
15188
965.8
1060
9.1
7.1
8.1
0.3382
10/20/01 0.0767
2.1200
0.8605
1174
965.8
1070
7.7
5.7
6.7
0.3381
10/20/01 0.0767
2.1200
0.8605
566
965.8
1075
5.9
5.1
5.5
0.3280
1020/01
0.0767
2.1200
0.8605
2050
965.8
1080
6
4.6
5.3
0.3342
10/20/01 0.0767
2.1200
0.8605
1075
965.8
1090
5
3.6
4.3
0.3337
10120/01 0.0767
2.1200
0.8605
453
965.8
1100
3.8
2.8
3.3
0.3293
10/20/01 0.0767
2.1200
0.8605
8756
965.8
1125
2.15
1.5
1.825
0.3272
10/20/01 0.0767
2.1200
0.8605
5810
965.8
1150
1.45
0.8
1.125
0.3337
10/20/01 0.0767
2.1200
0.8605
8718
965.8
1175
0.9
0.25
0.575
0.3309
10/20/01 0.0767
2.1200
0.8605
7376
965.8
1200
1
0.3
0.65
0.3671
10/20/01 0.0767
2.1200
0.8605
7665
965.8
850
141.3
137.3
139.3
0.4562
11/17/01 0.1534
2.1747
1.3828
17
965.8
900
101.9
97.9
99.9
0.4152
11/17/01 0.1534
2.1747
1.3828
2
965.8
950
67.5
63.5
65.5
0.3772
11/17/01 0.1534
2.1747
1.3828
11
965.8
995
42.6
38.6
40.6
0.3479
11/17/01 0.1534
2.1747
1.3828
784
965.8
1025
28.9
24.9
26.9
0.3252
11/17/01 0.1534
2.1747
1.3828
1539
965.8
1050
20.4
17.4
18.9
0.3153
11/17/01 0.1534
2.1747
1.3828
648
965.8
1075
14.3
11.3
12.8
0.3066
11/17/01 0.1534
2.1747
1.3828
1043
965.8
1100
8.5
7.5
8
0.2951
11/17/01 0.1534
2.1747
1.3828
1165
965.8
1125
6
5.1
5.55
0.2969
11/17/01 0.1534
2.1747
1.3828
2038
965.8
1150
3.9
2.9
3.4
0.2912
11/17/01 0.1534
2.1747
1.3828
645
965.8
1175
2.65
1.75
2.2
0.2913
11/17/01 0.1534
2.1747
1.3828
1313
965.8
1200
2
1.1
1.55
0.2965
11/17/01 0.1534
2.1747
1.3828
3997
965.8
1225
1.55
0.65
1.1
0.3019
11/17/01 0.1534
2.1747
1.3828
2729
965.8
1250
1.25
0.35
0.8
0.3081
11/17/01 0.1534
2.1747
1.3828
5140
965.8
1275
1.05
0.15
0.6
0.3151
11/17/01 0.1534
2.1747
1.3828
99
965.8
1300
1.2
0.3
0.75
0.3441
11/17/01 0.1534
2.1747
1.3828
1466
965.8
1325
1.05
0.15
0.6
0.3520
11/17/01 0.1534
2.1747
1.3828
174
965.8
800
190.6
186.6
188.6
0.4384
12/22/01 0.2493
2.2495
1.6462
1784
965.8
900
111.2
107.2
109.2
0.3743
12/22/01 0.2493
2.2495
1.6462
2155
965.8
950
77.9
73.9
75.9
0.3461
12/22/01 0.2493
2.2495
1.6462
9
965.8
975
63.6
59.6
61.6
0.3340
12/22/01 0.2493
2.2495
1.6462
3
965.8
1025
38.5
34.5
36.5
0.3045
12/22/01 0.2493
2.2495
1.6462
5877
965.8
1050
29
25.3
27.15
0.2937
12/22/01 0.2493
2.2495
1.6462
7097
965.8
1060
26.2
22.2
24.2
0.2914
2.2495
1.6462
455
965.8
1075
21.6
18.6
20.1
0.2873
12/22/01 0.2493 12/22/01 0.2493
2.2495
1.6462
3926
965.8
1100
15.8
12.8
14.3
0.2798
12/22/01 0.2493
2.2495
1.6462
4371
276
Appendices SX
Bid
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
965.8
1150
8
6.7
7.35
0.2746
12/22/01 0.2493
2.2495
1.6462
8558
965.8
1175
5.5
4.1
4.8
0.2680
12/22/01 0.2493
2.2495
1.6462
4721
965.8
1200
3.6
2.6
3.1
0.2632
12/22/01 0.2493
2.2495
1.6462
15940
965.8
1225
2.6
1.7
2.15
0.2636
12/22/01 0.2493
2.2495
1.6462
7937
965.8
1250
1.95
13
1.625
0.2682
12/22/01 0.2493
2.2495
1.6462
19339
965.8
1275
1.6
0.7
1.15
0.2698
12/22/01 0.2493
2.2495
1.6462
6948
965.8
1300
1.5
0.6
1.05
0.2816
1222101 0.2493
2.2495
1.6462
21389
965.8
1325
1.25
0.35
0.8
0.2855
1222/01 0.2493
2.2495
1.6462
10051
965.8
1350
1.05
0.15
0.6
0.2886
1222101 0.2493
2.2495
1.6462
8999
965.8
1400
0.9
0.25
0.575
0.3137
12/22/01 0.2493
2.2495
1.6462
15367
965.8
900
125.7
121.7
123.7
0.3189
03/16/02 0.4795
2.3326
1.8399
31
965.8
950
94
90
92
0.3016
03/16/02 0.4795
2.3326
1.8399
54
965.8
995
69.4
65.4
67.4
0.2866
03/16/02 0.4795
2.3326
1.8399
289
965.8
1025
55.9
51.9
53.9
0.2794
03/16/02 0.4795
2.3326
1.8399
509
965.8
1050
45.8
41.8
43.8
0.2725
03/16/02 0.4795
2.3326
1.8399
2300
965.8
1075
37.1
33.1
35.1
0.2663
03/16/02 0.4795
2.3326
1.8399
71
965.8
1100
28.7
24.7
26.7
0.2560
03/16/02 0.4795
2.3326
1.8399
1225
965.8
1125
22.2
19.2
20.7
0.2511
03/16/02 0.4795
2.3326
1.8399
1191
965.8
1150
17.4
14.4
15.9
0.2472
03/16/02 0.4795
2.3326
1.8399
1352
965.8
1175
13.5
10.5
12
0.2433
03/16/02 0.4795
2.3326
1.8399
1291
965.8
1200
10
890.2403
03/16/02 0.4795
2.3326
1.8399
6416
965.8
1225
7.8
5.8
6.8
0.2388
03/16/02 0.4795
2.3326
1.8399
1598
965.8
1250
5.9
4.5
5.2
0.2385
03/16/02 0.4795
2.3326
1.8399
4369
965.8
1275
4.4
3
3.7
0.2351
03/16/02 0.4795
2.3326
1.8399
3082
965.8
1280
3.9
2.9
3.4
0.2338
03/16/02 0.4795
2.3326
1.8399
2
965.8
1300
3.3
2.3
2.8
0.2353
03/16/02 0.4795
2.3326
1.8399
4352
965.8
1325
2.55
1.65
2.1
0.2353
03/16/02 0.4795
2.3326
1.8399
2003
965.8
1350
2.05
1.15
1.6
0.2361
03/16/02 0.4795
2.3326
1.8399
3928
965.8
1375
1.7
0.8
1.25
0.2379
03/16102 0.4795
2.3326
1.8399
800
965.8
1400
1.3
0.65
0.975
0.2395
03/16/02 0.4795
2.3326
1.8399
9296
965.8
1425
1.1
0.2
0.65
0.2363
03/16/02 0.4795
2.3326
1.8399
294
965.8
1450
0.95
0.05
0.5
0.2376
03/16/02 0.4795
2.3326
1.8399
2006
965.8
1500
0.9
0.25
0.575
0.2592
03/16/02 0.4795
2.3326
1.8399
1162
965.8
850
174.1
170.1
172.1
0.3031
06/22/02 0.7479
2.4342
1.8602
116
965.8
950
109.5
105.5
107.5
0.2792
06/22/02 0.7479
2.4342
1.8602
6
965.8
995
85.2
81.2
83.2
0.2682
06/22/02 0.7479
2.4342
1.8602
451
965.8
1050
60.6
56.6
58.6
0.2567
06/22/02 0.7479
2.4342
1.8602
1471
965.8
1100
42.3
38.3
40.3
0.2453
06/22/02 0.7479
2.4342
1.8602
1796
965.8
1125
35.3
31.3
33.3
0.2417
06/22/02 0.7479
2.4342
1.8602
105
965.8
1150
28.8
24.8
26.8
0.2366
0622102 0.7479
2.4342
1.8602
4348
965.8
1175
23
20
21.5
0.2326
06/22/02 0.7479
2.4342
1.8602
160
965.8
1250
11.6
9.6
10.6
0.2235
0622102 0.7479
2.4342
1.8602
6911
965.8
1300
7.6
5.6
6.6
0.2208
06/22/02 0.7479
2.4342
1.8602
6893
965.8
1325
5.8
4.4
5.1
0.2190
0622/02
0.7479
2.4342
1.8602
1016
965.8
1350
4.7
3.3
4
0.2183
0622/02
0.7479
2.4342
1.8602
5425
965.8
1400
2.85
1.95
2.4
0.2166
06/22/02 0.7479
2.4342
1.8602
14585
965.8
1425
2.3
1.4
1.85
0.2160
06/22/02 0.7479
2.4342
1.8602
325
965.8
1450
1.9
1
1.45
0.2160
06/22/02 0.7479
2.4342
1.8602
9377
965.8
1500
1.25
0.35
0.8
0.2133
06/22/02 0.7479
2.4342
1.8602
8297
965.8
1525
1.05
0.15
0.6
0.2125
06/22/02 0.7479
2.4342
1.8602
310
965.8
900
165.1
159.1
162.1
0.2692
12/21/02 1.2466
2.6133
1.8624
568
965.8
950
135.1
129.1
132.1
0.2595
12/21/02 1.2466
2.6133
1.8624
1
277
Appendices Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
965.8
995
1113
105.3
108.3
0.2522
1221/02
1.2466
2.6133
1.8624
254
965.8
1050
87.1
81.1
84.1
0.2464
1221/02
1.2466
2.6133
1.8624
1877
965.8
1100
67.4
61.4
64.4
0.2386
12/21/02 1.2466
2.6133
1.8624
2217
965.8
1150
50.5
44.5
47.5
0.2297
12/21/02 1.2466
2.6133
1.8624
2886
965.8
1200
36.9
30.9
33.9
0.2214
12/21/02 1.2466
2.6133
1.8624
5798
965.8
1225
31.7
25.7
28.7
0.2187
12/21/02 1.2466
2.6133
1.8624
253
965.8
1250
26.8
20.8
23.8
0.2151
12/21/02 1.2466
2.6133
1.8624
4593
965.8
1300
18.5
15.5
17
0.2123
1221/02
1.2466
2.6133
1.8624
4077
965.8
1350
13.7
10.7
12.2
0.2110
12/21/02 1.2466
2.6133
1.8624
2007
965.8
1400
9.1
7.1
8.1
0.2066
12/21/02 1.2466
2.6133
1.8624
4870
965.8
1450
6
4.6
5.3
0.2029
1221/02
1.2466
2.6133
1.8624
5516
965.8
1500
4.4
3
3.7
0.2027
12/21/02 1.2466
2.6133
1.8624
4738
965.8
1550
3.7
2.55
3.125
0.2092
1221/02
1.2466
2.6133
1.8624
1577
965.8
1600
2.6
1.7
2.15
0.2083
12/21/02 1.2466
2.6133
1.8624
6531
965.8
1650
2.7
1.8
2.25
0.2207
1221/02
1.2466
2.6133
1.8624
1070
965.8
1700
2.3
1.4
1.85
0.2247
1221/02
1.2466
2.6133
1.8624
4214
965.8
1800
1.4
0.5
0.95
0.2237
12/21/02 1.2466
2.6133
1.8624
12361
965.8
1900
1
0.2
0.6
0.2282
1221/02
2.6133
1.8624
6545
278
1.2466
Appendices
B. 5 October 19,2001 Call Options
SX
Bid
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1073.5
900
176.8
174.8
175.8
0.3676
11/17/01 0.0767
2.2900
0.9318
18
1073.5
950
130.3
128.3
129.3
0.3564
11/17/01 0.0767
2.2900
0.9318
1846
1073.5
1025
66.7
64.7
65.7
0.3087
11/17/01 0.0767
2.2900
0.9318
3616
1073.5
1050
48.7
46.7
47.7
0.2913
11/17/01 0.0767
2.2900
0.9318
8617
1073.5
1075
33.5
31.5
32.5
0.2764
11/17/01 0.0767
2.2900
0.9318
10462
1073.5
1100
20.7
19.2
19.95
0.2590
11/17/01 0.0767
2.2900
0.9318
16096
1073.5
1125
12.1
10.6
11.35
0.2481
11/17/01 0.0767
2.2900
0.9318
10061
1073.5
1150
6.5
5.5
6
0.2415
11/17/01 0.0767
2.2900
0.9318
10757
1073.5
1175
3.2
2.5
2.85
0.2354
11/17/01 0.0767
2.2900
0.9318
5605
1073.5
1200
1.3
1.2
1.25
0.2314
11/17/01 0.0767
2.2900
0.9318
9984
1073.5
1225
0.7
0.4
0.55
0.2315
11/17/01 0.0767
2.2900
0.9318
5318
1073.5
800
277.4
275.4
276.4
0.3604
1222/01 0.1726
2.2364
1.4118
2902
1073.5
900
184
182
183
0.3492
1222/01
0.1726
2.2364
1.4118
3092
1073.5
950
140.1
138.1
139.1
0.3277
1222/01
0.1726
2.2364
1.4118
1361
1073.5
995
103.7
101.7
102.7
0.3082
1222/01
0.1726
2.2364
1.4118
9588
1073.5
1025
81
79
80
0.2910
12/22/01 0.1726
2.2364
1.4118
10857
1073.5
1050
64.1
62.1
63.1
0.2790
12/22/01 0.1726
2.2364
1.4118
16233
1073.5
1060
57.8
55.8
56.8
0.2740
1222/01
0.1726
2.2364
1.4118
6559
1073.5
1100
36.2
34.2
35.2
0.2566
1222/01
0.1726
2.2364
1.4118
23692
1073.5
1150
17.4
15.9
16.65
0.2394
1222/01
0.1726
2.2364
1.4118
17883
1073.5
1175
11.6
10.1
10.85
0.2338
12/22/01 0.1726
2.2364
1.4118
6475
1073.5
1300
1.1
0.65
0.875
0.2248
1222/01
0.1726
2.2364
1.4118
21540
1073.5
1325
0.75
0.3
0.525
0.2263
1222/01
0.1726
2.2364
1.4118
9694
1073.5
900
195
193
194
0.2859
03/16/02 0.4027
2.1778
1.8128
51
1073.5
950
154.4
152.4
153.4
0.2755
03/16/02 0.4027
2.1778
1.8128
1079
1073.5
995
120.5
118.5
119.5
0.2635
03/16/02 0.4027
2.1778
1.8128
1774
1073.5
1025
100.2
98.2
99.2
0.2569
03/16/02 0.4027
2.1778
1.8128
1639
1073.5
1050
84.3
82.3
833
0.2499
03/16/02 0.4027
2.1778
1.8128
7888
1073.5
1075
69.9
67.9
68.9
0.2434
03/16/02 0.4027
2.1778
1.8128
1879
1073.5
1100
56.4
54.4
55.4
0.2352
03/16/02 0.4027
2.1778
1.8128
12447
1073.5
1125
45.1
43.1
44.1
0.2296
03/16/02 0.4027
2.1778
1.8128
2128
1073.5
1150
35.2
33.2
34.2
0.2235
03/16/02 0.4027
2.1778
1.8128
5713
1073.5
1175
27.1
25.1
26.1
0.2186
03/16/02 0.4027
2.1778
1.8128
2059
1073.5
1200
20
18.5
19.25
0.2130
03/16/02 0.4027
2.1778
1.8128
12048
1073.5
1225
14.7
13.2
13.95
0.2085
03/16/02 0.4027
2.1778
1.8128
2792
1073.5
1250
10.3
9.3
9.8
0.2040
03/16/02 0.4027
2.1778
1.8128
5990
1073.5
1275
7.3
6.3
6.8
0.2006
03/16/02 0.4027
2.1778
1.8128
3026
1073.5
1280
766.5
0.2016
03/16/02 0.4027
2.1778
1.8128
3
1073.5
1300
5
4.3
4.65
0.1980
03/16102 0.4027
2.1778
1.8128
5090
1073.5
1325
3.5
2.8
3.15
0.1961
03/16/02 0.4027
2.1778
1.8128
2003
1073.5
1350
2.3
2
2.15
0.1952
03/16/02 0.4027
2.1778
1.8128
4697
1073.5
1375
1.65
1.2
1.425
0.1940
03/16102 0.4027
2.1778
1.8128
805
1073.5
1400
1.1
0.65
0.875
0.1912
03/16/02 0.4027
2.1778
1.8128
8865
1073.5
1425
0.8
0.35
0.575
0.1909
2.1778
1.8128
269
1073.5
1450
0.65
0.2
0.425
0.1936
03/16/02 0.4027 03/16/02 0.4027
2.1778
1.8128
2006
1073.5
850
248.1
246.1
247.1
0.2635
06/22/02 0.6712
2.2282
1.8690
121
279
Appendices SX
Bid
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield
Dis. Div.
Open Interest
1073.5
950
168.7
166.7
167.7
0.2529
06/22/02 0.6712
(%) 2.2282
1073.5
995
136.8
134.8
135.8
0.2455
0622/02
0.6712
2.2282
1.8690
2849
1073.5
1050
101.4
99.4
100.4
0.2341
0622/02
0.6712
2.2282
1.8690
3390
1073.5
1100
74.1
72.1
73.1
0.2245
06/22/02 0.6712
2.2282
1.8690
1767
1073.5
1125
62.5
60.5
61.5
0.2206
0622/02
0.6712
2.2282
1.8690
560
1073.5
1150
51.8
49.8
50.8
0.2158
06/22/02 0.6712
2.2282
1.8690
5458
1073.5
1200
34.7
32.7
33.7
0.2084
06/22/02 0.6712
2.2282
1.8690
6395
1073.5
1250
22.1
20.1
21.1
0.2014
06/22/02 0.6712
2.2282
1.8690
8208
1073.5
1300
13.6
12.1
12.85
0.1970
0622/02
0.6712
2.2282
1.8690
7132
1073.5
1325
10.3
9.3
9.8
0.1947
0622/02
0.6712
2.2282
1.8690
1118
1073.5
1350
7.8
6.8
7.3
0.1921
06/22/02 0.6712
2.2282
1.8690
5471
1073.5
1400
4
3.3
3.65
0.1849
0622/02
0.6712
2.2282
1.8690
12348
1073.5
1425
2.95
2.25
2.6
0.1829
0622/02
0.6712
2.2282
1.8690
329
1073.5
1450
2.15
1.7
1.925
0.1826
0622/02
0.6712
2.2282
1.8690
8702
1073.5
1500
1.2
0.75
0.975
0.1805
0622/02
0.6712
2.2282
1.8690
8287
1073.5
1525
0.9
0.45
0.675
0.1793
0622/02
0.6712
2.2282
1.8690
320
1073.5
1550
0.7
0.25
0.475
0.1786
06/22/02 0.6712
2.2282
1.8690
3458
1073.5
1600
0.65
0.2
0.425
0.1898
0622/02
0.6712
2.2282
1.8690
6212
1073.5
900
228.6
224.6
226.6
0.2362
12/21/02 1.1699
2.4130
1.8768
568
1073.5
950
193.1
189.1
191.1
0.2337
12/21/02 1.1699
2.4130
1.8768
152
1073.5
995
163.4
159.4
161.4
0.2299
1221/02
1.1699
2.4130
1.8768
2801
1073.5
1050
129.9
125.9
127.9
0.2231
1221/02
1.1699
2.4130
1.8768
2677
1073.5
1100
103.3
99.3
101.3
0.2173
1221/02
1.1699
2.4130
1.8768
4743
1073.5
1150
80.4
76.4
78.4
0.2119
1221/02
1.1699
2.4130
1.8768
2692
1073.5
1200
60.9
56.9
58.9
0.2061
12/21/02 1.1699
2.4130
1.8768
7779
1073.5
1225
52.4
48.4
50.4
0.2030
12/21/02 1.1699
2.4130
1.8768
253
1073.5
1250
45.1
41.1
43.1
0.2008
12/21/02 1.1699
2.4130
1.8768
4646
1073.5
1300
32.4
28.4
30.4
0.1951
12/21/02 1.1699
2.4130
1.8768
4476
1073.5
21.8
19.8
20.8
0.1901
1221/02
1.1699
2.4130
1.8768
2210
1073.5
1350 " 1400
15
13
14
0.1863
1221/02
1.1699
2.4130
1.8768
6233
1073.5
1450
9.5
8.5
9
0.1821
1221/02
1.1699
2.4130
1.8768
5648
1073.5
1475
7.6
6.6
7.1
0.1800
12/21/02 1.1699
2.4130
1.8768
15
1073.5
1500
6.1
5.1
5.6
0.1782
1221/02
1.1699
2.4130
1.8768
4782
1073.5
1550
3.7
2.9
3.3
0.1740
12/21/02 1.1699
2.4130
1.8768
1677
1073.5
1600
2.3
1.8
2.05
0.1725
12/21/02 1.1699
2.4130
1.8768
6443
1073.5
1650
1.55
1.05
1.3
0.1721
12/21/02 1.1699
2.4130
1.8768
1070
1073.5
1700
1
0.5
0.75
0.1700
1221/02
2.4130
1.8768
3574
280
1.1699
1.8690
24
Appendices
B. 6 November 16,2001 Call Options
Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield
Open Interest
Dis. Div.
1138.65
700
440.8
438.8
439.8
0.6922
11/16/01 0.0959
(%) 2.0147 0.9460
1138.65
800
341.5
339.5
340.5
0.5605
11/16/01 0.0959
2.0147
0.9460
2832
1138.65
850
291.9
289.9
290.9
0.4930
11/16/01 0.0959
2.0147
0.9460
1
1138.65 900
242.6
240.6
241.6
0.4346
11/16/01 0.0959
2.0147
0.9460
3093
1138.65 910
232.8
230.8
231.8
0.4240
11/16/01 0.0959
2.0147
0.9460
9
950
193.7
191.7
192.7
0.3791
11/16/01 0.0959
2.0147
0.9460
1388
1138.65 960
184.1
182.1
183.1
0.3709
11/16/01 0.0959
2.0147
0.9460
1068
1138.65 970
172.5
172.4
172.45
0.3369
11/16/01 0.0959
2.0147
0.9460
2906
1138.65 980
164.8
162.8
163.8
0.3493
11/16/01 0.0959
2.0147
0.9460
156
1138.65
990
155.2
153.2
154.2
0.3384
11/16/01 0.0959
2.0147
0.9460
1123
1138.65
995
150.5
148.5
149.5
0.3345
11/16/01 0.0959
2.0147
0.9460
10175
1138.65
1010
136.3
134.3
135.3
0.3192
11/16/01 0.0959
2.0147
0.9460
7596
1138.65
1025
122.4
120.4
121.4
0.3059
11/16/01 0.0959
2.0147
0.9460
12530
1138.65
1050
99.8
97.8
98.8
0.2842
11/16/01 0.0959
2.0147
0.9460
18344
1138.65
1060
91
89
90
0.2754
11/16/01 0.0959
2.0147
0.9460
10750
1138.65
1070
82.3
80.3
81.3
0.2660
11/16/01 0.0959
2.0147
0.9460
6
1138.65
1080
74.2
72.2
73.2
0.2602
11/16/01 0.0959
2.0147
0.9460
2963
1138.65
1090
65.9
63.9
64.9
0.2504
11/16/01 0.0959
2.0147
0.9460
6423
1138.65
1095
62
60
61
0.2468
11/16/01 0.0959
2.0147
0.9460
1338
1138.65
1100
58.1
56.1
57.1
0.2426
11/16/01 0.0959
2.0147
0.9460
29062
1138.65
1110
50.7
48.7
49.7
0.2353
11/16/01 0.0959
2.0147
0.9460
512
1138.65
1115
47.6
45.6
46.6
0.2352
11/16/01 0.0959
2.0147
0.9460
1952
1138.65
1120
44.1
42.1
43.1
0.2312
11/16/01 0.0959
2.0147
0.9460
3347
1138.65
1140
31
29
30
0.2137
11/16/01 0.0959
2.0147
0.9460
6608
1138.65
1150
25.2
23.9
24.55
0.2077
11/16/01 0.0959
2.0147
0.9460
31822
1138.65
1160
21
19.5
20.25
0.2059
11/16/01 0.0959
2.0147
0.9460
1653
1138.65
1175
14.8
13.3
14.05
0.1974
11/16/01 0.0959
2.0147
0.9460
9606
1138.65
1300
0.6
0.25
0.425
0.1980
11/16/01 0.0959
2.0147
0.9460
21895
1138.65 900
244.8
242.8
243.8
0.3566
11/16/01 0.1726
1.9825
1.3711
2
1138.65
950
197.3
195.3
196.3
0.3272
11/16/01 0.1726
1.9825
1.3711
122
1138.65
1050
107.4
105.4
106.4
0.2647
11/16/01 0.1726
1.9825
1.3711
1085
1138.65
1075
87.4
85.4
86.4
0.2521
11/16/01 0.1726
1.9825
1.3711
3829
1138.65
1100
68.9
66.9
67.9
0.2400
11/16/01 0.1726
1.9825
1.3711
7780
1138.65
1125
52.2
50.2
51.2
0.2280
11/16/01 0.1726
1.9825
1.3711
9967
1138.65
1150
37.4
35.4
36.4
0.2149
11/16/01 0.1726
1.9825
1.3711
7697
1138.65
1175
25.7
23.7
24.7
0.2054
11/16/01 0.1726
1.9825
1.3711
1981
1138.65
1200
16.3
14.8
15.55
0.1959
11/16/01 0.1726
1.9825
1.3711
3582
1138.65
1225
9.8
8.8
9.3
0.1894
11/16/01 0.1726
1.9825
1.3711
1559
1138.65
1250
5.6
4.9
5.25
0.1846
11/16/01 0.1726
1.9825
1.3711
5556
1138.65
1300
1.65
1.2
1.425
0.1785
11/16/01 0.1726
1.9825
1.3711
26
1138.65
1350
0.6
0.15
0.375
0.1789
11/16/01 0.1726
1.9825
1.3711
203
1138.65
750
393.2
391.2
392.2
0.3154
11/16/01 0.3260
1.9835
1.8099
32
1138.65 900
249.4
247.4
248.4
0.2936
11/16/01 0.3260
1.9835
1.8099
51
1138.65 950
203.7
201.7
202.7
0.2779
11/16/01 0.3260
1.9835
1.8099
1078
1138.65 975 1138.65 995
1815
179.5
180.5
0.2690
1.9835
1.8099
2
164.3
162.3
163.3
0.2627
11/16/01 0.3260 11/16/01 0.3260
1.9835
1.8099
2113
1138.65
281
189
Appendices Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1138.65
1025
139.3
137.3
138.3
0.2524
11/16/01 0.3260
1.9835
1.8099
1818
1138.65
1050
119.7
117.7
118.7
0.2451
11/16/01 0.3260
1.9835
1.8099
8661
1138.65
1075
100.8
98.8
99.8
0.2359
11/16/01 0.3260
1.9835
1.8099
3924
1138.65
1100
83.4
81.4
82.4
0.2278
11/16/01 0.3260
1.9835
1.8099
13324
1138.65
1125
67.1
65.1
66.1
0.2184
11/16/01 0.3260
1.9835
1.8099
7102
1138.65
1150
53.2
51.2
52.2
0.2120
11/16/01 0.3260
1.9835
1.8099
8548
1138.65
1175
40.5
38.5
39.5
0.2037
11/16/01 0.3260
1.9835
1.8099
3118
1138.65
1200
29.7
27.7
28.7
0.1953
11/16/01 0.3260
1.9835
1.8099
16011
1138.65
1225
21.2
19.7
20.45
0.1897
11/16/01 0.3260
1.9835
1.8099
3922
1138.65
1250
14.9
13.4
14.15
0.1852
11/16/01 0.3260
1.9835
1.8099
11200
1138.65
1275
10.3
9.3
9.8
0.1833
11/16/01 0.3260
1.9835
1.8099
3051
1138.65
1280
9.3
8.3
8.8
0.1811
11/16/01 0.3260
1.9835
1.8099
602
1138.65
1300
6.7
5.7
6.2
0.1784
11/16/01 0.3260
1.9835
1.8099
6105
1138.65
1325
4.2
3.5
3.85
0.1750
11/16/01 0.3260
1.9835
1.8099
2241
1138.65
1350
2.7
2
2.35
0.1727
11/16/01 0.3260
1.9835
1.8099
4966
1138.65
1375
1.6
1.15
1.375
0.1704
11/16/01 0.3260
1.9835
1.8099
789
1138.65
1400
1.25
0.8
1.025
0.1755
11/16/01 0.3260
1.9835
1.8099
8718
1138.65
1425
0.9
0.45
0.675
0.1770
11/16/01 0.3260
1.9835
1.8099
269
1138.65
1450
0.65
0.2
0.425
0.1776
11/16/01 0.3260
1.9835
1.8099
1986
1138.65
850
303.6
301.6
302.6
0.2564
11/16/01 0.5945
2.1262
1.9409
121
1138.65
950
215.1
213.1
214.1
0.2437
11/16/01 0.5945
2.1262
1.9409
28
1138.65
995
178
176
177
0.2350
11/16/01 0.5945
2.1262
1.9409
2670
1138.65
1050
136.3
134.3
135.3
0.2248
11/16/01 0.5945
2.1262
1.9409
3398
1138.65
1075
118.7
116.7
117.7
0.2196
11/16/01 0.5945
2.1262
1.9409
1226
1138.65
1100
102.4
100.4
101.4
0.2150
11/16/01 0.5945
2.1262
1.9409
5919
1138.65
1125
86.9
84.9
85.9
0.2092
11/16/01 0.5945
2.1262
1.9409
3837
1138.65
1150
73.2
71.2
72.2
0.2051
11/16/01 0.5945
2.1262
1.9409
5729
1138.65
1200
48.8
46.8
47.8
0.1942
11/16/01 0.5945
2.1262
1.9409
8264
1138.65
1250
30.7
28.7
29.7
0.1856
11/16/01 0.5945
2.1262
1.9409
6669
1138.65
1300
18.4
16.9
17.65
0.1803
11/16/01 0.5945
2.1262
1.9409
7224
1138.65
1325
14
12.5
13.25
0.1778
11/16/01 0.5945
2.1262
1.9409
1669
1138.65
1350
10
9
9.5
0.1740
11/16/01 0.5945
2.1262
1.9409
5481
1138.65
1400
5.1
4.4
4.75
0.1689
11/16/01 0.5945
2.1262
1.9409
12147
1138.65
1425
3.7
3
3.35
0.1676
11/16/01 0.5945
2.1262
1.9409
329
1138.65
1450
2.75
2.05
2.4
0.1672
11/16/01 0.5945
2.1262
1.9409
8703
1138.65
1500
1.35
0.9
1.125
0.1651
11/16/01 0.5945
2.1262
1.9409
8537
1138.65
1525
1.1
0.65
0.875
0.1675
11/16101 0.5945
2.1262
1.9409
320
1138.65
1550
0.8
0.35
0.575
0.1662
11/16/01 0.5945
2.1262
1.9409
3465
1138.65
1600
0.6
0.15
0.375
0.1722
11/16/01 0.5945
2.1262
1.9409
6212
1138.65
800
362.5
359.5
361
0.2031
11/16/01 1.0932
2.4733
1.9580
1
1138.65
900
276.8
273.8
275.3
0.2195
11/16/01 1.0932
2.4733
1.9580
568
1138.65 950
237.4
234.4
235.9
0.2206
11/16/01 1.0932
2.4733
1.9580
1700
1138.65
995
203.5
200.5
202
0.2176
11/16/01 1.0932
2.4733
1.9580
4077
1138.65
1050
164.7
161.7
163.2
0.2118
11/16101 1.0932
2.4733
1.9580
3602
1138.65
1100
132.5
129.5
131
0.2054
11/16101 1.0932
2.4733
1.9580
7891
1138.65
1150
104.9
101.9
103.4
0.2011
11/16/01 1.0932
2.4733
1.9580
3436
1138.65
1200
80.4
77.4
78.9
0.1953
11/16/01 1.0932
2.4733
1.9580
10029
1138.65
1225
69.5
66.5
68
0.1922
2.4733
1.9580
311
1138.65
1250
59.6
56.6
58.1
0.1892
11/16/01 1.0932 11/16/01 1.0932
2.4733
1.9580
6659
1138.65
1300
42.4
39.4
40.9
0.1824
11/16/01 1.0932
2.4733
1.9580
5822
1138.65
1350
29.6
26.6
28.1
0.1774
2.4733
1.9580
2160
1138.65
1400
18.9
17.4
18.15
0.1715
11/16/01 1.0932 11/16/01 1.0932
2.4733
1.9580
6950
282
Appendices Bid
SX
Ask
Mid
BS Imp. Vol.
Exp.
Maturity
Yield (%)
Dis. Div.
Open Interest
1138.65
1450
12.2
10.7
11.45
0.1672
11/16/01 1.0932
2.4733
1.9580
5670
1138.65
1475
9.5
8.5
9
0.1654
11/16/01 1.0932
2.4733
1.9580
14
1138.65
1500
7.9
6.9
7.4
0.1658
11/16/01 1.0932
2.4733
1.9580
4788
1138.65
1550
4.7
4
4.35
0.1622
11/16/01 1.0932
2.4733
1.9580
1591
1138.65
1600
2.8
2.1
2.45
0.1588
11/16/01 1.0932
2.4733
1.9580
6376
1138.65
1650
1.6
1.15
1.375
0.1566
11/16/01 1.0932
2.4733
1.9580
1070
1138.65
1700
1
0.55
0.775
0.1552
11/16/01 1.0932
2.4733
1.9580
3568
283
Appendices
B. 7 Theoretical Delivery Price for Demeterfi et al. Variance Swap Model The delivery price is given by: K=
"°'
2{rT
s° S,
T
logs"
e'T i')
1
+e'rýs`
so
o KZ
P(K)dK+e'r
1 S'Kz
C(K)dK
The appropriate option portfolio weights for a finite set of call and put strikes, K,, and K,, c p are given by: (Sr )'2T g
SrS' S'
S,
log S'
ri g(Kr. j.c)g(Kr. c) w(Kt, c ) w'(Kr c)=, ý hr+ý.  Kr. c c J0
%s{K,. p)=
S(Kv+,.r) Kr.
S(Kt. r)
Kr+i. Pp
_Iw(Kjp) u 1=0
284
for calls
for puts
Appendices
C.1 Characteristic Functions For SV Model ä2 
+(1+ K,.
21 1
)pQv](1eCs) 7v
fisV
= exp
[ý"YKv +(1+iO)pQv]z+iOra+io
z
In [S(t)]
v
+
Q2
iO(iO +1)(1es°s)V(t) 2ýý [ýv K,
ý e? 
Qý
+2ýv
(1+i¢)Pav
7*
=
{[KrirPQrJ2
I (1e's. =)
xy +iýpQ, ](1e'sýf) ýý
21 1
f S`'=exp
+(1+io)Pc,
]2 iý(io
[ý"Yx'y +iopu iO(iO 1)(1
' xy [ý,
]z+iorz+io
ln[S(t)]
f e's. )y(t)
+iOPQ,
+1)vv I`
ý2 iýqo1)ory
285
](1eC"j)
Appendices
C.2 Characteristic Functions For SVJ Model
J_J2'n('ýz
+(1+ý)PQý](1e'ýs) Y
v
[ý', Ký, +(1+io)pQv]t+iorz+io
2
fisv" = expý
ln[S(t)]
v
iO(iO +1)(1eý, s)V(t)
+
2; [ýy x, +(1+iO)PQv](1e's. t) y +ý,(1+ýt, )z[(1+ýC, )'4e('4'zx1+ra)a; 1]Aioju,
I
ar
2
[C, xy +iOpQ,,](1ef't) 21n112ýv \1 e"
fsvi 2y
= exp
+
+i opQ',,]z +i orz +io ln[S (t)]
Q2
iO(iO 1)(1 err )V (t) 2ýý [ýy Kv +iopav](1eC: f ) uý )i4e(iq,
+As[(1+,
ýr
JY2
tKr (1+io)Porr]2

ýý _
{[x,,
2>(iq1)aý
14y¢+1)Q2
]= i¢(i¢1)aý i¢pa,,
1
286
1] Ai, u, z
Z
Appendices
D. 1 MATLAB Optimisation Toolbox Settings TrustRegion Reflective QuasiNewton Method
AssetRangeFactor Local Volatility Knots Asset Knots Time Knots Lower Volatility Bound
Upper Volatility Bound FunctionTolerance
F=2 , p
1=1
AssetLevels
1x 103 M=200
Time Levels
N=50
PCG Bandwidth
0
287