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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 (Unpublished Doctoral thesis, City University London)

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Essays on the Modelling of S&P 500 Volatility

Kar-Hei 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 Term-Structure 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 No-Arbitrage Approach 2.1.1.1 Black-ScholesFormula 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

Continuous-Time Process Self-Replicating 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 GARCH-type Models for Long-Run 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 Ornstein-UhlenbeckProcess

47 47 47 48 48

2.3.2 Hull-White Model

49

2.3.3 Johnson-ShannoModel

50

2.3.4 Stein-Stein 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 Breeden-LitzenbergerMethod 2.4.2.2 Multi-Log-Normality Method 2.4.2.3 Approximating the Risk-Neutral 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 Derman-Kani 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 Time-series 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 Term-Structure 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 1983-1998 3.4.2 Interpretation of the Implied Term-Structure Results 3.4.3 Charactersof At-the-Money Implied Volatility Term-Structure 3.4.3.1 Variability of Implied Volatility 3.4.3.2 Mean-Reversionof Implied Volatility 3.4.3.3 Consistencyof Implied Volatility Tem-Structure 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 Time-Series 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 In-Sample Analysis: Maximum Likelihood Estimations of APARCH Parameters 4.4.2.1 In-Sample 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 In-Sample Results 4.4.3 Are Conditional Volatility Models Prone to the State of Volatility? 4.4.3.1 Student-t 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 In-Sample Analysis: EGARCH and Statistical Loss Functions 4.4.4.1 Inclusion of EGARCH 4.4.4.1.1 In-Sample Results for EGARCH 4.4.4.1.2 Discussion for In-Sample 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 In-Sample Statistical Errors 4.4.4.2.2 Resultsfor Alternative Statistical Loss Functions 4.4.4.2.3 Commentson Results for In-Sample Statistical Loss Functions 4.4.5 Out-of-Sample 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 Delta-Neutral 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 Time-Series Statistics

142 142 143

4.4.5.5 Results of Trading At-the-Money Straddles 4.4.5.5.1 Preliminary Statistics for Directional Trading Signals 4.4.5.5.2 Profit and Loss: Trading At-the-Money 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 Options-basedVariance Swap Framework 5.2.2.1 Log-Contract 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 Jump-Diffusion Models 5.2.3.2.1 Justification for the Jump-Diffusion 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 One-FactorModel 5.2.3.3.3 Coleman et at. Approach 5.2.3.4 Ad hoc Black-ScholesModel 5.2.4 Time-Series Models for ForecastingVariance 5.2.4.1 Justification for the Conditional Volatility Approach 5.2.4.2 GARCH-Variance 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 1999-2002

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 Trust-Region Reflective Quasi-Newton Method 5.4.1.2.2 Calibrations for Absolute Diffusion Process 5.4.1.2.3.. Finite-Difference Settings

214 216 217 217 219

5.4.1.3 Calibrations for Ad Hoc Black-ScholesModel 5.4.1.4 Calibrations for Time-Series Models

219 219

5.4.2 Empirical Results 5.4.2.1 Calibration Results for Options-basedModels 5.4.2.1.1 Calibration Resultsfor StochasticVolatility Models with/without Jumps 5.4.2.1.2 Calibration Results for ad hoc Black-ScholesModel 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 Options-based Variance Swap Model 5.4.2.2.2 Out-of-Sample Test: Variance Forecastability 5.4.2.2.3 Comments on Out-of-Sample Results 5.4.2.2.4 In-Sample Fit for Option Pricing Models

226 226 227 228 230

5.4.2.3 Consistencywith the Time-seriesPropertiesof Volatility 5.4.2.3.1 CIR Square-RootProcess 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 Term-Strucutre of Implied Volatility

240

6.4.2 Conditional HeteroskedasticModels

241

6.4.3 Time-series and Options-basedVariance 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 In-Sample 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 Dickey-Fuller 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

At-the-Money

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

Cameron-Martin-Girsanov

CUBS

City University BusinessSchool

DIVF

Dynamic Implied Volatility Function

DVF

Deterministic Volatility Function

DF

Dickey-Fuller Unit Root Test

DTB

Deutsche Terminborse

EGARCH

Exponential GARCH

X

EGARCH-M

Exponential GARCH in Mean

EMS

European Monetary System

ERM

Exchange-Rate Mechanism

EWMA

Exponentially Weighted Moving Average

FIGARCH

Fractional Integrated GARCH

FIEGARCH

Fractional Integrated EGARCH

FESE-100

Financial Times Stock Exchange 100

GARCH

GeneralisedARCH

GARCH-M

GARCH in mean

GBM

Geometric Brownian Motion

GED

GeneralisedError Distribution

GMLE

Guassianquasi-MLE

GMM

GeneralisedMethod of Moment

HKSE

Hong Kong Stock Exchange

HIS

Hang SengINdex

HMSE

Heteroskedasticity-adjustedMSE

HW

Hull & White Model

III)

Independent& Identically Distributed

IM M

International Money Market

ISD

Implied StandardDeviation

ITM

In-the-Money

IV

Implied Volatility

IVF

Implied Volatility Function

KS

Kolmogorov-Smirnov

LEAPS

Long-term Equity Anticipation Securities

statistic

R1

LHS

Left-Hand Side

LIFFE

London Financial Futures Exchange

LL

Logarithmic Loss

LLR

Log-Likelihood

LM

Larange-Multipler Test

LR

Likelihood-Ratio

LRNVR

Locally Risk-Neutral Valuation Relationship

LTCM

Long Term Capital Management

MA

Moving Average

MAE

Mean-Absolute Error

MAPE

Mean-Abolute PercentError

MMEO

Mean-Mixed Error (over prediction)

MMEU

Mean-Mixed Error (under prediction)

MONEP

Marche des Options Negociablesde Paris

MSE

Mean-SquareError

NARCH

Nonlinear ARCH

NLS

Nonlinear Least-Square

NYSE

New York Stock Exchange

OLS

Ordinary Least Squares

OTC

Over-the-Counter

OTM

Out-of-the-Money

PIDE

Partial Intergro-Differential

PDE

Partial Differential Equation

PDF

Probability Density Function

PHLX

Philadelphia Exchange

Ratio

Test

Equation

X11

QMLE

Quasi-Maximum Likelihood Estimation

RHS

Right-Hand Side

RMAE

Root Mean Absolute Error

RMAPE

Root Mean Absolute PercentError

RMSE

Root Mean SquaredError

RND

Risk-Neutral 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

TS-GARCH Taylor's and Schwert's GARCH VAR

Vector AutoregressiveModel

VIX

CBOE's Volatility Index

VOLAX

DTB's Volatility Index Futures

VX1

MONEP's 31-day Short-Term Implied Volatility Index

WISD

Weighted Implied StandardDeviation

Xlii

Important Symbols ß

Elasticity Requirement

C

Call Option

x2 (m)

m`"degreeChi-statistics

D(

Any Distribution

8

Correlation for Wiener Processes

Ho

Null Hypothesis

H,

Alternative Hypothesis

h,

Conditional Variance basedon information up to time t-1

I(0)

CovarainceStationary Process

I(1)

Non-Stationary Process

it

Information set up to time t

L

Lag Operator

,u

Distributional Mean

N(

Normal Distribution

P

Put Option

Q2(m)

Ljung-Box Statistics at lag m for Serial Correlation

Q(m)

Box-Pierce Statistics at lag m for Serial Correlation

p(m)

SampleAutocorrelation Function at lag m

at

StochasticVolatility

T()

Student-t Distribution

W

Wiener Processor Brownian Motion

xiv

Tables Table 1: Time-to-Maturity and Moneyness Groups 86 ...................................................................................... Table 2: Normalised Data Groups

.................................................................................................................

87

Table 3: t-statistics for equal means but unequal variances for at-the-money calls 96 ...................................... Table 4: t-statistics for equal means but unequal variances for at-the-money puts 96 ....................................... Table 5: Curve-fitting estimations for Average Call Implied Volatility from 1983-1998

............................

98

Table 6: Curve-fitting estimations for Average Put Implied Volatility from 1983-1998 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 Ljung-Box 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 Out-of-Sample Buy and Sell Signals 162 ........................................................ xv

Table 26: Statistics for Forecasts of Volatility Changes 162 ............................................................................. 163 Table 27: Before-transactions-costs Statistics for 1985-1986 without Filter .............................................. 163 Table 28: Before-transactions-costs Statistics for 1989-1990 without Filter .............................................. 164 Table 29: Before-transactions-costs Statistics for 1993-1994 without Filter .............................................. 164 Table 30: Before-transactions-costs Statistics for 1997-1998 without Filter .............................................. 165 Table 31: Before-transactions-costs Statistics for 1985-1986 with ± 3% Delta Filter .............................. 165 Table 32: Before-transactions-costs Statistics for 1989-1990 with ± 3% Delta Filter .............................. 166 Table 33: Before-transactions-costs Statistics for 1993-1994 with ± 3% Delta Filter .............................. 166 Table 34: Before-transactions-costs Statistics for 1997-1998 with ± 3% Delta Filter .............................. Table 35: Before-transactions-costs Statistics for 1997-1998 with ± 3% Delta Filter (Excluding One Data Point)

..........................................................................................................................................

167

167 Table 36: After-transactions-costs Statistics for 1985-1986 with ± 3% Delta Filter ................................ 168 Table 37: After-transactions-costs Statistics for 1989-1990 with ± 3% Delta Filter ................................ 168 Table 38: After-transactions-costs Statistics for 1993-1994 with ± 3% Delta Filter ................................ Table 39: After-transactions-costs Statistics for 1997-1998 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 Black-Scholes Model

223 ...........................................................

224 for EGARCH Parameters Estimated Table 46: ........................................................................................... Table 47: Estimated Parameters for GARCH-Variance 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 Mean-Square Price Errors and Model Rankings for 3M, 6M and 9M Variance Swap 229 Contracts .................................................................................................................................... Table 51: In-Sample 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: 1983-1998 ......................................................................................... 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: 21-70 Calls with Sixth-Order Polynomial and Linear Trend ..................................................... 111 Figure 14: 21-70 Puts with Sixth-Order 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: 1983-1998 ............................................................................................. 170 170 for 1983-1998 20: Autocorrelations Figure r2: ............................................. ............................................... Figure 21: Autocorrelations for Irl: 1983-1998 ........................................................................................... 170 Figure 22: Autocorrelations for r. 1983-1986 ............................................................................................. 171 Figure 23: Autocorrelations for r2: 1983-1986..............:............................................................................. 171 Figure 24: Autocorrelations for Irk: 1983-1986 ........................................................................................... 171 172 for 1987-1990 Autocorrelations 25: Figure r: ............................................................................................. Figure 26: Autocorrelations for r2: 1987-1990............................................................................................ 172 xviii

Figure 27: Autocorrelations for Irl: 1987-1990

172 ...........................................................................................

Figure 28: Autocorrelations for r: 1991-1994 173 ............................................................................................. Figure 29: Autocorrelations for r2: 1991-1994 173 ............................................................................................ Figure 30: Autocorrelations for Irl: 1991-1994

...........................................................................................

173

Figure 31: Autocorrelations for r: 1995-1998 174 ............................................................................................ Figure 32: Autocorrelations for r2: 1995-1998 174 ........................................................................................... Figure 33: Autocorrelations for Irl: 1995-1998

174 ...........................................................................................

Figure 34: 3-State SWARCH-L(3,2) - High Volatility Regions 175 ................................................................ Figure 35: 3-State SWARCH-L(3,2) - Low Volatility Regions 175 ................................................................. Figure 36: Cumulative Rate of Return From Straddles Trading (1993-1994) 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: 1999-2002 ................................................................................... 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: Term-Structure 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 term-structure of implied volatility and examines the performance of different specifications of time-series and options-basedvolatility forecasting models under the influence of the observed market biases. Our researchis basedprimarily upon the use of S&P 500 data for the period 1982-2002. There are three self-containedbut seemingly related projects in this dissertation. The objectives of this researchare: 1) to characterisethe term-structureof implied volatility; 2) to compare the performance of asymmetric power ARCH and EGARCH models; 3) to evaluate the forecasting performance of time-series and options-based variance swap valuation models. The observedmarket anomalies in the term-structure of implied volatility of S&P 500 futures options are investigated between 1983and 1998. Term-structure evidence indicates that short-term options are most severely mispriced by the Black-Scholes formula. We find evidence that option prices are not consistent with the rational expectations under a mean-reverting 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 jump-diffusion 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. Log-likelihood 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 3-state is the that the support models model notion regime-switching 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. Options-basedvolatility trading rankings in EGARCH GARCH that profit and can generatestatistically significant ex-ante 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 time-series and options-based 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 option-based 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 time-series 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 options-basedstrategy cannot produce enough variance term-structurepatterns; 3) distributional dynamics implied by option parametersis not consistent with its time-series data as likelihood to by Future the needs the research maximum estimation of square-rootprocess. 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 time-series specifications of of and options-based 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 1982-2002. 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 Black-Scholes formula, implied volatility exhibits leverage the factors Many both smile effects and term-structure 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 fat-tails 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 term-structure of volatility, modelling of volatility and volatility derivatives.

1.2.1 Term-Structure of Volatility The modelling of the term-structure 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 sub-periods, implying that skewness of the risk-neutral density changed over time. Subsequently, numerous efforts have been made to investigate the mean-reverting process and term-structure of implied volatility.

Stein (1989) pioneered the examination of the term-structure of the average at-the-

implied volatility using two maturities on S&P 100 index options. By using a money options' mean-reverting volatility

model, evidence suggested that long-maturity

"overreact" to changes in the implied volatility

options tended to

investors because of short-maturity 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 term-structure of implied volatility has also been discussed by Heynen, Kemna and Vorst (1994). Basing their results upon Duan (1995), Heynen et al. derived the term-structures of implied volatility for EGARCH, GARCH and a in time-toStein Only (1989). two of to values stochastic model a similar way mean-reverting best investigated Heynen the EGARCH gave and et al. concluded that maturity were description of asset prices of the term-structure of implied volatility.

Xu and Taylor (1994)

also studied at-the-money currency options and used a mean-reverting volatility

model to

for between longimplied any volatility and short-term expectations of establish relationships implied Xu 's T. of the et al. model could explain time-varying 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 at-the-money contracts. Since the term-structure of implied volatility reflects the time-varying 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 option-pricing models in order to cope with the observed term-structure 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 time-varying 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 real-world

continuous-time

Besides, estimations of data are recorded at

discrete intervals. Following the path-breaking paper by Engle (1982), an alternative literature has focused on discrete-time

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) fat-tailed; than equity returns equally positive returns; volatility equity returns are (persistence refers to volatility clustering). persistent are

This class of discrete-time models

hypothesises that both smile effects and term-structure patterns can be explained by allowing the underlying asset's volatility to obey a stochastic process. There is a voluminous literature discrete time-varying 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 long-memory models is based on the observations of the so-called "stylised facts". For example, Ding et al. (1993) invented the APARCH models that used the Box-Cox 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 long-memory 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 delta-risk. the reduction on of always strategy

Once the underlying index moves,

however, a delta-neutral trade can become long or short delta. Rehedging becomes necessary to maintain a delta-neutral 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 options-based 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 closed-form 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 well-studied 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 options-based its hedged Despite such as straddle name, a variance or strategies call/put volatility options. swap is actually an over-the-counter 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

n-1

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 options-based strategy.

levels, it is considered a

It allows counterparties to exchange

for to fixed floating Counterparties swap variance a variance variance. use can cash-flows 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 mean-reverting 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, time-series. 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 time-series and biases. forecasting influence models under the of the observed market options-based volatility In order to present the results in a meaningful and manageable manner, three self-contained 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 Term-Structure 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 term-structure 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 mean-reverting 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 log-likelihood-based 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 log-likelihood ratio tests;

1 Seeappendix A. 1 for thesenestedmodels.

7

Chapter 1: Introduction to the Study

ii)

To provide evidence that the in-sample performance of asymmetrical and symmetrical conditional volatility models are prone to the state of volatility by using a 3-state regime switching volatility

conditional model to separate high and low volatility

states;

iii)

To compare the in-sample performance of EGARCH (Nelson, 1991) with APARCH models basedon aggregateAIC statistics;

iv)

To illustrate the quality of different conditional volatility forecasts by predicting the one-step ahead changes of implied volatility and conducting ex-ante (out-of-sample) 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 time-series 500 S&P forecasting the variance swap models' on and options-based volatility performance index from June 2001 to November 2001. After the terrorist attacks on September 11,2001, the longer-termed forward variance has become more volatile than the shorter-termed forward Based on six well-selected contract days, we design the three-, six- and nine-month variance. different for day by contracts swap each them contract evaluating variance and analyse implied Our different in time-series 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 out-of-sample results;

ii) To explore whether there is any systematic difference in volatility forecasting between time-series and options-basedvariance swap valuation models. performance It is intended to investigate whether options-basedmodels, which are forward-looking, historical discrete-time 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 term-structure 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. capitalisation-weighted, representing the market value of market all outstanding common shares of the 500 large-capitalisation 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 self-containedprojects individually.

1.4.1 Significance of the First ResearchProject The 3 in futures 500 S&P the Chapter studies observedmarket anomalies the options market. term-structure of implied volatility reflects the time-varying market expectations of asset different horizons. Despite the investigation time evidence the and over extensive volatility has far implied term-structure ever the thus on of volatility, no past study accumulated Prior S&P to large 500 implied the term-structure. study of empirical a volatility considered have always examined the term-structure of implied volatility only past papers this research, for particular at-the-money 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 term-structure 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 at-the-money maturity group. The use of relative implied volatility allows the measurementof broad degrees in implied the term-structure 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 term-structureof implied volatility shows any specific patterns then some models, such as stochasticvolatility models or GARCH-type 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 long-memory 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 continuous-time stochastic volatility time literature. Whilst the research on discrete and continuous-time models has evolved independently, many continuous-time models can be thought of as the limits of GARCH-type 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 options-based and time-series 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 time-series 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 options-based 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 self-contained 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 term-structure 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 in-sample 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 preference-free quality 11

Chapter 1: Introduction to the Study

best out-of-sample model. Chapter 5 adopts a financial engineering approach to evaluate the performance of different time-series and options-based variance swap valuation models on the S&P500 index under the influence of term-structure 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 No-Arbitrage Approach The Black-Scholes 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 Black-Scholes following: the were proposed i) Markets are frictionless, efficient and complete; ii) Constantinterest rate2and volatility; iii)

Portfolio rebalancedcontinuously;

iv) No-arbitrage and tradesare self-financing; follows S The asset underlying geometric Brownian motion3 (GBM). v) The underlyinng assetdynamics is given by: dS =/ Sdt + oSdW

2 The original Black-Scholes 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 Black-ScholesFormula According to the Black-Scholes 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

discrete-time version of the diffusion model is given by: AS = PSet + aSAW

(1)

The above discrete-time 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) = CSAS-AC

(3)

Substituting (1) and (2) into (3), one gets: A v(S, t) =-CtOt -2 CssQ zSsOt

(4)

4 SeeHull (2000) pp.235-236 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 risk-free portfolio in discrete time. Under no-arbitrages 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 Black-Scholes derivation is that equations (6) and (7) are independent of instantaneousstock return p; one only needs to know the risk-free 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(--) -(r-d X2

+12Q)(T -t)

Q T-t d2=d,

-Q

T-t

hedge ) N(d, the distribution function, is N(") the cumulative r the risk-free rate, where T d the dividend S X the and the the price the stock volatility, yield, strike price, parameter, o maturity. In the Black-Scholes options model, prices are always a non-decreasing function of the in is distributed Furthermore, log-normally. distribution The change of stock price volatility. ln(S) betweentime 0 and T is given by:

SThe no-arbitragetheorem 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

4u-2

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 risk-neutral measure. According to Merton (1976), option prices rely on put-call parity to enforce the internal consistency of option pricing.

The put-call parity is a no-arbitrage 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 non-dividend paying stock, this relationship is given by:

C-P=S-Xe"' If put-call parity is violated, arbitrage will arise. Note that put-call parity is true regardless distribution is log-normal. It does not, however, hold for American the price asset whether formula instant Black-Scholes The be can rearrangedsuch that a stock option at any options. can be thought of a weighted portfolio of risky stock and riskless zero-couponbonds: 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 risk-neutral 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 risk-neutral world. 16

Chapter 2: Review of the Literature

2.1.1.2 Other Variations There are many variations of the Black-Scholes model - so many that it is cumbersome to Black-Scholes (1973) derived it. For instance, Merton the survey of provide a comprehensive formula independently based on a three-asset 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 jump-diffusion 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 Barone-Adesi

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 Black-Scholes 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 Black-Scholes 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 Black-Scholes 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 cash-flow under the risk-neutral measure. Furthermore, the Black-Scholes formula can also be obtained from the Martingale approach. The following sub-sectionsdiscuss 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 Black-Scholes 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 non-random 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 risk-free rate via noP is the in Thus the risk-free 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 risk-free 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 Discrete-TimeProcess 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 tick-time 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 path-probability 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, self-financing 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 self-financing strategy to construct a contingent claim X if. i) Both (0,, (pi) are known by time i-1; 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 Q-Martingales. and

2.1.2.3 Continuous-TimeProcess 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 P-Brownian motion. One important tool of measure is the Radon-Nikodym

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 Radon-Nikodym

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, ), s-0 ? ao >0, a, /i=lpla,

The GARCH(p, q) model is given by:

j=ý

1Ir-i - 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 long-run 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 (non-negativity 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; =1-2, 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 multi-step 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=

1-2

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 long-term 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+kt-l

hf+k-a

= al +(al

E(h+k)=a2

-a)+

+ßl)k(h2

(hr+k-1

-a)

-a2)

1, the k-day 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, -, I-c(e, ß, b) }+ h, h, = ao + a, h, -b -, -, -, (er-i

)

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 Ornstein-UhlenbeckProcess The stochastic process a, is random and not observable. One of the most studied and celebratedcontinuous-time stochasticvolatility models is the Ornstein-Uhlenbeckprocess: dS, / S, = adt + QdW, d(Ina) =A(ý-1nQ)dt+W2 dW,dW2 = &It A for Wiener 8 W, is the long-term W2 the processes and the and correlation mean where ,ý the speedof the mean-revertingprocess. 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 discrete-time traded are models are approximations of stocks or commodities their continuous counterparts.

The discrete-time 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 discrete-time 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 discrete-time model is that the ARCH models' innovations depend on the past information discrete-time 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 Hull-White 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 well-known stochastic volatility model is the are applications Hull-White model (1987). This model is basedupon the following continuous-time process: dS = SßSdt + aSdW, da2 =, u(x-Q2)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

Hull-White price is the mean Black-Scholes price, evaluatedover the conditional distribution of averagevariance: C=ýc(V)g(V)dV

is is is Black-Scholes V the the the the of value variance rate, average price and c g where risk-neutral V in Furthermore, derived distribution Hull White of a world. an and probability based for European-style Taylor-series option on expansion. In this case,the solution analytic Hull-White model can be written as a combination of Black-Scholes 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 longer-term did had lower implied than that near-the-money call options concluded it is Finally, GARCH(1,1) that the noted model can be written as a shorter-term options. discrete-time approximation to the diffusion processof the Hull-White model. Hull and White attempted to use their model to explain Rubinstein's (1985) findings on the term-structure 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 Hull-White 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 Johnson-Shanno 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 Johnson-Shannonmodel 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 risk-free hedge could be formed by longing one share of J and shorting option.

(ap)C of

Johnson et al. used Monte-Carlo simulation to solve for a numerical solution and

found that their model could account for some term-structure of the implied volatility for the Johnson and Shannon concluded that: 1) they could not assert options. call out-of-the-money 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 1976-1978.

2.3.4 Stein-Stein Model Stein and Stein (1991) derived a closed-form option-pricing solution via inverse Fourier transformation. Stein and Stein (1991) formulated the evolution of stochasticvolatility based on the Ornstein-Uhlenbeckprocess: dS, = aS, dt + QS,dW, dQ=-8(Q--x)dt+&IW2

where dW,, dW2 are uncorrelated. The Stein-Stein model is more general than that of the Hull-White becauseit doesnot rely on Taylor-series expansion to solve explicitly for the option price. Simulations suggestthat this U-shape has However, the the as strike price was this varied. a model model exhibits disadvantage that it cannot capture skewness effects that arise from returns-volatility 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 closed-form solution for the price of a European-styleoption on an Ornstein-Uhlenbeck followed the This is first its the process. stochastic variance assetwith for that solution can account closed-form 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[8-Q2]dt+ýQdW2

where x is the speed which Q2 reverts to its long-term mean 0. As opposed to the Hull-White 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 in-the-money options relative to out-ofthe-money 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 non-diversifiable. 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 Black-Scholes 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, no-arbitrage argument cannot be invoked to price options. The jump-diffusion

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 jump-diffusion

patterns observed for currency models, Bakshi et al. (1997)

had improved that stochastic models new some showed pricing performance relative to the Black-Scholes 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 two-dimension 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 Hull-White process. to was those time similiar model of continuous Black-Scholes integral formula and the distribution function for the of that the solution was Scott computed option prices via Monte-Carlo simulations price. the variance of the stock 53

Chapter 2: Review of the Literature

better Black-Scholes 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 Ornstein-Uhlenbeck though typically the process. predictability was under the drift, which did not enter the option pricing formula under the no-arbitrage framework, Lo linked did Black-Scholes 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 Black-Scholes 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 Hull-White model and the Heston dollar-mark days forward data. twenty at-the-money of option using model

Gesser et al.

found that the Heston model was superior to the Hull-White 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 Hull-White

model's poor performance was possibly caused by the low-order Taylor-

Hull White in derivation its Despite that the and used process. success series approximation in accurately reproducing term-structure 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 non-zero 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 non-zero correlation version of the Heston model outperformed the

Black-Scholes

in both terms of out-of-sample model,

pricing

and hedging.

Nandi

be directed towards developing simpler stochastic future could research that acknowledged to estimate. that easier were models Hull-White 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 mean-reverting 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 Black-Scholes model in out-of-sample assessment. Madan et al. (1998) used the variance gamma process to price European options that allowed for skewness and excess kurtosis in a risk-neutral framework.

In contrast to traditional

Brownian motion, the variance-gamma 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

Closed-form formula

solutions for European options were derived and the new option

nested the Black-Scholes

Maden et al.

model as a special case.

demonstrated that the Black-Scholes model could be rejected in favour of the variance-gamma model. Das and Sundaram (1999) derived closed-form solutions for the conditional and unconditional important kurtosis two classes of of models: stochastic volatility and skewness jump-diffusion Poisson reversion and

with mean-

processes. Das et al. found that each model exhibited

fundamentally inconsistent in that the were those term-structure patterns with observed some market and neither class of models constituted an adequate explanation of the empirical jump-diffusions 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 at-the-money options under stochastic volatility patterns were possible better jump-diffusion 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 no-arbitrage 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 non-observable 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 two-dimensional partial differential equations and requires the use of Monte-Carlo 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 time-series 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 risk-neutral 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 log-normal those that than extent are consistent a with underlying asset price indicates degree the to which the market risk-neutral the the curve smile of convexity of distribution

function differs from the Black-Scholes' 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 risk-neutral causes normal density function.

Moreover, the sign of the slope in the volatility

distribution: the the market risk-neutral of skew reflects

a positively

smile curve also

(negatively) sloped

implied volatility smile curve results in a risk-neutral distribution that is more (less) positively from distribution flat log-normal that the smile curve. than risk-neutral 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 risk-neutral options market. if Since be distributions true "beliefs" the only risk-neutral equal and will researcher. to the of investors are truly risk-neutral, 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, risk-neutral 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 Breeden-Litzenberger 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 risk-neutral 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 Black-Scholes 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 log-normal

tails onto the observable part of the implied risk-neutral 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 risk-neutral 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 risk-reversal 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 risk-neutral distribution

and more accurate volatility

estimates for wing options.

2.4.2.2 Multi-Log-Normality

Method

The use of log-normal 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 multi-log-normal

method. Melick et al. applied this framework to options on crude

functions. Bahra log-normal (1997) for futures techniques three reviewed various with oil function distribution from the of an underlying prices of the asset price risk-neutral estimating framework for distribution two-log-normal derived the the estimating risk-neutral 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 two-log-normal currency options.

58

Chapter 2: Review of the Literature

In similar spirits to Bahra (1997), Dinenis et al. (1998) investigated the implied risk-neutral distribution around the exit of Sterling in 1992 and Gemill et al. (1999) studied the FTSE 100 index options over the 1987-1997 period. Dinenis et al. suggested that the two-log-normal framework was able to provide critical information in regard to the exit of Sterling whilst Gemill et al. found that although the two-log-normal model fitted the data significantly better than the Black-Scholes model, the out-of-sample performance was only marginally better. Gemill et al. also tested the "usefulness" of their model during elections and a number of forward-looking 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 risk-neutral Later, Campa et al. (1998) studied implied exchange rate distributions of European Monetary System cross-rates using three smoothing methods: implied binomial, two-log-normal

and

found Campa distributions fluctuated that et al. risk-neutral widely approaches. cubic spline from week to week without apparent reason. distribution

They stipulated that the two-log-normal

might impose too rigid a structure on the resultant risk-neutral distribution and

little two-log-normal that the approach made economic sense. argued

2.4.2.3 Approximating the Risk-Neutral Density Distribution Another vital development in recovering risk-neutral distribution is specialized to the problem be distribution, log-normal, if the underlying security can not of option valuation where distributed log-normally (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 Black-Scholes formula by approximating integration. direct prevent the true distribution

distribution log-normal and the resulting option pricing equation with

Black-Scholes linear the combination of be solution plus some adjustment viewed as a could between discrepancies log-normal 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

Black-Scholes formula significantly improved accuracy and consistency for pricing deep inthe-money and out-of-the-money

options.

Following

Jarrow et al. 's footstep, Rubinstein

(1998) applied Edgeworth expansion directly to discretise risk-neutral distribution and valued options in conjunction with the method of implied binomial tree. Investor's opinions about introduced be kurtosis to the risk-neutral 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 no-arbitrage 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 over-the-counter instruments even if the products included in the hedge may not be correctly priced. Breeden and Litzenberger

(1978) demonstrated that risk-neutral

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 tree-based 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 risk-neutral

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 Black-Scholes 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 risk-neutral

path

probabilities, Arrow-Debreu 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 no-arbitrage 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)e-r(T-t)

_N t=1

_

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-(S) er(T-t)

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 risk-neutral claim contingent the underlying asset and then discounting at the risk-free rate: ('°'V (s) f (s)ds H(t) = e''

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n

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p

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ý

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M 2

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ý

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ý

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O

n 00 N

ý n

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ý O Ö

O Ö

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M

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C

C

O

c

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O

ý

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Q

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a

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O Ö



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v

H

ý

H

pG C7

v

O C

v

O O

ä

Q

H

W

Chapter 4: An Empirical

Comparison of APARCH Models

L2

Table 17: 12thorder Ljung-Box statistics for

and

`2 t

1983-1998

1983-1986

14.2359 0.2859

10.3005 0.5896

Q12 Qý 2

12.8218

ARCH

APARCH Q12

Q12



1991-1994

1995-1998

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

1987-1990

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

TSGARCH-I Ql2

Q

lz

TSGARCH-1I

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 p-values are reported in italic.

156

Chapter 4: An Empirical Comparison of APARCH Models

Table 18: Model Rankings for the AIC Metric (Excluding EGARCH) 1983-1998 1983-1986 1987-1990 1991-1994 1995-1998 AIC

AIC

AIC

AIC

AIC

APARCH

1

4

1

4

2

ARCH

6

6

6

7

7

GARCH

5

1

5

1

5

TSGARCH-I

3

3

4

5

4

TSGARCH-H

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)

1983-1998 1983-19861987-19901991-19941995-1998 AIC

AIC

AIC

AIC

AIC

APARCH

2

4

2

5

3

ARCH

7

7

7

8

8

GARCH

6

1

6

1

6

TSGARCH-I

4

3

5

6

5

TSGARCH-II

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

TSGARCH-I

19

6

TSGARCH-II

13

2

GJR

13

2

TARCH

30

7

EGARCH

11

1

Note: Score is the sum of the rank for each model in each sub-period.

Table 21: Model Rankings for MMEU and MMEO Criteria 1983-1998

1983-1986

1987-1990

1991-1994

1995-1998

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

TSGARCH-I

5

6

7

1

4

4

4

5

5

5

TSGARCH-II

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

N



N

00

\D

V'1

N

00

\D

W)

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et

r-

W,



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00

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N

Vl

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[-

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00

M

n

M

'I7

00

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00

M

t-

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00

v

l

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v

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°\ .-. ri

W

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n

M

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M

n

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N

00

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h

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r-

ei

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t-

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w

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cc

ä

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u ¢

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ý

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h

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vl

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vl

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n

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n

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w

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00

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[z1

a

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00

w ý

Vl

l-

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ä

Q

V1

N

O

M

W

N

%lO

n

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M

00

W ý

M

00

vl

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V

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º. ý

U

¢

¢

ý=

a cý

v

H

l-

ý

a C7

H



v

1:4

H

Chapter 4: An Empirical Comparison ofAPARCH Models

Table 24: Aggregated Rankings for Statistical Loss Functions

APARCH ARCH GARCH TSGARCH-I TSGARCH-II 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 sub-period.

161

Chapter 4: An Empirical Comparison of APARCH Models

Table 25: Correlations Between Out-of-Sample Buy and Sell Signals 1985-1986 EGARCH

EGARCH

1.000

GARCH

GARCH

1989-1990

ARCH

2-STAGE

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

2-STAGE

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

2-STAGE

Table 26: Statistics for Forecasts of Volatility

Changes 1989-1990

EGARCH

GARCH

104

104

104

31.643

3.862

-29.058

#.samples

2-STAGE

1.000

1985-1986

Min

406 .

1997-1998

EGARCH

GARCH

2-STAGE

1.000

1993-1994

Max

ARCH

1.000

2-STAGE

EGARCH

GARCH

ARCH

2-STAGE

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

2-STAGE

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

.

1993-1994

EGARCH

GARCH

1997-1998

ARCH

2-STAGE

EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costs Statistics for 1985-1986 without Filter EGARCH Rate of Returns

0.606881

Std. R. of Returns

GARCH

ARCH

2-STAGE

-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: Before-transactions-costs Statistics for 1989-1990 without Filter EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costs Statistics for 1993-1994 without Filter EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costs Statistics for 1997-1998 without Filter EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costs Statistics for 1985-1986 with ± 3% Delta Filter EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costs Statistics for 1989-1990 with ± 3% Delta Filter EGARCH

GARCH

ARCH

2-STAGE

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: Before-transactions-costsStatistics for 1993-1994with ± 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

2-STAGE

Table 34: Before-transactions-costs Statistics for 1997-1998 with ± 3% Delta Filter EGARCH Rate of Returns

0.061182

Std. R. of Returns

GARCH

ARCH

2-STAGE

-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: Before-transactions-costs

Statistics

for

1997-1998 with

± 3%

Delta Filter

(Excluding One Data Point) EGARCH

GARCH

ARCH

2-STAGE

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: After-transactions-costs EGARCH

Statistics for 1985-1986 with ± 3% Delta Filter GARCH

ARCH

2-STAGE

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: After-transactions-costs Statistics for 1989-1990 with ± 3% Delta Filter EGARCH Rate of Returns

GARCH

ARCH

2-STAGE

-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: After-transactions-costs EGARCH

Statistics for 1993-1994 with ± 3% Delta Filter GARCH

ARCH

2-STAGE

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: After-transactions-costs

Statistics

for

1997-1998 with

(Excluding One Data Point) EGARCH

GARCH

ARCH

2-STAGE

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 (1983-1998)

0.30 1 0.25 = 0.20 ý 0.15

0.10, 0.05 : 0.00-0.05 -0.10-0.15 -0.20

1

51

101

151

201

251

301

351

401

301

351

401

Figure 20: Autocorretations for T (1983-1998) 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 (1983-1998)

0.30 0.25 -ý 0.20 0.15 0.10-1 0.05 ý 0.00 -0.05 -0.10 -0.15 -0.20

201

W"ktl~

51

101

151

201

251

170

10 LAAW&AVWUlZW--uw

301

351

401

Chapter 4: An Empirical Comparison ofAPARCHModels

Figure 22: Autocorrelations

for r (1983-1986)

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 r-2(1983-1986)

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 (1983-1986)

0.15 ,

0.10 1 0.05

ýýý

0.00 0.00-0 -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 (1987-1990)

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 (1987-1990)

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 (1987-1990)

0.40 ý 0.30 0.20 0.10-11 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 (1991-1994)

0.15 1 0.10

51

1

101

Figure 29: Autocorrelations

151

201

251

301

351

401

for rz (1991-1994)

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 (1991-1994)

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 (1995-1998)

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 - (1995-1998)

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 (1995-1998)

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: 3-State SWARCH-L(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: 3-State SWARCH-L(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 (1993-1994) With 25

bps Transactions Costs and ± 3% Delta Filter 70 50

EGARCH

30

GARCH

-

ARCH

10

-

-10 -30

176

2-Stage

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 time-series evaluate approach S&P 500 index. Pricing the valuation swap models on variance a variance swap can and options-based 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 longer-termed 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 nine-month three-, and variance the sixswap contracts analysing implied is Our design time-series different of and models. research rich enough to specifications using including: 1) hoc Black-Scholes 2) models prominent ad stochastic of model; number a admit GARCH 4) local 6) jump-diffusion 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 well-selected contract days, we illustrate that the options-based 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. arbitrage-free 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. time-series underperform for hedging; 2) implied volatility is predominantly a monotonically decreasing function of maturity 3) strategy cannot term-structure produce patterns; enough variance and therefore options-based distributional dynamics implied by option parameters is not consistent with its time-series data as likelihood to Future by the estimation need the of maximum square-root 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 closed-form solutions for volatility

framework. derived mean-reversion 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 delta-risk. focus In Chapter 4 we have demonstratedthe the on reduction of strategy always insufficiency of a delta-neutral hedge in the event of large market moves. Once the however, delta-neutral delta. become long index trade a moves, can short or underlying Rehedging becomes necessary to maintain a delta-neutral position as the market moves. Since transaction and operational costs generally prohibit continuous rehedging, residual from is It the ultimately arises underlying options-based 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 options-based 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 over-the-counter 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

n-1

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 options-based strategy. volatility allows counterpartiesto exchange on cleaner for fixed Counterparties floating to variance. variance swap can variance use cash-flows 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 mean-reverting 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%

a-1 ,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 longer-maturity to two for contract a one-year 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 longer-termed forward variance has become more volatile than the shorter-termed forward variance.

We analyse the three-, six- and nine-month variance swap contracts by

implied in different different time-series of specifications and models at points evaluating time.

The underlying hypotheses of this project are that if options-based Demeterfi et al.

(1999) framework is mathematically correct then:

i) Each generalisationof the benchmark Black-Scholes model should be able to improve the volatility forecastability of the options-basedpricing model; ii) If option prices are indeed representativeof their underlying time-series and forwardlooking then the forecastability of options-based variance swap models should be superior to their time-seriescounterparts. 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 out-of-sample results; whether ii)

To investigate whether there may be any systematic difference in variance forecasting between time-series and options-based variance swap valuation models. performance It is intended to explore whether options-based models, which are forward-looking, are discrete-time 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 time-series 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 options-based 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 time-series and used variance the criteria of review We for implied framework then the models. swap variance outline options-based 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 time-series 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) In-sample analysis. In view of option-pricing, it refers to the ability of each generalisation of Black-Scholes 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) Out-of-sample analysis. It comparesthe variance forecastability of all six time-series and options-based 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 time-series and options-basedvariance swap model for each of the three maturity months, i. e. three-, six- and nine-month contracts; iii)

Consistency of options-implied

distributional

dynamics and time-series properties. Maximum likelihood estimation of a square-root process is used in order to identify potential inconsistency between options-implied dynamics and time-series data by looking into the estimated structural parameters.

It should be noted that our results are based on the use of eighteen well-designed 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 options-based forward-looking do be to and not use historical data. supposed

5.2.2 The Options-based Variance Swap Framework The original Black-Scholes 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 at-the-money of

60Aggregate rank is defined as the sum of the rank for each model in each sub-period. 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 log-contract to provide an accurateand flexible volatility hedge. Since then log-contract has becomean indispensablecomponent for volatility research. 5.2.2.1 Log-Contract Neuberger (1994) demonstratedthat by dynamically hedging the log-contract 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 log-contract 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 log-contract with appropriate 1/F, The delta for log-contract futures is to and until maturity. contracts equal amounts of independentof volatility. If traders' view on volatility is Kv01# QR, the value of log-contract 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"°ý-Qx)

T

life implied KY01 the is the over the volatility of the contract and volatility realised where QR 0. log-contract time at the in the price of log-contract, it is hedging the clear that one can replicate the cash-flows of By dynamically But is log-contract 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 log-contract, 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 log-contract, one can take a Taylor-series expansion derivatives logarithm to the second-order 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 log-contract, 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 cash-flows on the RHS, in particular the log payoff. 5.2.2.2 Demeterfi et al. Framework Since log-contract is non-traded 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 Black-Scholes 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 38-41 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 at-the-money 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 log-contract: 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 Black-Scholes 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,) _U-I S`

S, ) d(1og =2Qzdt t22T LS' ST Jo log gyý _TE S S-t0

:.

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



Max(K-ST, ))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 risk-free interest rate r and some arbitrary at strike struck boundary S' separatingactively traded out-of-the 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 r-t j=o

is: the strikes where the order of Kr-I. P < < K3,P < K2.P < KI.P < S* = Ko < Kl. < K2, < K3, < < Kr-i,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 log-contract.

Clearly the essence of this derivation is that log payoff can be decomposed into

a portfolio consisting of a forward contract and out-of-the-money 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 Black-Scholes 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 cash-flows, they will capture less than the true realised variance. According to Little and Pant (2001), this reduction is greater for the longer-maturity 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 short-dated 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 arbitrage-free 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 model-independent prices of variance swaps could process. volatility be inferred from the market prices of European-style 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 term-structure 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 in-the-money The degree, lesser termto options. presence of skews, smiles and, a out-of-the-money and basic it Black-Scholes 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 Black-Scholes 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 jump-diffusion model; (1973)

63There is not an equivalent framework for assetthat follows a jump-diffusion 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 two-dimensional 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

sub-sections 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 time-state-dependent There This the so-called 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, (heavy-tailed) distributions, leverage it returns effect or could simulate simulate non-Gaussian (2000) Bakshi as a stationary, mean-reverting suggested volatility process. et al. and capture diffusion inadequate inconsistency to models were explain pricing that one-dimensional After for 500 S&P in time-decay 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, Black-Scholes that volatility stochastic models the model outperformed evidence provided is in In moneyness-maturity all groups. almost volatility other words, stochastic significantly Black-Scholes' log-normal the describing of far-reaching 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 closed-form 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 out-of-

both to the zero correlation version of the stochastic when compared the-money options volatility

Black-Scholes 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 closed-form solution models

European options and at the same time allows a non-zero risk premium for volatility as well between One asset correlation returns and volatility. can also use the arbitrary as an long in time-series or the options market to calibrate model a information contained in-sample in thereafter context and compute out-of-sample 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 in-the-money

options relative

to out-of-the-money

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 from-the-money in the turn versus only affects pricing near-the-money of returns, which options.

Since options are usually traded near-the-money and the Black-Scholes formula

for identical to the stochastic models at-the-money 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 square-root process with mean-reversion. 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 , long-run 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 T-t T) (t, the B1 price pays zero-coupon a unit periods. of currency where from be Ramaswamy Since the obtained European and can put put-call parity. of a Bakshi (1993) Scott (1997) found that the stochasticinterest rate (1985), and et al. Sundaresan improve the Black-Scholes 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,

e-t4m(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 Jump-Diffusion Models 5.2.3.2.1 Justification for the Jump-Diffusion Approach The explanation that volatility

smile is the sole consequence of time-state-dependent 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 long-term options and cannot take a well-known by the effects smile exhibited strong short-term 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 diffusion-based models explaining shortsmile effects, particularly pure Bakshi et al. concluded that the Poisson-type 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 Black-Scholes could model rejected in favour of the variance-gamma model. that the Furthermore, empirical investigations of time-series 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 risk-neutral 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 jump-risk premia, stochastic pure model varying

194

Chapter 5: Empirical Performance ofAlternative Variance Swap Valuation Models

diffusion models. Pan concluded that introducing volatility-risk

premia in addition to jump-

risk premia would not result in any significant improvement in the goodness of fit. Qualitatively,

jump-diffusion

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 jump-diffusion

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 long-maturity 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, log-contract 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 non-continuous asset dynamics on variance swap pricing, we Demeterfi jump-diffusion 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 time-state-dependentapproach. 5.2.3.2.2 Bakshi et al. Model We adopt the closed-form jump-diffusion the jump-analysis.

(1997) for developed by Bakshi al. option model et

Following Baskshi et al., this risk-neutral jump-diffusion

known to variations6S of the Black-Scholes admit many enough including:

1) Black-Scholes model: A=0

The jump-diffusion

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 log-normally, 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, long-run diffusion, V, the coefficient of mean, and variation . The advantageof modelling volatility as a square-rootprocessis 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 put-call 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 Black-Scholes 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 Black-Scholes 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 Black-Scholes 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 continuous-time market option by discrete-time 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 Derman-Kani

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 implied-tree is primarily based on a discretisation of the asset price process, the finite-difference

focuses on developing a discrete-time model by discretising the approach

fundamental no-arbitrage partial differential scheme to the volatility Osher

(1997),

equation.

The application of finite-difference

smile problem has been studied by many authors, e.g. Lagnado and

Andersen

and

Brotherton-Ratcliffe

(1998),

Coleman

et

al.

(1999),

Chryssanthakopoulos (2001) and Little and Pant (2001). Whilst somewhat more complicated however, finite-difference calibrate, to evaluate and

scheme is shown to exhibit much better

implied-trees finite-difference because than properties convergence and stability not involve

explicit

adjustments of branching probabilities

time-partitioning. the and stockof prescription be shown to be similar to a trinomial implicit

or Crank-Nicolson

method does

and allows for independent

The explicit finite-difference 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 One-FactorModel The inspiring research by Breeden et al. (1978) stated that the risk-neutral probability distributions could be recoveredfrom European-style 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 risk-neutral world, Dupire's local volatility model is assumedto evolve according to the following one-factor diffusion model: continuous-time

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. risk-neutral 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 constantrisk-freerate. In the continuous-timelimit whenrisk-free 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[

acr-P (rr + gK T+ 11

- g)Ck7

a2cKr K2

1 1

;)V2

The major advantage of the above one-factor continuous model, as compared to the jumpdiffusion or stochastic model, is that no non-traded 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 risk-neutral 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 risk-neutral and prices can Consequently,the completenessof this one-factor 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 continuous-time results have been supplemented by a number of finite-difference methods. For example, Zou and Derman (1997) applied the "pseudo-analytical"

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 finite-difference to extract local volatility consistent with the equity option volatility implicit interest using rate of

and Crank-Nicholson

demonstrated its application by pricing

down-and-out

Coleman et al. (1999) developed a Crank-Nicholson

lattice

smile and term-structure

lattices68; Andersen et al. also knock-out

options.

scheme to "optimise"

In addition, local volatility

"smoothness" in introducing by the Black-Scholes 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 finite-difference method. In construct Little (2001) Coleman to et al., et al. also approximated a variance swap by using additional the Crank-Nicholson method in an extended Black-Scholes framework that was based on a decomposition of a two-dimensional problem into the solving of a set of onecleverly dimensional Black-Scholes partial differential equations. At a glance, Little et al.'s method because finite-difference 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 assumption-free 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 finite-difference the equation using arbitrage partial

method. Given Si,,i,, r, q and

cr(S, t) and under the no-arbitrage condition, the option value must satisfy the Black-Scholes

67 See also pp. 8-10 of Andersen and Brotherton-Ratcliffe

(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]

=e-ecT-t>

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 S-1 as 2I S=O as 2I

=0

Before applying finite-difference

method to calculate option prices, Q(S, t) needs to be

lack Due to of market option price data, i. e. non-continuum of strikes, this can approximated. be regarded as a well-known

but ill-posed 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 one-factor 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 well-behaved 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 p-vector 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 ex-dividend 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 ex-dividend 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 ex-dividend 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 T-t `S es-dividend(t)

-

Sclose (t)-

e-ý

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 risk-free 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 Newton-Raphson method to the Black-Scholes call option formula:

(t)N(d, )- Xe-rcr-t)N(d2 ) C(t, T) = Sex-eevraend (t) /X]+ (r + 0.5Q2)(T t) ln[Sex-divJdene dl QT -t d2 =d, -Q T-t 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

(out-of-the-money) in-the-money in given a category are quite similar to the implied options in the (in-the-money) opposing options out-of-the-money put of category regardless volatility term-to-expiration. or period sample of

For a fixed term-to-expiration,

call and put options

imply the same U-shaped volatility pattern across strike prices. Such similarities in pricing between due call and put to the working of the put-call 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 Black-Scholes, 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 June-November 2001 variance swap contracts is neither incidental nor arbitrary.

Many significant global macroeconomic and political events occurred during

the 2001-2002 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 Israel-Palestine 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 19-year 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 2000-2002 period was the worst three-year

1929-1931 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 2000-2002. turbulent over ride a indeed endured

for Call Options Statistics Descriptive 5.3.3 and S&P 500 Index in 1999-2002 index 500 data S&P for option are shown in table 40. Table 40 reveals that Basic statistics is higher in post-September 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. 1-B.6 exhibit the specifications and for daily Causal inspection input parameters our option contracts. of various option 1) lower have higher implied 2) B1-B6 that: strikes a volatility; reveals volatility appendices for is near-term 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 Kurtosis-3 Q(10) Jarque-Gerastat. #. Obs.

Pre-9/11 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: 1999-2002 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 nine-month 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 Jarque-Bera 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 post-9/11 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 mean-reverting property. In addition, the that between the 3- and 9-month 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 longer-termed 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 shorter-termed 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 June-November 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 term-structure relationship correctly during this period.

In

different how to variance swap models can predict the changing term-structure of assess order included have three-, six- and nine-month variance swap contracts which are variance, we Money International Market (IMM) the compatible with

rulebook76. The specifications for

the three-, six- and nine-month 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 Pre-9/11 Maturity

June 2001 Start

End

August 2001

July 2001 Start

Start

End

End

3-Month

15/06/01

20/09/01

20/7/01

18/10/01

17/08/01

15111101

6-Month

15/06/01

20/12101

20/7/01

17/01/02

17/08/01

14/02/02

9-Month

15/06/01

14/03/02

20/7/01

18/04/02

17/08/01

16/05/02

Post-9/11 Maturity

September2001 Start

End

November 2001

October 2001 Start

End

Start

End

3-Month

21/09/01

20/12/01

19/10/01

17/01/02

16/11/01

14/02/02

6-Month

21/09/01

14/03/02

19/10/01

18/04/02

16/11/01

16/05/02

9-Month

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 time-series and options-basedvariance swap models: i) Out-of-Sample Analysis. The out-of-sample 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) In-Sample Analysis. In-sample 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. In-sample 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 Options-based by time-series models are calibrated whilst call option variance. data Calibration historical to their estimate on structural parameters. rely models both inby and out-of-sample analysis; results are shared

Estimation Likelihood of the square-root process. We apply this procedure Maximum iv) dynamics implied by the that underlying illustrate options are not consistent with to data. time-series

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 out-of-sample 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 5.4.1.1-4 discuss the econometric

and numerical

Black-Scholes, 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 GARCH-VS (Javaheri et al., 2002) models.

5.4.1.1 Calibrations for Stochastic Volatility withlwithout jump Estimation of stochastic processes on discrete-time data is difficult.

Since volatility

is not

directly observable, many parameter estimation methods have relied either on time-series 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 cross-section 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 risk-neutral 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 asset-volatility jump-diffusion model. in a parameters

model, and into unique jump distribution

for are available solutions our selected stochastic models, a natural Since closed-form the risk-neutral 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 non-linear 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/jump-diffusion 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 non-stationary but option-asset hypothesised However, under most this metric would stationary are processes. price ratios lead to a more favourable treatment of cheaper options, e.g. out-of-the-money options at the long-term have Based in-the-money 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 finite-difference

method in MATLAB

using a trust-region optimisation algorithm

(Coleman et al., 1999) with a partial differential equation (PDE) approach's to directly solve for local volatility

Q(S, t).

The Black-Scholes partial differential

log-spacing. S-dimension the with along

Crank-Nicholson

discretised is equation

finite-difference

method is used

for solving the Black-Scholes partial differential equation because it improves the stability finite-difference 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 pre-determined 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 = jM-1,

j0,...,

AS =[f

*S;

-(1/f)*S;,, tt

u]l(M-1)

We use backwards difference to approximate a2C

ac

N-1

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

S-dimension. In boundary the the along conditions values addition, option time-zero gives a zC Is-u into the finite difference scheme by setting their incorporated 0 are =az S=L= 21 as as to zero. Further descriptions of finite-difference 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 Trust-Region Reflective Quasi-Newton Method Bicubic

important in implementation is the the element most spline

of finite-difference

determined is by Its solving a constrained non-linear 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 S-direction.

The reader is referred to Dierckx (1995) for discussions of

interpolation spline schemes that are smooth in both T- and S-directions. 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

built-in

minimisation "optimset"

MATLAB

Optimisation

is "lsgnonlin".

Through

Toolbox

function

the MATLAB

Large Scale Algorithm the options: select we

for

non-linear

Optimisation

least squares

Toolbox

function

ON, Jacobian OFF, and Function

Tolerance lx 10-3. Pre-conditioned Conjugate Gradient is left to the default value of zero. These settings refer to, respectively, the "trust-region

reflective

quasi-Newton"

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 finite-difference prices equal call values the European option provided routines. market index be initial let S;,,,,= 100, risk-free 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 twenty-four 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 Crank-Nicholson

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.. Finite-Difference 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,...,
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