A Time Series Approach to Option Pricing (eBook)

Notation 6
Contents 10
List of Figures 12
List of Tables 16
1 Introduction 18
References 25
2 The Time Series Toolbox for Financial Returns 27
2.1 Stylized Facts 27
2.1.1 Stationarity, Ergodicity and Autocorrelation 28
2.1.2 Time-Varying Volatility and Leverage Effects 31
2.1.3 Semi Fat-Tailed Distributions 32
2.2 Symmetric GARCH Models 34
2.2.1 From Linear to Non-linear Models 34
2.2.2 Definitions 37
2.2.3 Stationarity Properties 39
2.2.4 Covariance Structure of the Squares 43
2.2.5 Why We Need More: Kurtosis and Asymmetry in a GARCH(1,1) Model 47
2.3 Asymmetric Extensions 49
2.3.1 GJR Model 51
2.3.2 EGARCH Model 53
2.3.3 APARCH Model 55
2.3.4 Concluding Remarks 57
2.4 Conditional Distribution of Returns 58
2.4.1 Generalized Hyperbolic Distributions 58
2.4.2 Mixture of Two Gaussian Distributions 61
2.4.3 Some Practical Remarks 63
2.5 GARCH in Mean 64
2.6 Dealing with the Estimation Challenge 65
2.6.1 Maximum Likelihood 66
2.6.2 Quasi Maximum Likelihood 67
2.6.3 Recursive Estimation 68
2.6.4 Empirical Finite Sample Properties of the Three Estimation Methodologies 69
2.7 From GARCH Processes to Continuous Diffusions 74
2.7.1 Convergence Toward Hull1987 Diffusions 75
2.7.2 Convergence Toward Diffusions with Deterministic Volatilities 77
2.7.3 Convergence Toward the ch2:Heston1993 Model 77
References 78
3 From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach 83
3.1 Description of the Economy Under the Historical Probability 84
3.2 Option Pricing in Discrete Time 86
3.2.1 Arbitrage-Free Price of a European Contingent Claim 86
3.2.2 The Stochastic Discount Factor 89
3.2.3 Economic Interpretation: The CCAPM Model 91
3.3 The Extended Girsanov Principle 96
3.3.1 Definition and Properties 96
3.3.2 Risk-Neutral Dynamics for Classical Distributions 99
3.4 The Conditional Esscher Transform 100
3.4.1 Definition and Properties 100
3.4.2 Risk-Neutral Dynamics for Classical Distributions 105
The Generalized Hyperbolic Distribution 106
Mixture of Gaussian Distributions 109
Gaussian Jumps 110
3.5 Second Order Esscher Transform 111
3.6 The Empirical Martingale Simulation Method 115
3.7 Remarks on Closed-Form Option Pricing Formulas 117
3.8 Proofs of Chapter 3 119
References 127
4 Empirical Performances of Discrete Time Series Models 130
4.1 Historical Dynamics of Option Prices 131
4.2 The Heston and Nandi Case: Calibration vs. Estimation 144
4.3 Empirical Performances of Heavy Tailed Models 164
4.3.1 Estimation Strategies 164
4.3.2 Pricing Performances 177
4.4 Conclusion 187
References 188
Mathematical Appendix 190
Gaussian Random Variables 190
Conditional Expectation 191
Monte Carlo Methods 195
Convergence of Discrete Time Markov Processes to Diffusions 198
From Moment Generating Functions to Option Prices 199
References 200
Index 201