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City, University of London Institutional Repository Citation: Lam, K. H. (2004). Essays on the Modelling of S&P 500 Volatility. (Unpublished Doctoral thesis, City University London)
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[email protected] Essays on the Modelling of S&P 500 Volatility
KarHei LAM
Thesis Submitted for the Degree of Doctor of Philosophy in Finance
City University Cass Business School Department of Finance
February 2004
Contents x
Abbreviations Important
AV
Symbols
xv
Tables
xviii
Figures Acknowledgements
xx
Declaration
xxi
Abstract
xxii
1
CHAPTER 1 Introduction to the Study 1.1 Introduction
1
1.2 Background of the Study 1.2.1 TermStructure of Volatility
1 2
1.2.2 Modelling of Volatility
3
1.2.3 Volatility Derivatives
4
1.3
6
The Problem Statement 1.3.1 Objectives of the First Research Project
7
1.3.2 Objectives of the Second Research Project
7
1.3.3 Objectives of the Third Research Project
8
1.3.4 Delimitations of the Study
9
1.4 The Significance of the Study 1.4.1 Significance of the First ResearchProject 1.4.2 Significance of the SecondResearchProject 1.4.3 Significance of the Third ResearchProject
9 9 10 11
1.5 Organisation of the Dissertation
11
CHAPTER 2 Review of the Literature
13 13
2.1 Option Pricing Theories
13 14 17 17
2.1.1 NoArbitrage Approach 2.1.1.1 BlackScholesFormula 2.1.1.2 Other Variations 2.1.1.3 Implied Volatility 2.1.2 Martingale Approach 2.1.2.1 Underlying Concepts 2.1.2.2 DiscreteTime Process
18 18 19 ii
2.1.2.3 2.1.2.4 2.1.2.5 2.1.2.6 2.1.2.7
ContinuousTime Process SelfReplicating Strategy Kolomogorov Equation Market Price of Risk Summary of the Martingale Approach
2.2 Conditional
Heteroskedastic
20 23 25 26 26 27
Models
2.2.1 Underlying Concepts 2.2.1.1 Random Walk 2.2.1.2 Skewness and Kurtosis 2.2.1.3 Unconditional and Conditional Variances
27 27 27 28
2.2.2 Autoregressive Conditional Heteroskedasticity Models 2.2.2.1 ARCH Model 2.2.2.1.1 Implications 2.2.2.1.2 Maximum Likelihood Estimation
29
2.2.2.2 GARCH Model 2.2.2.2.1 Implications 2.2.2.3 EGARCH Model 2.2.2.3.1 Implications 2.2.3 Long Memory and Asymmetric Models 2.2.3.1 Underlying Concepts 2.3.3.1.1 Stylised Facts 2.3.3.1.2 Inadequacyof GARCHtype Models for LongRun Effects 2.2.3.2 ARFIMA Model 2.2.3.3 News Impact Curve 2.2.3.4 APARCH Models 2.2.3.5 HentschelFramework 2.2.3.6 Other Asymmetric Models
32 32 34 35
2.2.4 GARCH Option Models
43
2.2.5 Other Developments
46
2.3 Stochastic Volatility Models
47
29 31 31
36 36 36 37 38 39 40 41 42
2.3.1 Underlying Concepts 2.3.1.1 Wiener Process 2.3.1.2 StochasticProcess 2.3.1.3 StochasticDifferential Equation 2.3.1.4 OrnsteinUhlenbeckProcess
47 47 47 48 48
2.3.2 HullWhite Model
49
2.3.3 JohnsonShannoModel
50
2.3.4 SteinStein Model
51
2.3.5 Heston Model
51
2.3.6 Merton Model
52
2.3.7 Other Developments
53 56
2.4 Implied Methodology 2.4.1 Underlying Concepts
56
2.4.2 Direct Approach
57
111
2.4.2.1 BreedenLitzenbergerMethod 2.4.2.2 MultiLogNormality Method 2.4.2.3 Approximating the RiskNeutral Density Distribution 2.4.3 Indirect Approach 2.4.3.1 Implied Tree Assumptions 2.4.3.2 Rubinstein Model 2.4.3.3 Dupire Model Model 2.4.3.4 DermanKani Model
57 58 59
2.4.4 Other Developments 2.4.4.1 Direct Approach 2.4.4.2 Indirect Approach
64 64 65
2.5 Factors Influencing
60 60 61 62 63
65
Option Pricing
66
2.5.1 Underlying Concepts 2.5.1.1 Observed Biases 2.5.1.2 Historical Volatility versus Implied Volatility 2.5.1.3 Timeseries Properties
66 66 67
2.5.2 OverreactionHypothesis
68
2.5.3 Information Content 2.5.3.1 Evidence Supporting the Significance of Implied Volatility 2.5.3.2 Evidence Against the Significance of Implied Volatility 2.5.3.3 Other Developments
69 69 70 70
2.5.4 Negative Relationship Between Returns and Volatility 2.5.4.1 Evidence Supporting Leverage Effect as Sole Explanation for Asymmetries 2.5.4.2 Evidence Against LeverageEffect as Sole Explanation for Asymmetries
71 71 72
2.5.5 Persistencyof Volatility Shocks 2.5.5.1 Evidence Against Persistencyof Volatility Shocks 2.5.5.2 Structural Changeas Explanation of Persistency 2.5.5.3 Identifying Structural Breaks
73 73 73 74
2.5.6 Market Efficiency 2.5.6.1 Volatility Trading
75 75
2.6 Common Diagnostic Tests
76
2.6.1 Test for Stationarity
77
2.6.2 Test for Independence
77
2.6.3 Test for Normality
78
2.6.4 Hypothesis Tests for Dependence
79 81
2.7 Summary CHAPTER 3A
Report on the Properties of the TermStructure of S&P 500 Implied Volatility82
Abstract
82
3.1 Introduction
82
3.1.1 Background of the Study
82
3.1.2 The Problem Statement
83
iv
3.1.3 The Significance of the Study
84
3.1.4 Organisation
85 85
3.2 Methodology 3.2.1 Relative Implied Volatility
85
3.2.2 Futures Options versus Spot Index Options
87
3.2.3 Strategies
88
3.2.4 Summary of the Methodology
89 89
3.3 Data Description 3.3.1 S&P 500 Futures and Futures Options
89
3.3.2 Contract Specifications
89
3.3.3 Approximating
Implied Volatility
for American Options
90 91
3.3.4 Filtering
3.4 Results and Analysis 3.4.1 Financial and Political Events for 19831998 3.4.2 Interpretation of the Implied TermStructure Results 3.4.3 Charactersof AttheMoney Implied Volatility TermStructure 3.4.3.1 Variability of Implied Volatility 3.4.3.2 MeanReversionof Implied Volatility 3.4.3.3 Consistencyof Implied Volatility TemStructure 3.4.4 Option Pricing Under Asymmetric Processes 3.4.4.1 SkewnessPremiums 3.4.4.1.1 Underlying Concepts 3.4.4.1.2 Data Construction 3.4.4.2.2 Resultsof Distributional Hypothesis
92 92 93 94 95 96 98 101 101 101 102 103 104
3.5 Summary CHAPTER 4 An Empirical Comparison of APARCH Models
114
Abstract
114
4.1 Introduction
114
4.1.1 Background of the Study
114
4.1.2 The Problem Statement and Hypotheses
116
4.1.3 The Significance of the Study
117
4.1.4 Organisation
117
117
4.2 Methodology 4.2.1 PerformanceCriteria
118
4.2.2 Analytical Procedures 4.2.3 Conditional Volatility Models 4.2.3.1 APARCH Specification
118 119 119
V
4.2.3.2 Lag Structure of APARCH Models 4.2.3.3 EGARCH
121 121 122
4.2.4 Summary of the Methodology
122
4.3 Data Description 4.3.1 Rollover of S&P 500 Futures Contracts 4.3.2 Partitioning and Descriptive Statisticsfor TimeSeries 4.3.2.1 Summary of Descriptive Statistics
122 123 125 125
4.4 Results & Analysis 4.4.1 Rationale for AR(1) Return Process
125
4.4.2 InSample Analysis: Maximum Likelihood Estimations of APARCH Parameters 4.4.2.1 InSample Results from Maximum Likelihood Estimations 4.4.2.2 Are APARCH Specifications Effective? 4.4.2.2.1 LLR Test: Is Power Transformation Effective? 4.4.2.2.2 LLR Test: Is Asymmetric ParameterisationEffective?
126 127 128 128 129
4.4.2.3 Discussions for APARCH InSample Results 4.4.3 Are Conditional Volatility Models Prone to the State of Volatility? 4.4.3.1 Studentt SWARCH(3,2)L Model 4.4.3.2 Detecting Structural Breaks in S&P 500 Futures Series
4.4.3.2.1 Interpretationsof Estimated SWARCH(3,2)L Parameters 4.4.3.3 Implications of Results from SWARCH(3,2)L Model 4.4.4 Additional InSample Analysis: EGARCH and Statistical Loss Functions 4.4.4.1 Inclusion of EGARCH 4.4.4.1.1 InSample Results for EGARCH 4.4.4.1.2 Discussion for InSample Resultsbasedon AIC 4.4.4.1.3 Plausible Explanation for the Poor Performanceof APARCH 4.4.4.2 Inclusion of Alternative Statistical Loss Functions 4.4.4.2.1 Proceduresfor Calculating InSample Statistical Errors 4.4.4.2.2 Resultsfor Alternative Statistical Loss Functions 4.4.4.2.3 Commentson Results for InSample Statistical Loss Functions 4.4.5 OutofSample Analysis: Trading S&P 500 Straddles 4.4.5.1 Background 4.4.5.2 Volatility ForecastingModels 4.4.5.3 Trading Methodology 4.4.5.3.1 Why Trading DeltaNeutral Straddles? 4.4.5.3.2 Trading Assumptions 4.4.5.3.3 Trading Strategy 4.4.5.3.4 Why Not Other Trading Strategy?
130 130 131 132
132 133 134 134 135 135 136 136 136 137 137 138 138 138 139 139 140 140 142
4.4.5.4 Trading Database 4.4.5.4.1 Weekly Straddles 4.4.5.4.2 Weekly TimeSeries Statistics
142 142 143
4.4.5.5 Results of Trading AttheMoney Straddles 4.4.5.5.1 Preliminary Statistics for Directional Trading Signals 4.4.5.5.2 Profit and Loss: Trading AttheMoney Straddles 4.4.5.5.3 Trading Summary
143 143 145 147 147
4.5 Summary
V1
CHAPTER 5 Empirical Performance of Alternative Variance Swap Valuation Models
177
Abstract
177
5.1 Introduction
177
5.1.1 Background of the Study 5.1.1.1 New Way of Trading: Variance Swap 5.1.1.2 Usageof Variance Swap 5.1.1.3 Variance Swap Example
177 178 179 180
5.1.2 The Problem Statementand Hypotheses
181
5.1.3 The Significance of the Study
182
5.1.4 Organisation
182
5.2 Methodology
182
5.2.1 PerformanceCriteria
182
5.2.2 The OptionsbasedVariance Swap Framework 5.2.2.1 LogContract 5.2.2.2 Demeterfi et al. Framework 5.2.2.2.1 Derivation of Demterfi et at. Framework 5.2.2.2.2 Implementation Issueswith Demeterfi et at. Framework 5.2.3 Option Models for Variance Swaps 5.2.3.1 StochasticVolatility Models 5.2.3.1.1 Justification for the StochasticVolatility Approach 5.2.3.1.2 Heston Model 5.2.3.2 JumpDiffusion Models 5.2.3.2.1 Justification for the JumpDiffusion Approach 5.2.3.2.2 Bakshi et al. Model 5.2.3.3 Local Volatility Models 5.2.3.3.1 Justification for the Local Volatility Approach 5.2.3.3.2 OneFactorModel 5.2.3.3.3 Coleman et at. Approach 5.2.3.4 Ad hoc BlackScholesModel 5.2.4 TimeSeries Models for ForecastingVariance 5.2.4.1 Justification for the Conditional Volatility Approach 5.2.4.2 GARCHVariance Swap 5.2.4.3 EGARCH Simulations
183 184 185 187 189 190 191 191 192 194 194 196 197 197 198 200 203 204 204 204 205 206
5.2.5 Summary of the Methodology
206
5.3 Data Description 5.3.1 Specifications and Filtering 5.3.1.1 Dividends 5.3.1.2 Calibration Using Call Options
206 207 208
5.3.2 Financial and Political Events 5.3.3 Descriptive Statistics for Call Options and S&P 500 Index in 19992002
209 209
5.3.4 Contract Specifications 5.3.4.1 Design of Variance Swap Contracts
211 212
vii
213
5.4 Results & Analysis
214
5.4.1 Calibration Procedures 5.4.1.1 Calibrations for Stochastic Volatility with/without Jump 5.4.1.2 Calibrations for Local Volatility Model 5.4.1.2.1 TrustRegion Reflective QuasiNewton Method 5.4.1.2.2 Calibrations for Absolute Diffusion Process 5.4.1.2.3.. FiniteDifference Settings
214 216 217 217 219
5.4.1.3 Calibrations for Ad Hoc BlackScholesModel 5.4.1.4 Calibrations for TimeSeries Models
219 219
5.4.2 Empirical Results 5.4.2.1 Calibration Results for OptionsbasedModels 5.4.2.1.1 Calibration Resultsfor StochasticVolatility Models with/without Jumps 5.4.2.1.2 Calibration Results for ad hoc BlackScholesModel 5.4.2.1.3 Calibration Results for EGARCH and GARCH Variance Swap Models 5.4.2.1.4 Calibration Results for Local Volatility Model
220 221 221 223 224 225
5.4.2.2 Variance Swap Forecasting Results 5.4.2.2.1 Implementation Issues for Optionsbased Variance Swap Model 5.4.2.2.2 OutofSample Test: Variance Forecastability 5.4.2.2.3 Comments on OutofSample Results 5.4.2.2.4 InSample Fit for Option Pricing Models
226 226 227 228 230
5.4.2.3 Consistencywith the TimeseriesPropertiesof Volatility 5.4.2.3.1 CIR SquareRootProcess 5.4.2.3.2 Results of Maximum Likelihood Estimation
231 231 232 233
5.5 Summary CHAPTER 6 Summary, Discussion and Suggestions for Further Research
236
6.1 Introduction
236
6.2 Statement of the Problem
236
6.3 Summary of the Results
237
6.4 Discussions of the Results
239
6.4.1 TermStrucutre of Implied Volatility
240
6.4.2 Conditional HeteroskedasticModels
241
6.4.3 Timeseries and OptionsbasedVariance ForecastingModels 6.4.4 Final Comment
243
6.5 Recommendations for Further Research
244 244 247
EPILOGUE REFERENCES
248
APPENDICES
267
A. 1 APARCH Models
267
A. 2 InSample Model Selection Criteria
268
viii
B. 1 June 15,2001 Call Options B. 2 July 20,2001 Call Options B. 3 August 17,2001 Call Options B.4 September21,2001 Call Options B. 5 October 19,2001 Call Options B. 6 November 16,2001 Call Options B. 7 Theoretical Delivery Price for Demeterfi et al. Variance Swap Model C. 1 CharacteristicFunctions For SV Model C.2 CharacteristicFunctions For SVJ Model D. 1 MATLAB Optimisation Toolbox Settings
ix
Abbreviations ACF
Autocorrelation Function
ADF
Augmented DickeyFuller Unit Root Test
AIC
Akaike Information Criterion
APARCH
Asymmetric Power AutoregressiveConditional Heteroskedasticity
ARCH
AutoregressiveConditional Heteroskedasticity
ARFIMA
Autoregressive Fractionally Integrated Moving Average
ARIMA
Autoregressive Integrated Moving Average
ARMA
AutoregressiveMoving Avereage
ATM
AttheMoney
BDF
Brock, Dechert and ScheinkmanIndependenceTest
BHHH
Berndt, Hall, Hall and HausmanAlgorithm
BPI
Binomial Path Independence
BS
Black & Scholes Model
CBOE
Chicago Board Options Exchange
CBOT
Chicago Board of Trade
CDF
Cumulative Distribution Function
CME
Chicago Merchantile Exchange
CMG
CameronMartinGirsanov
CUBS
City University BusinessSchool
DIVF
Dynamic Implied Volatility Function
DVF
Deterministic Volatility Function
DF
DickeyFuller Unit Root Test
DTB
Deutsche Terminborse
EGARCH
Exponential GARCH
X
EGARCHM
Exponential GARCH in Mean
EMS
European Monetary System
ERM
ExchangeRate Mechanism
EWMA
Exponentially Weighted Moving Average
FIGARCH
Fractional Integrated GARCH
FIEGARCH
Fractional Integrated EGARCH
FESE100
Financial Times Stock Exchange 100
GARCH
GeneralisedARCH
GARCHM
GARCH in mean
GBM
Geometric Brownian Motion
GED
GeneralisedError Distribution
GMLE
GuassianquasiMLE
GMM
GeneralisedMethod of Moment
HKSE
Hong Kong Stock Exchange
HIS
Hang SengINdex
HMSE
HeteroskedasticityadjustedMSE
HW
Hull & White Model
III)
Independent& Identically Distributed
IM M
International Money Market
ISD
Implied StandardDeviation
ITM
IntheMoney
IV
Implied Volatility
IVF
Implied Volatility Function
KS
KolmogorovSmirnov
LEAPS
Longterm Equity Anticipation Securities
statistic
R1
LHS
LeftHand Side
LIFFE
London Financial Futures Exchange
LL
Logarithmic Loss
LLR
LogLikelihood
LM
LarangeMultipler Test
LR
LikelihoodRatio
LRNVR
Locally RiskNeutral Valuation Relationship
LTCM
Long Term Capital Management
MA
Moving Average
MAE
MeanAbsolute Error
MAPE
MeanAbolute PercentError
MMEO
MeanMixed Error (over prediction)
MMEU
MeanMixed Error (under prediction)
MONEP
Marche des Options Negociablesde Paris
MSE
MeanSquareError
NARCH
Nonlinear ARCH
NLS
Nonlinear LeastSquare
NYSE
New York Stock Exchange
OLS
Ordinary Least Squares
OTC
OvertheCounter
OTM
OutoftheMoney
PIDE
Partial IntergroDifferential
PDE
Partial Differential Equation
PDF
Probability Density Function
PHLX
Philadelphia Exchange
Ratio
Test
Equation
X11
QMLE
QuasiMaximum Likelihood Estimation
RHS
RightHand Side
RMAE
Root Mean Absolute Error
RMAPE
Root Mean Absolute PercentError
RMSE
Root Mean SquaredError
RND
RiskNeutral Distribution
S&P 100
Standard & Poor 100
S&P 500
Standard & Poor 500
SACF
SampleAutocorrelation Function
SBC
SchwarzBayesian Critereon
SDE
StochasticDifferential Equation
SPSE
Sum of Price SquareError
SV
StochasticVolatility
SWARCH
Switching ARCH
TSGARCH Taylor's and Schwert's GARCH VAR
Vector AutoregressiveModel
VIX
CBOE's Volatility Index
VOLAX
DTB's Volatility Index Futures
VX1
MONEP's 31day ShortTerm Implied Volatility Index
WISD
Weighted Implied StandardDeviation
Xlii
Important Symbols ß
Elasticity Requirement
C
Call Option
x2 (m)
m`"degreeChistatistics
D(
Any Distribution
8
Correlation for Wiener Processes
Ho
Null Hypothesis
H,
Alternative Hypothesis
h,
Conditional Variance basedon information up to time t1
I(0)
CovarainceStationary Process
I(1)
NonStationary Process
it
Information set up to time t
L
Lag Operator
,u
Distributional Mean
N(
Normal Distribution
P
Put Option
Q2(m)
LjungBox Statistics at lag m for Serial Correlation
Q(m)
BoxPierce Statistics at lag m for Serial Correlation
p(m)
SampleAutocorrelation Function at lag m
at
StochasticVolatility
T()
Studentt Distribution
W
Wiener Processor Brownian Motion
xiv
Tables Table 1: TimetoMaturity and Moneyness Groups 86 ...................................................................................... Table 2: Normalised Data Groups
.................................................................................................................
87
Table 3: tstatistics for equal means but unequal variances for atthemoney calls 96 ...................................... Table 4: tstatistics for equal means but unequal variances for atthemoney puts 96 ....................................... Table 5: Curvefitting estimations for Average Call Implied Volatility from 19831998
............................
98
Table 6: Curvefitting estimations for Average Put Implied Volatility from 19831998 98 .............................. Table 7: 6(K, T, P) for Calls 100 ................................................................................................................... Table 8: ß(K, T, P) for Puts
...................................................................... ...............................................
101
Table 9: Descriptive Statistics for r 149 ............................................................................................................. Table 10: Descriptive Statistics for r2 149 ......................................................................................................... Table 11: Descriptive Statistics for 14 150 ......................................................................................................... Table 12: Estimated Parameters for 1983  1998 151 ........................................................................................ Table 13: Estimated Parameters for 1983 152 1986.. .....................................................................................
Table 14:EstimatedParametersfor 1987 1990 153 ....................................................................................... Table 15: Estimated Parameters for 1991  1994 154 ....................................................................................... Table 16: Estimated Parameters for 1995  1998 155 ....................................................................................... z £' 82 Table 17: 12`horder LjungBox statistics for and ..... ......................... Table 18: Model Rankings for the AIC Metric (Excluding EGARCH) Table 19: Model Rankings for AIC Statistics (Including EGARCH)
......................................................
.........................................................
Table 20: Aggregated Rankings for AIC Statistics (Including EGARCH)
157 157
................................................. 158
Table 21: Model Rankings for MMEU and MMEO Criteria........ 158 ..............................................................
Table 22: Model Rankingsfor HMSE, GMLEand LL Criteria
159 ....................... ...........................................
Table 23: Model Rankings for MSE, MAE and MAPE Criteria Table 24: Aggregated Rankings for Statistical Loss Functions
................................................................
...................................................................
160 161
Table 25: Correlations Between OutofSample Buy and Sell Signals 162 ........................................................ xv
Table 26: Statistics for Forecasts of Volatility Changes 162 ............................................................................. 163 Table 27: Beforetransactionscosts Statistics for 19851986 without Filter .............................................. 163 Table 28: Beforetransactionscosts Statistics for 19891990 without Filter .............................................. 164 Table 29: Beforetransactionscosts Statistics for 19931994 without Filter .............................................. 164 Table 30: Beforetransactionscosts Statistics for 19971998 without Filter .............................................. 165 Table 31: Beforetransactionscosts Statistics for 19851986 with ± 3% Delta Filter .............................. 165 Table 32: Beforetransactionscosts Statistics for 19891990 with ± 3% Delta Filter .............................. 166 Table 33: Beforetransactionscosts Statistics for 19931994 with ± 3% Delta Filter .............................. 166 Table 34: Beforetransactionscosts Statistics for 19971998 with ± 3% Delta Filter .............................. Table 35: Beforetransactionscosts Statistics for 19971998 with ± 3% Delta Filter (Excluding One Data Point)
..........................................................................................................................................
167
167 Table 36: Aftertransactionscosts Statistics for 19851986 with ± 3% Delta Filter ................................ 168 Table 37: Aftertransactionscosts Statistics for 19891990 with ± 3% Delta Filter ................................ 168 Table 38: Aftertransactionscosts Statistics for 19931994 with ± 3% Delta Filter ................................ Table 39: Aftertransactionscosts Statistics for 19971998 with ± 3% Delta Filter (Excluding One Data Point)
..........................................................................................................................................
Table 40: Basic Statistics for S&P 500 Index Options
...............................................................................
169 210
Table 41: Descriptive Statistics for r ........................................................................................................... 210 213 Table 42: Contract Specifications for Variance Swaps ............................................................................... Table 43: Calibrated Parameters for Stochastic Volatility Model
...............................................................
222
223 for Stochastic Parameters Volatility Calibrated Model 44: Jump Table with ............................................. Table 45: Estimated Parameters for Ad Hoc BlackScholes Model
223 ...........................................................
224 for EGARCH Parameters Estimated Table 46: ........................................................................................... Table 47: Estimated Parameters for GARCHVariance Swap
....................................................................
225
227 500 Index S&P Variation 48: Table of ........................................................................................................ Table 49: Delivery Prices for 3M, 6M and 9M Variance Swap Contracts ................................................. 229 Table 50: Aggregate MeanSquare Price Errors and Model Rankings for 3M, 6M and 9M Variance Swap 229 Contracts .................................................................................................................................... Table 51: InSample Fit (SPSE) for Option pricing Models ....................................................................... 230 xvi
Table 52: Estimated & Implied Structural Parameters 232 ................................................................................
xvii
Figures Figure 1: Hentschel's Framework
.................................................................................................................
42
92 Figure 2: S&P500 Futures & Returns: 19831998 ......................................................................................... 106 Figure 3: Call Maturity = 21 70 Days ........................................................................................................ 106 Figure 4: Call Maturity = 71  120 Days ..................................................................................................... 107 Figure 5: Call Maturity = 121  170 Days .................................................................................................. 107 Figure 6: Call Maturity = 171 220 Days .................................................................................................. Figure 7: Call Maturity = 221+ Days .......................................................................................................... 108 Figure 8: Put Maturity = 21 70 Days ........................................................................................................ 108 Figure 9: Put Maturity = 71  120 Days ...................................................................................................... 109 109 Figure 10: Put Maturity = 121  170 Days .................................................................................................. Figure 11: Put Maturity = 171 220 Days ........................ 110 Figure 12: Put Maturity = 221+ Days ......................................................................................................... 111 Figure 13: 2170 Calls with SixthOrder Polynomial and Linear Trend ..................................................... 111 Figure 14: 2170 Puts with SixthOrder Polynomial and Linear Trend ...................................................... 112 Figure 15: Mean Implied Volatilities and Least SquaresFit for 21 70 Calls ........................................... 112 Figure 16: Mean Implied Volatilities and Least SquaresFit for 21 70 Puts ............................................ Figure 17: 0% Skewness Premium .............................................................................................................. 113 Figure 18: 4% Skewness Premium
..............................................................................................................
113
Figure 19: Autocorrelations for r: 19831998 ............................................................................................. 170 170 for 19831998 20: Autocorrelations Figure r2: ............................................. ............................................... Figure 21: Autocorrelations for Irl: 19831998 ........................................................................................... 170 Figure 22: Autocorrelations for r. 19831986 ............................................................................................. 171 Figure 23: Autocorrelations for r2: 19831986..............:............................................................................. 171 Figure 24: Autocorrelations for Irk: 19831986 ........................................................................................... 171 172 for 19871990 Autocorrelations 25: Figure r: ............................................................................................. Figure 26: Autocorrelations for r2: 19871990............................................................................................ 172 xviii
Figure 27: Autocorrelations for Irl: 19871990
172 ...........................................................................................
Figure 28: Autocorrelations for r: 19911994 173 ............................................................................................. Figure 29: Autocorrelations for r2: 19911994 173 ............................................................................................ Figure 30: Autocorrelations for Irl: 19911994
...........................................................................................
173
Figure 31: Autocorrelations for r: 19951998 174 ............................................................................................ Figure 32: Autocorrelations for r2: 19951998 174 ........................................................................................... Figure 33: Autocorrelations for Irl: 19951998
174 ...........................................................................................
Figure 34: 3State SWARCHL(3,2)  High Volatility Regions 175 ................................................................ Figure 35: 3State SWARCHL(3,2)  Low Volatility Regions 175 ................................................................. Figure 36: Cumulative Rate of Return From Straddles Trading (19931994) With 25 bps Transactions Costs 3% ± Delta Filter 176 and .............................................................................................................. Figure 37: Volatility vs. Variance Swap Payoffs Long 180 ............................................................................. Figure 38: Vega of Individual Strikes: 80,100,120
...................................................................................
Figure 39: Sum of the Vega contributions of Individual Strikes: 80,100,120
...........................................
186 186
Figure 40: Vega of Individual Strikes: 60 to 140 spaced 10 apart 186 .............................................................. Figure 41: Sum of the Vega contributions of Individual Strikes: 60 to 140 spaced 10 apart 186 ...................... 211 Figure 42: S&P 500 index and Returns: 19992002 ................................................................................... Figure 43: Realised Forward Variances
......................................................................................................
212
218 Figure 44: Calibrated Local Volatility Surfaces for Absolute Diffusion Process ........................................ Figure 45: Future Realised Variances for 3M, 6M & 9M Variance Swap
.................................................
220
221 Figure 46: TermStructure of Implied Volatility ......................................................................................... 226 Figure 47: Calibrated Local Volatility Surfaces ..........................................................................................
xix
Acknowledgements I would like to take this opportunity to express my sincere gratitude to my supervisor, Dr. Yannis Hatgioannides, for his invaluable guidance and helpful suggestions throughout the PhD process. I am grateful to him for the bursary I received at the initial stage of my studies for Mathematical N. Centre Finance CASS Business School, Dimitris the the the to at at and Chorafas Foundation for awarding me a scholorship in support of my PhD research. I would also like to thank Ron de Braber, Gordon Fong, Matt Jaume, Peter Nolan, Maurizio Pietrini and Sharon Woolf for their stimulating input, and Philipp Jokisch for introducing me to Cantor FitzgeraldleSpeed, thus adding a new chapter to my career in finance. I am also indebted to Demetri Chryssanthakopoulos for his computing support, and John Dillon and Tom Ley for reviewing some of the chapters.
Finally, I wish to add my appreciation to
Thierry Vongphanith for his professional advice and editorial assistance in preparing this manuscript. Special thanks are due to my family, in particular to my parents, and also to my sisters and have dissertation this come whose support not and encouragement would cousins, without into existence.
xx
Declaration I grant powers of discretion to the University Librarian to allow this dissertation to be copied in whole or in parts without further reference to me. This permission covers only single copies made for study purposes,subject to normal conditions of acknowledgement.
i
xxi
Abstract This dissertation studies the patterns of termstructure of implied volatility and examines the performance of different specifications of timeseries and optionsbasedvolatility forecasting models under the influence of the observed market biases. Our researchis basedprimarily upon the use of S&P 500 data for the period 19822002. There are three selfcontainedbut seemingly related projects in this dissertation. The objectives of this researchare: 1) to characterisethe termstructureof implied volatility; 2) to compare the performance of asymmetric power ARCH and EGARCH models; 3) to evaluate the forecasting performance of timeseries and optionsbased variance swap valuation models. The observedmarket anomalies in the termstructure of implied volatility of S&P 500 futures options are investigated between 1983and 1998. Termstructure evidence indicates that shortterm options are most severely mispriced by the BlackScholes formula. We find evidence that option prices are not consistent with the rational expectations under a meanreverting volatility process. In addition, skewnesspremiums results show that the degreesof anomalies in the S&P 500 options market have been gradually worsening since around 1987. As correlation may be responsible for skewness,our diagnostics suggest that leverage and jumpdiffusion models are more appropriate for capturing the observedbiasesin the S&P 500 futures options market. Sixteen years of daily S&P 500 futures series are employed to examine the performance of the APARCH models that use asymmetric parameterisation and power transformation on conditional volatility and its absolute residual to account for the slow decay in returns autocorrelations. No evidence can be found supporting the relatively complex APARCH models. Loglikelihood ratio tests confirm that power transformation and asymmetric parameterisationare not effective in characterising the returns dynamics within the context of APARCH specifications. Furthermore, results of a 3state is the that the support models model notion regimeswitching performance of conditional volatility EGARCH In AIC the the that to of volatility of state returns series. addition, prone statistics stipulate is best in "noisy" periods whilst GARCH is the top performer in "quiet" periods. Overall, aggregated for the AIC metric show that the EGARCH model is best. Optionsbasedvolatility trading rankings in EGARCH GARCH that profit and can generatestatistically significant exante exercisesalso reveal one out of four sample periods after transactions costs. When considering a stochastic volatility for be little incentive look beyond to volatility there to seems model, a simple model which allows leverage a effect. and clustering The volatility forecasting performance of different specifications of timeseries and optionsbased before S&P to 500 index is from after the three valuation models on months swap variance evaluated is framework By far, Demeterfi 9/11 the (1999) the optionbased attacks. et al. varianceswap valuation the most popular tool to price varianceswaps. This framework stipulates that pricing a variance swap in the implied be options the of as an exercise computing the weighted volatility viewed average of can influence the of volatility skew. Our research design offers a comprehensive under required even from Based six the on results relative merits of competing option pricing models. empirical study of illustrate future days, implied that variance and we contract chosen carefully models may overpredict The timeseries models. reasons could be: 1) the implied strategy was originally underperform developed for hedging; 2) implied volatility is predominantly a monotonically decreasingfunction of maturity and therefore optionsbasedstrategy cannot produce enough variance termstructurepatterns; 3) distributional dynamics implied by option parametersis not consistent with its timeseries data as likelihood to by Future the needs the research maximum estimation of squarerootprocess. stipulated findings. justify in larger to to our sampleset order establish a more statistically significant result use a Until then we have a strong reservation about the use of Demeterfi et al. methodology for variance forecasting.
xxii
To My Parents
Chapter 1: Introduction to the Study
CHAPTER I
Introduction to the Study
"Learning without thought is labour lost; thought without learning is perilous. " Confucius 
1.1 Introduction This dissertation is a quantitative
study whose primary
objective is to investigate the
forecasting different timeseries specifications of of and optionsbased volatility performance models under the influence of the observed market biases in the S&P 500 markets.
Our
is based primarily upon the use of futures, futures on options and index options work research data for the period 19822002. This first chapter of the dissertation introduces the background of the study, specifies the problems of the study and describes its significance.
The chapter
by dissertation. the the outlining structure of concludes
1.2 Background of the Study Volatility
of the underlying asset price is the primary determinant of option prices and many
instruments. derivatives related
An option pricing model that does not properly capture the
do with well that agree to not processes volatility can give of rise option prices evolution BlackThe in hedge investor's the to risk. market and can also reduce ability prices observed Scholes option pricing model is commonly used to price a wide range of options contract. known behaviour its is as documented, However, erratic empirical a phenomenon well "volatility
1992). (e. MacBeth Bollerslev 1979; Rubinstein, 1985; al., et g. et al., smile"
Contrary to the basic assumptions of the BlackScholes formula, implied volatility exhibits leverage the factors Many both smile effects and termstructure patterns. such as market have industrial etc cycles, serial correlated news arrival, market psychology effect, taxing, in biases in these the marketplace. causing observed roles crucial played very
As a result,
have to is dynamics yet distribution to the adequate specify researchers not and returns normal deal with fattails and excess kurtosis which form the basis of smile effects. Below we will
1
Chapter 1: Introduction to the Study
briefly discuss the three areas of interest in this dissertation, namely, the termstructure of volatility, modelling of volatility and volatility derivatives.
1.2.1 TermStructure of Volatility The modelling of the termstructure of implied
volatility
has been discussed by many
researchers, e.g. Rubinstein (1985), Stein (1989), Diz and Finucane (1993), Heynen, Kemna, and Vorst (1994) and Xu and Taylor (1994).
Rubinstein (1985) documented that implied
volatility of exchange traded call options between August 1976 and August 1978 exhibited a systematic pattern with respect to different maturities and exercise prices. Rubinstein's most intriguing result was that the direction of bias changed signs between subperiods, implying that skewness of the riskneutral density changed over time. Subsequently, numerous efforts have been made to investigate the meanreverting process and termstructure of implied volatility.
Stein (1989) pioneered the examination of the termstructure of the average atthe
implied volatility using two maturities on S&P 100 index options. By using a money options' meanreverting volatility
model, evidence suggested that longmaturity
"overreact" to changes in the implied volatility
options tended to
investors because of shortmaturity options
had a systematic tendency to overemphasise recent data at the expense of other information when making projections.
This result was disputed by Diz and Finucane (1993) following
their analysis of similar S&P 100 index data. The termstructure of implied volatility has also been discussed by Heynen, Kemna and Vorst (1994). Basing their results upon Duan (1995), Heynen et al. derived the termstructures of implied volatility for EGARCH, GARCH and a in timetoStein Only (1989). two of to values stochastic model a similar way meanreverting best investigated Heynen the EGARCH gave and et al. concluded that maturity were description of asset prices of the termstructure of implied volatility.
Xu and Taylor (1994)
also studied atthemoney currency options and used a meanreverting volatility
model to
for between longimplied any volatility and shortterm expectations of establish relationships implied Xu 's T. of the et al. model could explain timevarying crossovers number of maturity volatility
at different maturities but it did not emphasise the effects of volatility
smile.
Surprisingly little research has been done on the properties and evolution of implied volatility. Past research has mainly focused on "fitting"
biases the theoretical to observed a option model
in a particular options market from an arbitrarily short span of data for atthemoney contracts. Since the termstructure of implied volatility reflects the timevarying market expectations of
2
Chapter 1: Introduction to the Study
asset volatility over different time horizons, it is imperative to focus on a single market and gain a thorough understanding of its behaviour.
1.2.2 Modelling of Volatility Since the late 1980's many researchers have developed alternative optionpricing models in order to cope with the observed termstructure biases in the equity market. The latest onefactor implied models such as Derman and Kani (1994), Rubinstein (1994) and Dupire (1994) have created specifications that can implicitly
model volatility as a deterministic function of
time. However the major setback for "implied"
methods is that they all require substantial
"engineering" efforts to calibrate their lattice structures. These complex models are usually for the valuation of exotic options and are seldom used for volatility forecasts. On reserved the other hand, a more structural approach to improving the forecasting performance is to model volatility as a timevarying stochastic variable. Whilst stochastic models such as Hull and White (1987), Johnson and Shanno (1987), Scott (1987) and Stein and Stein (1991) provide another means to capture smile effects, many problems limit
the use of these
stochastic volatility models. The main problem associated with stochastic volatility models is that volatility
is not a traded asset and is therefore unobservable. models' parameters are problematic as realworld
continuoustime
Besides, estimations of data are recorded at
discrete intervals. 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 in increases larger by followed returns are strongly asymmetric, e.g. negative returns are large 3) 2) fattailed; than equity returns equally positive returns; volatility equity returns are (persistence refers to volatility clustering). persistent are
This class of discretetime models
hypothesises that both smile effects and termstructure patterns can be explained by allowing the underlying asset's volatility to obey a stochastic process. There is a voluminous literature discrete timevarying that suggesting
volatility
insightful. and models are practical
The
is implemented, ARCH is that such modelling volatility of readily predictable and usefulness ARCH models assume the presence of a serially correlated news arrival process and require As data. ARCH such, the of past models allow conditional variance to change over use only
3
Chapter 1: Introduction to the Study
time as a function of past conditional variance constant.
variance and error, whilst leaving unconditional
Most of the early research efforts focused on conditional models that
imposed symmetry on the conditional variance structure. In response to criticisms that the symmetric model may not be appropriate for modelling stock returns volatility, more recent in features leverage has transformation etc considered other such as effects, power research 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 models is based on the observations of the socalled "stylised facts". For example, Ding et al. (1993) invented the APARCH models that used the BoxCox transformation on conditional in its for decay the to the absolute residual account and variance slow of autocorrelations returns process. Subsequently, many researchers have also developed different specifications for the longmemory process (e.g. Baille, 1996; Bollerslev et al., 1996; Ding et al., 1996). Several papers have given the impression that their models are capable of accounting for empirical features such as volatility clustering and leptokurtosis in the distribution of returns. Despite the huge amount of effort researchers has put into modelling volatility, it is clear that for issues remain unexplored many of these more "elaborate" models. empirical
1.2.3 Volatility Derivatives Until now the conventional instruments for implementing a volatility
hedge remain rather
through The is most widely accepted achieved way of speculating usually crude. on volatility the purchase of European call and put options. Traditional techniques such as delta hedging focus deltarisk. the reduction on of always strategy
Once the underlying index moves,
however, a deltaneutral trade can become long or short delta. Rehedging becomes necessary to maintain a deltaneutral position as the market moves. Since transaction and operational costs generally prohibit continuous rehedging, residual exposure of the underlying ultimately been has from fact Despite there an that optionsbased volatility strategies. the arises increased interest in volatility products since the late 1990's, little research has been directed towards to the development of volatility
derivatives.
The first theoretical paper to value
but is by Grünbichler (1996). derivatives Grünbichler a simple et al. et al. presented volatility
4
Chapter 1: Introduction to the Study
technically complicated framework that used the equilibrium approach within which specific closedform solutions for volatility
futures and option prices were derived within a mean
reversion framework. 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 (2001), Heston Nandi Little Pant (2000), (2000b), Howison (2001), Long and and and et al. Carr and Madan (1999,2002), Javaheri et al. (2002) and Theoret et al. (2002) also researched in but derivatives, in invested the still pales amount products of research volatility volatility comparison with other wellstudied exotic derivatives products such as barrier and Asian options.
Volatility
risk has yet to be dealt with so that investors and traders can directly
express their views on future volatility. The arrival of variance swaps offers an opportunity for traders to take synthetic positions in volatility and hedge volatility risk. They were first introduced in 1998 in the aftermath of the Long Term Capital Management (LTCM)
melt down when implied stock index volatility
levels rose to unprecedented levels. These variance swap contracts are mostly based on equity indices and they were originally designed to be a replacement for traditional optionsbased its hedged Despite such as straddle name, a variance or strategies call/put volatility options. swap is actually an overthecounter forward contract whose payoff is based on the realised index. Their stated a equity of payoff at expiration is equal to: volatility /22 \ýR

Kvd)
N
is in N the the amount notional of swap some currency units per annualisedvariance where days) (n K,, life quoted the the are realised stock volatility over the contract of point, QR and 2j,
in annual term, i. e.
FI
n1
n ;to
S'+' S;
Sj
delivery fixed price, the and annualised volatility
is F factor. the annualisation appropriate respectively. Since a variance swap provides pure exposure on future volatility cleaner bet on volatility
than optionsbased strategy.
levels, it is considered a
It allows counterparties to exchange
for to fixed floating Counterparties swap variance a variance variance. use can cashflows future to between (fixed) (floating) implied or the volatility, spread realised and speculate
5
Chapter 1: Introduction to the Study
hedge the volatility exposure of other positions or businesses. According to Curnutt (2000), someof the possible strategiesusing variance swapsare: i) Speculating a directional view that implied volatility is too high or too low relative to anticipated realised volatility because 1) volatility follows a meanreverting process. In this model, high volatility decreasesand low volatility increases; 2) there is a negative correlation between volatility and stock or index level. The volatility stays high after large downward moves in the market; 3) volatility increaseswith the risk and uncertainty; ii) Implementing a view that the implied volatility in one equity index is mispriced relative to the implied volatility in anotherequity index; iii)
Trading volatility on a forward basis by purchasing a variance swap of one expiration and a variance swap of another expiration.
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 is far, By (1999) developed by Demeterfi the most the model et al. volatility exposure. popular tool to price variance swaps but, surprisingly, no researchhas ever consideredusing for its framework data This to test usefulness. stipulates that pricing a variance swap market implied in be the volatility of the as an exercise computing weighted of viewed average can information Therefore influence the the options required even under of volatility skew. the filtered is be through in directly having to option prices used without embedded the is long As the continuous, timeseries. the as movement of underlying asset underlying the hedging is the of independent choice of of variance contracts completely pricing and volatility process.
1.3 The Problem Statement This dissertation investigates the performance of different specifications of timeseries and biases. forecasting influence models under the of the observed market optionsbased volatility In order to present the results in a meaningful and manageable manner, three selfcontained but interrelated projects are included in this dissertation.
6
In this section we will state the
Chapter 1: Introduction to the Study
objectives for each of the three projects separately.
We end the section by noting the
delimitations of the study.
1.3.1 Objectivesof the First ResearchProject Chapter 3, entitled "A report on the Properties of the TermStructure of S&P 500 Implied Volatility",
is a descriptive study. It examines the empirical behaviour of S&P 500 futures
option's implied volatility
using daily data from 1983 through 1998.
We consider this
500 implied S&P the of most volatility termone extensive empirical work studies of research structure in literature to date. The primary objectives are:
i)
To observe, characterise and analyse the patterns of the termstructure of implied 500 S&P in the marketplace; volatility
ii)
To investigate whether option prices are in line with the rational expectations hypothesis under a meanreverting volatility assumption;
iii)
To identify what types of option models would be consistent with the observed moneynessbiasesin the S&P 500 options market.
Intermediate results obtained in Chapter 3 can also help facilitate our research efforts in modelling volatility in Chapters 4 and 5.
1.3.2 Objectives of the Second Research Project Chapter 4, entitled "An Empirical
Comparison
Models", APARCH of
investigates the
for the APARCH (Ding slow 1993) that of models account et al., can potentially performance decay in returns autocorrelations using daily S&P 500 futures series from 1983 through 1998. The use of the APARCH framework is convenient to evaluate different model specifications because loglikelihoodbased statistics can be used to directly test for the robustness of many Our primary objectives are: models'. nested
i) To check whether the unrestricted APARCH model is a good description of the driving by investigating the significance of asymmetric volatility process APARCH transformation of the context parameterisation and power within specifications using loglikelihood ratio tests;
1 Seeappendix A. 1 for thesenestedmodels.
7
Chapter 1: Introduction to the Study
ii)
To provide evidence that the insample performance of asymmetrical and symmetrical conditional volatility models are prone to the state of volatility by using a 3state regime switching volatility
conditional model to separate high and low volatility
states;
iii)
To compare the insample performance of EGARCH (Nelson, 1991) with APARCH models basedon aggregateAIC statistics;
iv)
To illustrate the quality of different conditional volatility forecasts by predicting the onestep ahead changes of implied volatility and conducting exante (outofsample) S&P 500 straddle trading exercises.
1.3.3 Objectivesof the Third ResearchProject The title of Chapter 5 is "Empirical Models".
Performance of Alternative
Variance Swap Valuation
The model developed by Demeterfi et al. (1999) is the most popular tool to price
but surprisingly, no research has ever considered using market data to test for swaps, variance its usefulness in forecasting volatility.
The pricing of variance swap can be viewed as the market
consensusof expectedfuture variance. Chapter 5 examines different specifications of timeseries 500 S&P forecasting the variance swap models' on and optionsbased volatility performance 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 Based on six wellselected contract days, we design the three, six and ninemonth variance. different for day by contracts swap each them contract evaluating variance and analyse implied Our different in timeseries time. of and primary goals models at specifications points
are: i)
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;
ii) To explore whether there is any systematic difference in volatility forecasting between timeseries and optionsbasedvariance swap valuation models. performance It is intended to investigate whether optionsbasedmodels, which are forwardlooking, historical discretetime are capable of outperforming processes, which use only information, in predicting future variance.
8
Chapter 1: Introduction to the Study
1.3.4 Delimitations of the Study Volatility
models and their forecasts are of interest to many types of economic agents, e.g.
options traders require asset volatility
to price options whilst portfolio
volatility forecasts to access risks of their portfolio.
managers need
Having the ability to estimate volatility
from have than that trading activities. others means accurately one could success more more Given the changing nature of volatility termstructure in the marketplace, it is important for us to focus on a single market and gain a thorough understanding of its behaviour. If the termimplied of volatility shows any specific pattern then some models, such as stochastic structure volatility
for heteroskedasticity be to account models or autoregressive used models, may
these imperfections in the market. In this dissertation, we have opted for the use of S&P500 500 index S&P is The data. capitalisationweighted, representing the market value of market all outstanding common shares of the 500 largecapitalisation firms listed in the U. S.A. This is of importance to investors because S&P 500 products are one of the most liquid contracts in the financial world. Liquidity is the ability of a market to efficiently absorb the execution of large purchases and sales. It is a key component to attracting investors and ensuring a by fact, In S&P benchmark 500 index has long been the the which market's success. 500 S&P its immense that size guarantees professionals measure portfolio performance and hedging ideal tool. as a are products
1.4 The Significance of the Study We will explain the researchsignificance for each of the selfcontainedprojects individually.
1.4.1 Significance of the First ResearchProject The 3 in futures 500 S&P the Chapter studies observedmarket anomalies the options market. termstructure of implied volatility reflects the timevarying market expectations of asset different horizons. Despite the investigation time evidence the and over extensive volatility has far implied termstructure ever the thus on of volatility, no past study accumulated Prior S&P to large 500 implied the termstructure. study of empirical a volatility considered have always examined the termstructure of implied volatility only past papers this research, for particular atthemoney contracts. The purpose of Chapter 3 is to fill this gap in the
9
Chapter 1: Introduction
to the Study
literature by utilising all available daily S&P 500 futures option prices from the inception of S&P 500 futures option in March 1983 to December 1998. Although descriptive in nature, we extend previous termstructure work in several ways:
i) The new aspectof this researchis that we define relative implied volatility as implied for by its implied each normalised corresponding volatility volatility atthemoney maturity group. The use of relative implied volatility allows the measurementof broad degrees in implied the termstructure of anomaly relative across a volatility moneynessrange; ii) Our sampleperiod is more extensive,making the results more statistically reliable. Our researchis of importance to institutional investors becauseS&P 500 products are one of the most liquid contractsin the financial world and their immensesize guaranteesthat they are ideal as a hedging tool. If the termstructureof implied volatility shows any specific patterns then some models, such as stochasticvolatility models or GARCHtype models, may be more for Blackby imperfections be to the that adjustments make market suitable cannot explained These adjustments could be important even for small levels of for longer especially maturity options. predictability, Scholes formula.
1.4.2 Significance of the SecondResearch Project In Chapter4 we comparethe performanceof the asymmetric power ARCH (Ding et al., 1993) EGARCH (Nelson, favours 1991) literature The the somerather with model. existing models little is but explored. their specifications usually empirical performance complex volatility Since the development of longmemory models in the early 1990's, there has been little investigate the 4 Chapter In the their we significance of specifications. research about importance of power transformation and asymmetric parameterisationwithin the context of APARCH specifications. The consequenceof this researchis not only significant to discretebut finance also potentially meaningful for continuoustime stochastic volatility time literature. Whilst the research on discrete and continuoustime models has evolved independently, many continuoustime models can be thought of as the limits of GARCHtype Wiggins (1991) Nelson For the to EGARCH(1,1) that example, showed converged processes. in limit. the Moreover, (1987) time that Duan (1997) of most continuous proved also model diffusion had been bivariate that models used to model assetreturns volatility could existing be representedas limits of a family of GARCH models. If it can be shown that there is not 10
Chapter 1: Introduction to the Study
much to gain from moving beyond a more parsimonious discrete specification such as EGARCH or GARCH, there seemsto be little incentive to look beyond a simple bivariate stochasticmodel which allows for volatility clustering and a leverageeffect such as the HullWhite model (1987) or the Heston model (1993).
1.4.3 Significance of the Third ResearchProject Chapter 5 inspects the pricing performance of optionsbased and timeseries variance swap valuation models on the S&P 500 index.
Variance swap is an exciting new product that
immunises traders' exposure into the ups and downs of volatility.
It is getting more popular
because it is one of the very few financial products to allow traders speculate on future volatility
levels. The Demeterfi et al. (1999) variance swap pricing methodology has been
but by little tested and scrutinised. practitioners widely accepted
Regrettably, no empirical
studies have ever used any market data to investigate the pricing performance of variance This models. research presents the first of any known attempts to use market swap valuation data to evaluate the effectiveness of the Demeterfi et al. framework.
In particular, it
first the study on variance swaps under alternative timeseries and competing represents It is known models. also not pricing whether and by how much each option model will option improve variance swap pricing.
Since implied volatility
can be regarded as the market's
the implication of any poor variance forecasting by for look is to that a such practitioners and academicians alike may need optionsbased models
future of realised volatility, expectation
historical integrate to and market information in a composite option pricing model. way
1.5 Organisation of the Dissertation Chapter 1 is the introduction. Chapter 2 but literature. The dissertation into is divided this three selfcontained rest of reviews the interrelated projects and each project is accompaniedby an abstract. Chapter 3 characterises The structure of this dissertation is as follows.
the termstructure of S&P500 implied volatility and examines empirical issues relating to 4 Chapter hypotheses distributional in S&P 500 futures the options market. rational and different in insample APARCH EGARCH the the performance of models with compares the daily It S&P futures 500 assesses sixteen years using of also series. volatility regimes the different to select criteria statistical and conducts approach of a preferencefree quality 11
Chapter 1: Introduction to the Study
best outofsample model. Chapter 5 adopts a financial engineering approach to evaluate the performance of different timeseries and optionsbased variance swap valuation models on the S&P500 index under the influence of termstructure biases found in Chapters 3 and 4. Chapter 6 summarises and discusses the results and suggests directions for future research.
12
Chapter 2: Review of the Literature
CHAPTER 2
Review of the Literature
This chapter will review the literature on issues related to option pricing, as a means of providing an intellectual background for the present dissertation. It will examine both the theoretical and empirical studies in these areas, giving special emphasis in volatility. The chapter organisesthe review by examining the six aspectsof finance literature: option pricing theories, conditional and stochasticvolatility models, implied methodology, market anomalies and diagnostic tests.
2.1 Option Pricing Theories 2.1.1 NoArbitrage Approach The BlackScholes option pricing formula (Black & Scholes, 1973) relates the price of an option to the underlying asset price, the volatility of the return of the underlying asset, dividend yield, interest rate and strike price. The main assumptions that BlackScholes following: the were proposed i) Markets are frictionless, efficient and complete; ii) Constantinterest rate2and volatility; iii)
Portfolio rebalancedcontinuously;
iv) Noarbitrage and tradesare selffinancing; follows S The asset underlying geometric Brownian motion3 (GBM). v) The underlyinng assetdynamics is given by: dS =/ Sdt + oSdW
2 The original BlackScholes paper assumesa constant interest rate. But this assumptioncan be relaxed and nolong interest be as applied as rate is deterministic. still arbitrage can 3 Unlike arithmetic Brownian, geometric Brownian motion does become to asset the not allow underlying limited is liability of stock ownership (Samuelson1965). to that consistent negative,a property
13
Chapter 2: Review of the Literature
where the percentage change from t to t+dt is normally distributed with mean ,udt and variance a2 dt; W is the Wiener process, and u and a are the instantaneous return drift and
volatility, respectively.
2.1.1.1 BlackScholesFormula According to the BlackScholes assumptions, one can apply Ito's lemma4 to show that it is hedged to synthetic create a portfolio possible in option. position a short and
v(S, t) that consists of a long position in stock
If rebalanced continuously, this hedged position can be
achieved independent of stock price movements and its instantaneous return drift.
The
discretetime version of the diffusion model is given by: AS = PSet + aSAW
(1)
The above discretetime relationship involves a small approximation. It assumesthat the drift S in In discrete time constant rate of remain a addition, the variance very short and period. is in the the of value option governedby the stochasticdifferential equation (SDE) that change lemma: Ito's the satisfies AC=
CS,uS+Ct+ýCSSQ2S2 At+CSQSOW
(2)
Css first derivative Cs is the C, the time, partial to and of option and price with respect where first and second partial derivative of option price with respect to stock price, respectively. In Cs long this hedged of value of the of shares and portfolio change one call option short, a discrete is: in time small period a portfolio
Ov(S, t) = CSASAC
(3)
Substituting (1) and (2) into (3), one gets: A v(S, t) =CtOt 2 CssQ zSsOt
(4)
4 SeeHull (2000) pp.235236 for the derivation of Ito's Lemma.
14
Chapter 2: Review of the Literature
Since the increments of the portfolio
are dependent on the same source of underlying
uncertainty, it is possible to form a riskfree portfolio in discrete time. Under noarbitrages condition, the return earned on it in a short discrete period must equal risk free rate r so that:
(5)
Ov(S,t) = rv(S, t)Ot Substituting (5) into (4) one can write the following SDE: 2 CSSQ2S C1 +ý rC = rCSS +
If O=
ac
ac
,Q= T,
as
and r=
azC
as
(6)
(6) can be rewritten as: equation ,
1I'Q2S2 2
rC=rOS+O+
(7)
The most striking feature of the BlackScholes derivation is that equations (6) and (7) are independent of instantaneousstock return p; one only needs to know the riskfree rate in heat backout C. By into transfer (6) the transforming to option price an equivalent order The it be in boundary to can conditions. physics, solved analytically subject problem Europeancall formula is expressedas follows6: C(S, t) = SN(d, ) Xe" N(d. ) a
f. 41
1n() (rd X2
+12Q)(T t)
Q Tt d2=d,
Q
Tt
hedge ) N(d, the distribution function, is N(") the cumulative r the riskfree rate, where T d the dividend S X the and the the price the stock volatility, yield, strike price, parameter, o maturity. In the BlackScholes options model, prices are always a nondecreasing function of the in is distributed Furthermore, lognormally. distribution The change of stock price volatility. ln(S) betweentime 0 and T is given by:
SThe noarbitragetheorem simply statesthat two equivalent assetsmust not be sold for different prices. 6 Wilmott (1997) give precisedetails in solving equation (6).
15
Chapter 2: Review of the Literature
d ln(S) _pSr
In
So
Jdt 2
+ QdW
1( 4)raff] p N i Q2
ln(S,.)N
ln(So)+
Ju 2
ICV[
I
The distributional result implies that the expected continuously compound return for In(S) is 2per
AS year whereas
4u2
is distributed as N(, utt, Q At) Under the real probability .
measure,the expectedforward price and its instantaneousvalue at time Tare given by: E(ST) = Soe"T ST

soe
(PZo2)T+oýe,
+
Er

N(0,1)
It is widely noted that option prices are not priced off the real measure but riskneutral measure. According to Merton (1976), option prices rely on putcall parity to enforce the internal consistency of option pricing.
The putcall parity is a noarbitrage condition which
European the that a value of call option with a certain exercise price and exercise date shows from deduced be the value of a European put option with the same exercise price and can date, and vice versa. For a nondividend paying stock, this relationship is given by:
CP=SXe"' If putcall parity is violated, arbitrage will arise. Note that putcall parity is true regardless distribution is lognormal. It does not, however, hold for American the price asset whether formula instant BlackScholes The be can rearrangedsuch that a stock option at any options. can be thought of a weighted portfolio of risky stock and riskless zerocouponbonds: C(S, t) = e [SN(d, )e"  XN(d )] Z N(d2) can be interpreted as the probability that the option will be exercisedin a riskneutral is XN(d2) the strike price times the probability that the strike price will be world whereas if SN(di)e" S. is Accordingly, the term the that equals expected value of a variable paid. Sr >X and zero otherwise in a riskneutral world. 16
Chapter 2: Review of the Literature
2.1.1.2 Other Variations There are many variations of the BlackScholes model  so many that it is cumbersome to BlackScholes (1973) derived it. For instance, Merton the survey of provide a comprehensive formula independently based on a threeasset riskless hedge model. Merton's model had the to developed be deterministic. Merton's interest taken that also paper rate was advantage such a set of restrictions for the rational pricing of European and American options without making any distributional
assumption and gave the solutions to perpetual American call and put
include jumpdiffusion Other the option pricing models prominent options.
model by Merton
(1976), the futures option model by Black (1976), the compound option formula by Geske (1979), the American
option pricing
model by BaroneAdesi
and Whaley (1986), the
is by by Hull White (1987) list This and no means exhaustive. model etc. stochastic volatility The use of any particular model should be judged on its own merits.
2.1.1.3 Implied Volatility data Historical formula but is BlackScholes In the the volatility parameter a all observable. be to however, be to employed techniques a, also estimate could many other used may & Corrado 1995; (e. Brenner & Subrahmanyam, 1988; Bharadia g. et al., approximate a Miller, 1996). On ther other hand, one may observe the market price of the option and invert the BlackScholes formula to determine a.
This market's assessment of the underlying is life the option, of asset's volatility, which reflects the average volatility over the remaining Newtonby Its is implied using known as volatility. calculation usually accomplished C', Raphson method, which uses the first derivative of the option price with respect to a,
to
According Figlewski (1989a), option implied an of to the volatility convergence. speed up believed that between prices demand. is It the supply equilibrium and generally will represent in the market reflect all available information affecting the value of a contract. In principal, implied If direct future implied volatility gives a reading of the market's volatility estimate.
7 This is referred to vega. Calculating vega from the BlackScholes model may seem strange becauseBlackScholesequation assumesthat volatility is constant. It would be theoretically and conceptually more correct to is be from to volatility where assumed a model stochastic. calculate vega
17
Chapter 2: Review of the Literature
volatility is low compared to volatility forecast, a trader will prefer to buy options, and vice versa.
2.1.2 Martingale Approach The essenceof the Martingale approachis to changethe probability measureso as to make the discounted stock a Martingale, therefore making its drift zero. The option price can be expressedas the discounted value of the expected cashflow under the riskneutral measure. Furthermore, the BlackScholes formula can also be obtained from the Martingale approach. The following subsectionsdiscuss the underlying concept of the probabilistic approachand illustrate how it can be used to solve for option prices8. 2.1.2.1 Underlying Concepts The BlackScholes formula can be derived via the probabilistic approach. Mathematicians have known for a while that to be random is not necessarily to be without some internal structure in nonrandom ways. The central theme of the probabilistic approach demonstrates that the arbitrage justified
contingent claim is the expectation of the discounted claim under
Q measure under which the discounted underlying process is also a Martingale. one special Under the Martingale measure Q, derivatives can be valued with the riskfree rate via noP is the in Thus the riskfree the measure rate real where readily available arbitrage, market. in follows Q is irrelevant. The have place the to underlying which necessity a new measure for asset valuation can best be illustrated in the following example. Suppose an analyst would like to calculate the price of an asset. One way to do this is to exploit the relation: E`r
1 (1 +R) t
SM
Sr =
By doing this requires a knowledge of the
by calculating the expectation on the LHS. distribution r=R,
. a.
of R, which requires knowing Yet it is usually difficult
the risk premium
U where riskfree rate
to obtain the risk premium before knowing the asset
8 The
(1996) for in Rennie the Baxter this from section probabilistic and used approach materials are extracted and Neftci (1996).
18
Chapter 2: Review of the Literature
price. On the other hand, it might be easierto transform the mean of R, without having to use the risk premium. If one can find a new probability measureQ without having to use the risk premium such that: 11 E`Q[(1ý,
)5l+l
=S,
for be it calculating the assetprice. where can very useful The above illustration implies that there is a separationof process and measureand only the its interrelation underlying movements affects the prices of derivatives, but the of size and does them achieving of not. probabilities
For example, the forward contract on stock
be is T but forward fair time the the enforceable9, may not contract price of maturing at S, exp(rT), which does not dependon the expectedvalue of the stock under its real measure.
2.1.2.2 DiscreteTimeProcess The use of probabilistic approachincludes the conceptsof Martingale, filtration F,, stock and bond processes. A stock processS is a Martingale with respect to an arbitrary measureP and a filtration F, if. EP(S1 I F, )= Si for all i: 5 j.
That meansthe processS has no drift under P, no bias up or down in its value
Ep. the operator expectation under A filtration F, is the history of the stock up until ticktime i; filtration fixes a history of idea IF, The (" ) the EQ of conditional expectation extends operator choices or paths. Q Q history tells F,. The us two to measure parameters under measure a expectation a in determining "probabilities" to use pathprobability and thus the expectation whilst which from later take to filtration expectations starting point rather than along the whole the serves Coupled binomial from the time with zero. use of representationtheorem, a noof a path
9 The
Kolmogorov's from law is: S0 the 0.50.2 ) strong price + u stock exp(, expected
19
Chapter 2: Review of the Literature
arbitrage, selffinancing hedging strategy can be constructed to price contingent claim in a ) binomial environment. Given a binomial tree model with a stock S and bond B, then (O; gyp; , is a selffinancing strategy to construct a contingent claim X if. i) Both (0,, (pi) are known by time i1; ii)
The change in value V of the portfolio defined by the strategy obeys the difference equation: AV, = O;AS, + cp;AB; where AS, and AB, are the changes in S and B from time i Ito
iii)
cTST
i, respectively;
+DpTBT= X, the final claim.
i, Sl Thus Binomial representationtheorem assuresthat Oj+1S, O; B1. time + tp; + gyp; at any = +1Bj the value of a claim X maturing at date T is B1EQ(BT'X I F) . It is also noted that both B,'X `S, B, are QMartingales. and
2.1.2.3 ContinuousTimeProcess The discrete models are only a rough approximation to the way that prices actually move. The binary choice of a single jump "up" or "down" only becomesmore important as the ticks fine In in be continuous a closer. process, values can expressed arbitrarily get closer and fractions and they cannot make instantaneousjumps. Two special tools are used for manipulating stochasticprocesses: i) If
dX, = p, dt + a, dW,
and
f (X, )
d(f(X, ) = (a, f (X, ))dW, +(, u, f'(X, )+
1af
is
twice
differentiable
then
"(X, ))dt
ii) If dX, = U,dt +Q, dW, and dY, = v, dt + p, dW, then , d(X, Y,) = X, dY, + Yd Y, + a, p, dt i) is referred Ito's formula. Its most immediate use is to generateSDE's from a functional in LHS is ii) It is final for the the term product the on rule. a process. noted that expression ii) is actually dX, dY, (following from (dW,)2 = dt ), marking the difference between Newtonian and stochasticcalculus. The above equations are a manipulation of differentials of Brownian motion, not a W, is but Brownian in its not a strictly measure. a right, of own motion manipulation
20
Chapter 2: Review of the Literature
Brownian motion with respect to some measure P, a PBrownian motion. One important tool of measure is the RadonNikodym
for manipulation
derivative of Q with respect to P, i. e. and
operate
P(A) >0a define
on
same sample
exists only if two measures P and Q are equivalent space and
dP and
dQ
up to time t given Fs where t>s,
ii) iii)
agree on
what
is
possible,
if P and Q are equivalent and Q could be extracted from P and
dQ E,, (X) EQ For X example, versa. vice = dP
i)
The RadonNikodym
i. e.
Q(A) > 0, where A is any event in the sample space. One can only uniquely
dQ dP
the
dQ,
derivative.
dQ dP
and ,
dQ if X To claim exists. price a contingent dP
the procedures are as follows:
1 F. dQ
S, = EP
EQ(XT) = Ep
dQ
X,
EQ(X, JF, )=Sj'E,
for all claims knowable by time T. ,
(S,X, I F, ), s0 ? ao >0, a, /i=lpla,
The GARCH(p, q) model is given by:
j=ý
1Iri  N(O,h, )
,j
+jj=lýj
but between higher A be bigger for is and zero one than to may usually one or equal where order models.
2.2.2.2.1 Implications GARCH models allow for clustering of periods with high and low volatility. A GARCH(p,q) is analogous to an ARMA(p, q) representation. It reverts to a longrun mean and is leptokurtic. Both ARCH and GARCH impose restrictions on coefficients to ensurea positive ARCH(p) GARCH(p, to In If the the q) a process reduces process. q=0, addition, variance". degree for be ARCHpj) by to any a of accuracy approximated a stationary process can both j. Furthermore, ARCH large of value and GARCH models are symmetric sufficiently lag GARCH ARCH, for longer flexible Compared to allows a memory and a more models. be justified Finally, description. the therefore as a more parsimonious may and structure
12 For ARCH(p) and GARCH(1,1) the Bollerslev inequality constraints (nonnegativity of parameters) are derived for by Nelson Cao (1992), constraints was A some relaxing of more and allows set which sufficient. for higher GARCH. in order negativeparameters estimation
32
Chapter 2: Review of the Literature
EWMA
can be viewed
model
as a particular
case of
GARCH(1,1)
where
ao =0, a; =12, ß=A. The use of GARCH models is widespread. The GARCH(1,1) specification has proven to be for most financial time series. In order to understand the nature of an adequate representation persistence in variance under the GARCH(1,1) model one can write it as follows:
hl = ao +aleý +ßlh? 1 , h2 = ao + ýi.hz 1+ alv? 1 vt2 ll=
Ct2

ht 2
ý=(al+A) where v, is serially uncorrelatedwith mean zero. The parametersof GARCH are meaningful. al can be viewed as a "news" coefficient, with a higher value implying that recent news has a Engle impact changes. on price and Bollersleve (1986) shows that conditional greater kurtosis of a distribution of multistep returns dependsupon a,.
Higher a, implies higher
conditional kurtosis and the coefficient of kurtosis is K= 6a, (1 ß;  2a, ß,  3cri
which
is leptokurtic. Just as a, reflects the impact of recent news, ß, can be thought of as reflecting the impact of "old news", picking up the impact of news which arrived before yesterday (Antoniou and Holmes, 1995). If one believes that "old news" will have less impact on today's price A fall then relative to a,. should changes,
By repeatingly substituting v, into the
h? and eliminating equation the variance one can variance express conditional unconditional , as: h12 =Q2+a,
(v12 +Avl _,
z+Av;,
+...
)
a0 _2_
Q=
12
The above expressions make clear the dependence of the persistence of volatility shocks v, A. A If GARCH 1 from below the parameters, the the effects of past shocks on of sum on > A For become is the stronger. =1, process said to be integrated in variance current variance 1986). Bollerslev, In (Engle this case, shocks do not decay over time and IGARCH and or This does behaviour not exist. extreme variance of the IGARCH process may unconditional
33
Chapter 2: Review of the Literature
for because IGARCH its asset pricing attractiveness reduce
assumptions could make the
initial for longterm the to sensitive conditions. very contracts pricing
The GARCH(1,1)
be written as: also can model
ht h2 =(I_ 2)a2 +a, Eý ,+ß, 222222
ßl
(ý+ktl
hf+ka
= al +(al
E(h+k)=a2
a)+
+ßl)k(h2
(hr+k1
a)
a2)
1, the kday forecast will be stable as k increases. This variance forecasts
If (a, +A)
h, space. In this paper, Engle et at. pointed out that two broad decisions needed _, to be made: about the "shift/position" and "rotation/shape" of such a curve. The mapping framework is given by: E, = e,h, b) c(e, bI .f I , {Ie, , Ic(e, ß, b) }+ h, h, = ao + a, h, b , , , (eri
)
e,
where oo_0) such that S, can be written as: ' fo Sr = So+ asdW, +f pads Fprevisible processes such that random where or, p are
f(a
+I ks J)ds is finite for all
The differential form be I. the also of can above stochastic process times t with probability
written as: dS, = Q,dW, + p, dt
47
Chapter 2: Review of the Literature
The behaviour of St fluctuates around a straight line with slope U,.
The size of or,
determines the extent of the fluctuations around this line. In particular, these fluctuations do Given larger become time as passes. Q,,,u, and S,,, the process S is unique. not
2.3.1.3 StochasticDifferential Equation depend on W only through S,, such as a, =Q(S, t), the In the special case when o and ,u (SDE) for is differential S equation given by: stochastic
dS, =Q(S,, t)dW, +, u(S,, t)dt Regrettably, there are few soluble SDE's. One of them is geometric Brownian motion. The SDE for geometric Brownian motion is dS1 = S, (QdW, +, udt).
This setup gives asset prices
that fluctuate randomly around an exponential trend. Its solution is:
S, = Soexp((,u
2Q2)t+QWý)
2.3.1.4 OrnsteinUhlenbeckProcess The stochastic process a, is random and not observable. One of the most studied and celebratedcontinuoustime stochasticvolatility models is the OrnsteinUhlenbeckprocess: dS, / S, = adt + QdW, d(Ina) =A(ý1nQ)dt+W2 dW,dW2 = &It A for Wiener 8 W, is the longterm W2 the processes and the and correlation mean where ,ý the speedof the meanrevertingprocess. The continuous models are intrinsic
in understanding theoretical finance.
This model
introduces a correlation in the formulation of volatility process. In practical world, however, discretely. The discretetime traded are models are approximations of stocks or commodities their continuous counterparts.
The discretetime model of the corresponding continuous
process is:
48
Chapter 2: Review of the Literature
ln(S, )= ln(S, )+p+a, U, _, , ln(Q, )= a+ O[ln(Q, ) a)] + 9?7, _, 8. bivariate According to Taylor (1994), the U, 17, are normal with and correlation where in the mean equation of the discretetime process is the Euler , approximation of its continuous time counterpart. However, it can be argued that a more lagged volatility
q,
be: would simplification natural ln(S, ) = ln(S, ) +, u + a, U, _, Therefore, the main difference between the ARCH model and discretetime model is that the ARCH models' innovations depend on the past information discretetime stochastic volatility
set I,,
whilst in the case of
models, they are independent of the returns history I,
,. The ARCH models tell that past information can be used to predict the future but the discretetime stochastic volatility models imply that this information is irrelevant for future volatility.
2.3.2 HullWhite Model Continuous time stochastic volatility models endogenisethe volatility patterns and may be hedging. They in directly and valuation are largely theoretical and usually their used intensive. A computational wellknown stochastic volatility model is the are applications HullWhite model (1987). This model is basedupon the following continuoustime process: dS = SßSdt + aSdW, da2 =, u(xQ2)dt+ýQ2dWZ
ý 0, Wiener W. W,, U, K, and processes are constant. are where This model stipulatesthat the variance rate has a drift to pull it back to a level K at rate u. is the volatility of the volatility and it is possible to estimate ý by examining the changes in Since by implied prices. volatility option volatility
is not a traded asset, it is not possible to
form a hedge portfolio that eliminates all the risk.
If W, and W. are not correlated so that
volatility
is not correlated with stock price and the volatility
is uncorrelated with aggregate
i. (zero no risk risk, preferences, systematic e. constant risk premia), then the consumption
49
Chapter 2: Review of the Literature
HullWhite price is the mean BlackScholes price, evaluatedover the conditional distribution of averagevariance: C=ýc(V)g(V)dV
is is is BlackScholes V the the the the of value variance rate, average price and c g where riskneutral V in Furthermore, derived distribution Hull White of a world. an and probability based for Europeanstyle Taylorseries option on expansion. In this case,the solution analytic HullWhite model can be written as a combination of BlackScholes solution with adjustment Their main empirical result was that different "asymmetric" patterns could be ý by the/. [, and the sign of correlation parameters. Hull and White changing generated
terms.
European longerterm did had lower implied than that nearthemoney call options concluded it is Finally, GARCH(1,1) that the noted model can be written as a shorterterm options. discretetime approximation to the diffusion processof the HullWhite model. Hull and White attempted to use their model to explain Rubinstein's (1985) findings on the termstructure of implied volatility. But Rubinstein's results from comparing implied different in It times to to across consistent maturity. not was posit were necessary volatility the HullWhite model that, from one year to the next, the correlation between stock prices and No sign. reversed reason could be found to justify such a change of the associatedvolatility sign.
2.3.3 JohnsonShanno Model Johnson and Shanno (1987) applied an equilibrium approach to derive an option pricing between that the to changing explain sign volatility and of correlation attempted model and for in biases 1985 in Rubinstein's the switch exercise responsible results. was return processes The JohnsonShannonmodel model is given by: dS = Sdt + aS°dZ
(a > 0)
d0' =, uPQdt+QPo'ßdZP
(ß ? 0)
Johnsonand Shannonassumedthere existed a traded assetJ that had the samerandom term as the variance of the stock:
50
Chapter 2: Review of the Literature
di =, u, Jdt+Q, J6dZP Thus a riskfree hedge could be formed by longing one share of J and shorting option.
(ap)C of
Johnson et al. used MonteCarlo simulation to solve for a numerical solution and
found that their model could account for some termstructure of the implied volatility for the Johnson and Shannon concluded that: 1) they could not assert options. call outofthemoney that the switch in bias in Rubinstein's paper was caused by an upward shift in correlation; 2) they could not point to any macroeconomic event that would indicate a change of correlation in Rubinstein's study period of 19761978.
2.3.4 SteinStein Model Stein and Stein (1991) derived a closedform optionpricing solution via inverse Fourier transformation. Stein and Stein (1991) formulated the evolution of stochasticvolatility based on the OrnsteinUhlenbeckprocess: dS, = aS, dt + QS,dW, dQ=8(Qx)dt+&IW2
where dW,, dW2 are uncorrelated. The SteinStein model is more general than that of the HullWhite becauseit doesnot rely on Taylorseries expansion to solve explicitly for the option price. Simulations suggestthat this Ushape has However, the the as strike price was this varied. a model model exhibits disadvantage that it cannot capture skewness effects that arise from returnsvolatility Stein derived Nevertheless, Fourier the the way et al. transformation solution via correlation. for look for to more complex stochasticvolatility models. researchers openeda new way
2.3.5 Heston Model Heston (1993a) derived a closedform solution for the price of a Europeanstyleoption on an OrnsteinUhlenbeck followed the This is first its the process. stochastic variance assetwith for that solution can account closedform with correlation between volatility volatility model is by: Heston The given model and assetreturns. 51
Chapter 2: Review of the Literature
dS = OSdt+ aSdWl du2 =
K[8Q2]dt+ýQdW2
where x is the speed which Q2 reverts to its longterm mean 0. As opposed to the HullWhite model where risk premium was zero, Heston specified a volatility 2(S, 2a2. to the t) Q2, proportional variance: =
risk premium that was
Using Ito's lemma and standard arbitrage
arguments, Heston (1993a, 1993b) showed that the price of a European call was given by: c(S, a, t) = Sp,  KB(t, T)p2
where p,, p2 and B(t, T) are the conditional probabilities that can be calculated from formulas, and the price of a pure discount bond at time t with maturity of T, respectively". Heston's model has the advantagethat it allows arbitrary correlation between volatility and assetreturns. It can link any type of bias to the dynamics of the spot price and the distribution of spot returns. Heston suggestedthat this model might be able to explain some option biases that changed through time by Rubinstein (1985). In addition, the Heston model can possibly incorporate stochastic interest rates in pricing formula. Heston found that: 1) correlation between volatility and the spot price was necessaryfor explaining skewnessand strike price biases; a positive correlation results in high variance when the spot assetrises and this spreads the right tail of the probability density relative to the left tail, and vice versa; 2) skewnessin the distribution of spot returns affected the pricing of inthemoney options relative to outofthemoney options. Without this correlation, it is generally known that stochastic volatility J. kurtosis through only changesthe
2.3.6 Merton Model Apart from the Wiener process, researchers have also tried other processes to model risks. One of the pioneer works was by Merton (1976). Merton suggested a model where the asset Brownian jumps had a upon geometric superimposed motion. price
In this seminal paper,
Merton used two different sources to represent risks: 1) Wiener process to model daily news from the market and are diversifiable; 2) Poison process to that randomly come and risks
17The details can be found in the Heston's (1993a) appendix.
52
Chapter 2: Review of the Literature
describe jumps/shocks that capture the arrival of important news and are nondiversifiable. This model can be described by the following SDE:
dS = (a  Ak)Sdt + QSdW+ Adq where the parameter a is the instantaneousexpectedreturn on the stock, Cr the instantaneous A the rate of arrival, dW the Wiener process and dq the Poisson the volatility of returns, process. It is important to note that the size of Poisson outcomes does not depend on the infinitesimal interval dt. Instead, the probabilities associated with the outcomes are only a function of dt. The size of Brownian motion gets smaller as dt approaches zero. The BlackScholes model can be written as a special case of the Merton model when A= 0.
Due to the non
diversifiable risks presented in this model, noarbitrage argument cannot be invoked to price options. The jumpdiffusion
model can give rise to fatter left and right tail than the Black
Scholes model and is consistent with the implied volatility options.
In a study of stochastic volatility
and jumpdiffusion
patterns observed for currency models, Bakshi et al. (1997)
had improved that stochastic models new some showed pricing performance relative to the BlackScholes formula, but there was also evidence to suggest that the benefits derived from these mathematical parameterisations used for option pricing were not in proportion with the Nevertheless, the Merton model has successfully inspired many the models. complexity of for to alternative stochastic processes to price options. seek researchers
2.3.7 Other Developments Since Rubinstein (1985) documented the observed implied volatility patterns in relation to had to tried the diffusion twodimension use researchers many models to account moneyness, for thesebiases. We have selecteda few of them for discussions. In a study of stohcastic volatility option pricing model, Scott (1987) assumedthat volatility in diversified be changes and away volatility were uncorrelated with the stock risk could the for derive This equilibrium asset the to used pricing model study solutions return. Scott'solution diffusion HullWhite process. to was those time similiar model of continuous BlackScholes integral formula and the distribution function for the of that the solution was Scott computed option prices via MonteCarlo simulations price. the variance of the stock 53
Chapter 2: Review of the Literature
better BlackScholes found than the that the model at explaining was marginally model and actual option prices. Lo and Wang (1995) investigated the effect of predictability of asset return on option prices Even induced by OrnsteinUhlenbeck though typically the process. predictability was under the drift, which did not enter the option pricing formula under the noarbitrage framework, Lo linked did BlackScholes to the that the that was parameters enter predictability showed et al. option pricing formula.
In addition, Lo et al. constructed an adjustment for predictability to
the BlackScholes formula and demonstrated that this adjustment could be important even for for longer levels especially maturity options. of predictability, small Gesser and Poncet (1997) compared the performance of the HullWhite model and the Heston dollarmark days forward data. twenty atthemoney of option using model
Gesser et al.
found that the Heston model was superior to the HullWhite model because 1) correlation was allowed between volatility
and asset returns; 2) the market price of volatility
risk was not
in Heston's Gesser the the but to variance model. et al. also pointed out proportional constant that the HullWhite
model's poor performance was possibly caused by the loworder Taylor
Hull White in derivation its Despite that the and used process. success series approximation in accurately reproducing termstructure of volatility
and minimising volatility fitting errors,
failed to reproduce smile convexities as observed in foreign exchange Heston the model still market. Nandi (1998) studied how the incorporation of a nonzero correlation between asset returns and volatility
impacted pricing and hedging in the Heston model. The data that Nandi used
in 1992. index The 500 instantaneous S&P days 126 unobservable volatilities were of were invariant to jointly time parameters moment using other of generalised method with estimated in Blackfound inconsistency Nandi the the that estimation process. avoid any potential Scholes model outperformed the zero correlation version of the Heston model in terms of pricing.
However, the nonzero correlation version of the Heston model outperformed the
BlackScholes
in both terms of outofsample model,
pricing
and hedging.
Nandi
be directed towards developing simpler stochastic future could research that acknowledged to estimate. that easier were models HullWhite the (1998) Su model to study the stochastic process implied by used Corrado and Su's Corrado and index paper provided evidence that observed option 500 options. the S&P
54
Chapter 2: Review of the Literature
prices on the S&P 500 index corresponded to a meanreverting stochastic volatility process, where return volatility was strongly negatively correlated with changes in stock index levels. Corrado et al. also showed that a stochastic volatility
option pricing model provided a
significant improvement over the BlackScholes model in outofsample assessment. Madan et al. (1998) used the variance gamma process to price European options that allowed for skewness and excess kurtosis in a riskneutral framework.
In contrast to traditional
Brownian motion, the variancegamma process is a pure jump process with an infinite arrival has finite This jumps. variation and a random time change that can be written process of rate increasing difference two the processes each giving separately the market up and down of as moves. valuation
Closedform formula
solutions for European options were derived and the new option
nested the BlackScholes
Maden et al.
model as a special case.
demonstrated that the BlackScholes model could be rejected in favour of the variancegamma model. Das and Sundaram (1999) derived closedform solutions for the conditional and unconditional important kurtosis two classes of of models: stochastic volatility and skewness jumpdiffusion Poisson reversion and
with mean
processes. Das et al. found that each model exhibited
fundamentally inconsistent in that the were those termstructure patterns with observed some market and neither class of models constituted an adequate explanation of the empirical jumpdiffusions Furthermore, this that study showed evidence.
could only generate realistic
but implied smile at short maturities volatility not at long maturities. and sharp stochastic volatility
In contrast,
models were not capable of generating high levels of skewness and
kurtosis at short maturities under "reasonable" parameterisations but the smile did not flatten increased. Das found implied that et al. maturity volatility as a variety of out appreciably for atthemoney options under stochastic volatility patterns were possible better jumpdiffusion than models were volatility that stochastic concluded
models and they models.
Overall, stochastic models take into account some of the characters of volatility.
This allows
in part the explanation of the "volatility smile". But many problems limit the use of stochastic is First, No instantaneously is traded traded not a volatility asset. asset volatility models. it is build hedge to so not possible volatility with a portfolio to eliminate correlated perfectly impossible it is by to Thus price options noarbitrage techniques without volatility risk. introducing as an exogenous parameter the market price of volatility risk. Second, estimations
55
Chapter 2: Review of the Literature
likelihood parameters using of several nonobservable maximum method are not valid in because discussed dependent distribution joint for time stock returns above are cases over and be difficult derive. Third, has to to make very would of observations a sample one usually questionable assumption that asset returns and volatility
is uncorrelated`$. Fourth, closed
form solution usually does not exist for solving of these twodimensional partial differential equations and requires the use of MonteCarlo simulation as well as advanced econometric and numerical techniques, which are computationally demanding. Last, there is no systematic way to determine the changing sign and magnitude of correlation, which is important in generating smile convexities.
The factors mentioned above make it very challenging to
evaluate more complex products.
2.4 Implied Methodology 2.4.1 Underlying Concepts Implied methodology refers to the methods to exploit information about the distribution of the future asset from the options market. The major innovation that implied models offer is the direct gain of market information embedded in traded option prices without having to be filtered through the underlying asset's properties.
Many studies have shown that options
found in future is for information the timeseries that underlying not useful predicting contain volatility 1993).
(e.g. Chiracs and Manster, 1978; Day and Lewis, 1992; Lamoureux and Lastrapes, There are two major approaches in extracting market information
framework:
in the implied
1) the direct approach makes assumptions about the distribution
neutral distribution;
of the risk
2) the indirect or implied approach does not make any distributional
but options are observed priced consistently not necessary correctly. accepts assumptions and Neither approach makes any assumptions about the stochastic process of the underlying asset be because to implied but are proven more general methods any given riskneutral price distribution is consistent with many different stochastic processes.
The primary reason for using market information is the existence of the observed options' biases. The volatility smile curve indicates that market participants make more complex Brownian the motion about than path of the underlying asset price. geometric assumptions
18Some assumethat the volatility risk is not priced.
56
Chapter 2: Review of the Literature
Consequently, market participants attach different probabilities
to terminal values of the
distribution. The lognormal those that than extent are consistent a with underlying asset price indicates degree the to which the market riskneutral the the curve smile of convexity of distribution
function differs from the BlackScholes' constant volatility
assumption.
Any
in by in the slope the the curve are changes smile corresponding mirrored of shape variations function19. In the pricing particular, the more convex the smile curve, call and convexity of This for to the the the attaches price. market the greater extreme outcomes asset probability logdistribution function have "fatter to than tails" a the with market riskneutral causes normal density function.
Moreover, the sign of the slope in the volatility
distribution: the the market riskneutral of skew reflects
a positively
smile curve also
(negatively) sloped
implied volatility smile curve results in a riskneutral distribution that is more (less) positively from distribution flat lognormal that the smile curve. than riskneutral would result a skewed
2.4.2 Direct Approach The direct approach corresponds to the way market information is explicitly extracted from distribution functions The are usually assigned a priori according riskneutral options market. if Since be distributions true "beliefs" the only riskneutral equal and will researcher. to the of investors are truly riskneutral, or if risk in the underlying security is not priced, the riskneutral distribution distribution.
embedded in option prices is usually different from that of the actual
From the pricing perspective, riskneutral distribution are sufficient statistics in
business information they all relevant summarise about preferences and an economic sense financial for securities. of pricing purposes conditions
2.4.2.1 BreedenLitzenberger Method Breeden and Litzenberger (1978) were first to show that the second partial derivative of the is directly the the to to function riskexercise price respect proportional with pricing call The slope and convexity of the smile curve could be translated function. distribution neutral into probability space to reveal the market's implied riskneutral distribution function for the intervals discretely Since prices are option only available spaced at observed asset price.
19SeeBahra (1997) for a more detailed discussionon these issues.
57
Chapter 2: Review of the Literature
rather than being continuous, some approximation for the second derivative is necessary and implied, distribution be depending on the approximation chosen. implied than could one more Shimko (1993) derived an analytic expression for the probability density functions under the parabolic implied volatility volatility
assumption by fitting a quadratic relationship between implied
and exercise price.
The BlackScholes formula was then used to invert the
into option prices, thus allowing the application of Breeden et al. 's results smoothed volatility straightforwardly.
However, Shimko's extrapolation procedure, which grafted lognormal
tails onto the observable part of the implied riskneutral distribution arbitrarily
assigned a constant volatility
function, was that it
structure to the smile outside of the traded strike
range. Therefore it was not always possible to ensure a smooth transition for the observable part of the distribution
to the tails.
In addition, nothing in the Shimko's approach could
prevent negative probabilities. Malz (1997) used the volatility
function technique to access the riskneutral distribution of
exchange rates. The estimate of the volatility smile was parameterised by the traded straddle, it did option prices so not require the construction of a cubic spline strangle and riskreversal function or regression on implied volatilities.
Unlike Shimko, Malz did not make special
fitted hence delta, for the tails to the the and allowed curve cover entire range of allowances the entire support for the probability
density function.
Malz concluded that this method
leaded to smoother estimates of the riskneutral distribution
and more accurate volatility
estimates for wing options.
2.4.2.2 MultiLogNormality
Method
The use of lognormal density function has also received a great deal of attention. Using the framework of Ritchey (1990), Melick and Thomas (1997) constructed implied distributions using the multilognormal
method. Melick et al. applied this framework to options on crude
functions. Bahra lognormal (1997) for futures techniques three reviewed various with oil function distribution from the of an underlying prices of the asset price riskneutral estimating framework for distribution twolognormal derived the the estimating riskneutral options and Subsequently, Dinenis data. (1999) (1998) Gemill also et al. al. and et market using observed framework in "usefulness" to the study events embedded of used this twolognormal currency options.
58
Chapter 2: Review of the Literature
In similar spirits to Bahra (1997), Dinenis et al. (1998) investigated the implied riskneutral distribution around the exit of Sterling in 1992 and Gemill et al. (1999) studied the FTSE 100 index options over the 19871997 period. Dinenis et al. suggested that the twolognormal framework was able to provide critical information in regard to the exit of Sterling whilst Gemill et al. found that although the twolognormal model fitted the data significantly better than the BlackScholes model, the outofsample performance was only marginally better. Gemill et al. also tested the "usefulness" of their model during elections and a number of forwardlooking Despite are options market crashes.
instruments, Gemill et al. concluded that
implied distributions did not anticipate various market crashes under study and suggested that help in "market during distribution telling only a story" elections. could riskneutral Later, Campa et al. (1998) studied implied exchange rate distributions of European Monetary System crossrates using three smoothing methods: implied binomial, twolognormal
and
found Campa distributions fluctuated that et al. riskneutral widely approaches. cubic spline from week to week without apparent reason. distribution
They stipulated that the twolognormal
might impose too rigid a structure on the resultant riskneutral distribution and
little twolognormal that the approach made economic sense. argued
2.4.2.3 Approximating the RiskNeutral Density Distribution Another vital development in recovering riskneutral distribution is specialized to the problem be distribution, lognormal, if the underlying security can not of option valuation where distributed lognormally (1982) Jarrow Rudd by were random variable. a and approximated first to derive a theoretical framework to include the influence of skewness and kurtosis in from fact large idea Their the that was motivated a class of valuation problems pricing option. itself distribution was a convolution where the underlying
of other distributions.
In such
its (e. distribution known be information the g. underlying concerning may situations, partial distribution but function itself be to the be tabulated) as complex so may moments may Jarrow et al. adjusted the BlackScholes formula by approximating integration. direct prevent the true distribution
distribution lognormal and the resulting option pricing equation with
BlackScholes linear the combination of be solution plus some adjustment viewed as a could between discrepancies lognormal kurtosis for the the skewness and of terms that accounted Later, Corrado 's distribution. Su (1997) Jarrow true al. et distribution and the and used 500 index S&P found the option that the investigate market and volatility smile to method
59
Chapter 2: Review of the Literature
was effectively flattened.
Corrado et al. concluded that skewness and kurtosis added to the
BlackScholes formula significantly improved accuracy and consistency for pricing deep inthemoney and outofthemoney
options.
Following
Jarrow et al. 's footstep, Rubinstein
(1998) applied Edgeworth expansion directly to discretise riskneutral distribution and valued options in conjunction with the method of implied binomial tree. Investor's opinions about introduced be kurtosis to the riskneutral distribution could skewness and
and this model
American be to as well as exotic options. value used could also
2.4.3 Indirect Approach Indirect/implied
approach employs the noarbitrage condition to price options.
The use of
implied approach is motivated by the "beliefs" that both exotic and vanilla instruments should be priced based on the same set of information and therefore they are expected to deviate by from theoretically the correct prices a similar amount. Consequently, traded consistently European call and put options can be used to hedge the more complicated overthecounter instruments even if the products included in the hedge may not be correctly priced. Breeden and Litzenberger
(1978) demonstrated that riskneutral
distributions
Since
could be
by butterfly derivative from pricing the spreads and of options expressed as second recovered the call option price with respect to the exercise price, recent developments have considered implied tree models that incorporate observed volatility incorporating the volatility Methods of process20.
structures into the option pricing
smile into treebased models have been
(1990), Rubinstein (1994), Derman Dupire (1994) Kani by Longstaff and and suggested (1994) for European options. The following sections discusses different types of implied tree models.
2.4.3.1 Implied Tree Assumptions The basic assumption for implied tree model is that riskneutral
distribution
assumes a
followed in form S by the functional stochastic the and process a riskstock price specific by: is governed neutral world
dS = rSdt + SQ(S,t)dz
20Jackworth (1999) and Flamouris (2001) provide a good review for the developmentof implied models.
60
Chapter 2: Review of the Literature
The above diffusion equation is closely related to the original BlackScholes model except that local volatility
a(S, t) is no longer constant but depends on stock price and time. It is
important to note that no functional form is prescribed for local volatility in the implied tree Instead, special rules are developed for
technique.
deducing
the riskneutral
path
probabilities, ArrowDebreu prices21 , and transition probabilities for stock price movements in the tree from one time level to the next in such a way that the market prices of options can be reproduced with the tree used in a noarbitrage fashion. Thus, given N different states, the time t price of a contingent claim expiring at time T is given by: N
V(s)p(s)
II(t) _
V(s)er(Tt)
_N t=1
_
EV(s)er(Tt),
(S) er(Tt)
r(S)
$=1
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When the state space is continuous, the price of a
integrating by derived is function density the the payoff over of riskneutral claim contingent the underlying asset and then discounting at the riskfree rate: ('°'V (s) f (s)ds H(t) = e''
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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.
180
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:
182
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 )
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 p0,6Z. 0,
> 0,i =1,...,p, i=1,...,