书名：Modeling and pricing of swaps for financial and energy markets with stochastic volatilities
责任者：Anatoliy Swishchuk | University of Calgary | Canada. | Svishchuk, A. V.
出版时间：2013
出版社：World Scientific,
摘要
Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities is devoted to the modeling and pricing of various kinds of swaps, such as those for variance, volatility, covariance, correlation, for financial and energy markets with different stochastic volatilities, which include CIR process, regime-switching, delayed, mean-reverting, multi-factor, fractional, Lévy-based, semi-Markov and COGARCH(1,1). One of the main methods used in this book is change of time method. The book outlines how the change of time method works for different kinds of models and problems arising in financial and energy markets and the associated problems in modeling and pricing of a variety of swaps. The book also contains a study of a new model, the delayed Heston model, which improves the volatility surface fitting as compared with the classical Heston model. The author calculates variance and volatility swaps for this model and provides hedging techniques. The book considers content on the pricing of variance and volatility swaps and option pricing formula for mean-reverting models in energy markets. Some topics such as forward and futures in energy markets priced by multi-factor Lévy models and generalization of Black-76 formula with Markov-modulated volatility are part of the book as well, and it includes many numerical examples such as S&P60 Canada Index, S&P500 Index and AECO Natural Gas Index.
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目录
Preface vii
Acknowledgments xi
1. Stochastic Volatility 1
1.1 Introduction 1
1.2 Non-Stochastic Volatilities 2
1.2.1 Historical Volatility 2
1.2.2 Implied Volatility 2
1.2.3 Level-Dependent Volatility and Local Volatility 3
1.3 Stochastic Volatility 3
1.3.1 Approaches to Introduce Stochastic Volatility 5
1.3.2 Discrete Models for Stochastic Volatility 6
1.3.3 Jump-Diffusion Volatility 6
1.3.4 Multi-Factor Models for Stochastic Volatility 6
1.4 Summary 7
Bibliography 8
2. Stochastic Volatility Models 11
2.1 Introduction 11
2.2 Heston Stochastic Volatility Model 11
2.3 Stochastic Volatility with Delay 12
2.4 Multi-Factor Stochastic Volatility Models 12
2.5 Stochastic Volatility Models with Delay and Jumps 13
2.6 Lévy-Based Stochastic Volatility with Delay 14
2.7 Delayed Heston Model 14
2.8 Semi-Markov-Modulated Stochastic Volatility 15
2.9 COGARCH(1,1) Stochastic Volatility Model 16
2.10 Stochastic Volatility Driven by Fractional Brownian Motion 16
2.10.1 Stochastic Volatility Driven by Fractional Ornstein-Uhlenbeck Process 16
2.10.2 Stochastic Volatility Driven by Fractional Vasićek Process 17
2.10.3 Markets with Stochastic Volatility Driven by Geometric Fractional Brownian Motion 17
2.10.4 Stochastic Volatility Driven by Fractional Continuous-Time GARCH Process 17
2.11 Mean-Reverting Stochastic Volatility Model (Continuous-Time GARCH Model) in Energy Markets 18
2.12 Summary 19
Bibliography 19
3. Swaps 21
3.1 Introduction 21
3.2 Definitions of Swaps 21
3.2.1 Variance and Volatility Swaps 21
3.2.2 Covariance and Correlation Swaps 23
3.2.3 Pseudo-Swaps 24
3.3 Summary 26
Bibliography 26
4. Change of Time Methods 29
4.1 Introduction 29
4.2 Descriptions of the Change of Time Methods 29
4.2.1 The General Theory of Time Changes 31
4.2.2 Subordinators as Time Changes 32
4.3 Applications of Change of Time Method 33
4.3.1 Black-Scholes by Change of Time Method 33
4.3.2 An Option Pricing Formula for a Mean-Reverting Asset Model Using a Change of Time Method 33
4.3.3 Swaps by Change of Time Method in Classical Heston Model 33
4.3.4 Swaps by Change of Time Method in Delayed Heston Model 34
4.4 Different Settings of the Change of Time Method 34
4.5 Summary 36
Bibliography 37
5. Black-Scholes Formula by Change of Time Method 39
5.1 Introduction 39
5.2 Black-Scholes Formula by Change of Time Method 39
5.2.1 Black-Scholes Formula 39
5.2.2 Solution of SDE for Geometric Brownian Motion using Change of Time Method 40
5.2.3 Properties of the Process W~(øt-1) 41
5.3 Black-Scholes Formula by Change of Time Method 42
5.4 Summary 43
Bibliography 43
6. Modeling and Pricing of Swaps for Heston Model 45
6.1 Introduction 45
6.2 Variance and Volatility Swaps 48
6.2.1 Variance and Volatility Swaps for Heston Model 51
6.3 Covariance and Correlation Swaps for Two Assets with Stochastic Volatilities 54
6.3.1 Definitions of Covariance and Correlation Swaps 54
6.3.2 Valuing of Covariance and Correlation Swaps 55
6.3.3 Variance Swaps for Lévy-Based Heston Model 57
6.4 Numerical Example: S&P60 Canada Index 58
6.5 Summary 61
Bibliography 61
7. Modeling and Pricing of Variance Swaps for Stochastic Volatilities with Delay 65
7.1 Introduction 65
7.2 Variance Swaps 67
7.2.1 Modeling of Financial Markets with Stochastic Volatility with Delay 68
7.2.2 Variance Swaps for Stochastic Volatility with Delay 72
7.2.3 Delay as A Measure of Risk 75
7.2.4 Comparison of Stochastic Volatility in Heston Model and Stochastic Volatility with Delay 75
7.3 Numerical Example 1: S&P60 Canada Index 77
7.4 Numerical Example 2: S&P500 Index 80
7.5 Summary 83
Bibliography 83
8. Modeling and Pricing of Variance Swaps for Multi-Factor Stochastic Volatilities with Delay 87
8.1 Introduction 87
8.1.1 Variance Swaps 87
8.1.2 Volatility 88
8.2 Multi-Factor Models 89
8.3 Multi-Factor Stochastic Volatility Models with Delay 91
8.4 Pricing of Variance Swaps for Multi-Factor Stochastic Volatility Models with Delay 93
8.4.1 Pricing of Variance Swap for Two-Factor Stochastic Volatility Model with Delay and with Geometric Brownian Motion Mean-Reversion 93
8.4.2 Pricing of Variance Swap for Two-Factor Stochastic Volatility Model with Delay and with Ornstein-Uhlenbeck Mean-Reversion 96
8.4.3 Pricing of Variance Swap for Two-Factor Stochastic Volatility Model with Delay and with Pilipovic One-Factor Mean-Reversion 98
8.4.4 Variance Swap for Three-Factor Stochastic Volatility Model with Delay and with Pilipovic Mean-Reversion 100
8.5 Numerical Example 1: S&P60 Canada Index 103
8.6 Summary 110
Bibliography 110
9. Pricing Variance Swaps for Stochastic Volatilities with Delay and Jumps 113
9.1 Introduction 113
9.2 Stochastic Volatility with Delay 114
9.3 Pricing Model of Variance Swaps for Stochastic Volatility with Delay and Jumps 117
9.3.1 Simple Poisson Process Case 118
9.3.2 Compound Poisson Process Case 121
9.3.3 More General Case 123
9.4 Delay as a Measure of Risk 126
9.5 Numerical Example 127
9.6 Summary 133
Bibliography 133
10. Variance Swap for Local Lévy-Based Stochastic Volatility with Delay 137
10.1 Introduction 137
10.2 Variance Swap for Lévy-Based Stochastic Volatility with Delay 139
10.3 Examples 141
10.3.1 Example 1 (Variance Gamma) 141
10.3.2 Example 2 (Tempered Stable) 142
10.3.3 Example 3 (Jump-Diffusion) 142
10.3.4 Example 4 (Kou's Jump-Diffusion) 143
10.4 Parameter Estimation 143
10.5 Numerical Example: S&P500 (2000-01-01-2009-12-31) 144
10.6 Summary 147
Bibliography 148
11. Delayed Heston Model: Improvement of the Volatility Surface Fitting 151
11.1 Introduction 151
11.2 Modeling of Delayed Heston Stochastic Volatility 153
11.3 Model Calibration 155
11.4 Numerical Results 158
11.5 Summary 159
Bibliography 159
12. Pricing and Hedging of Volatility Swap in the Delayed Heston Model 161
12.1 Introduction 161
12.2 Modeling of Delayed Heston Stochastic Volatility: Recap 163
12.3 Pricing Variance and Volatility Swaps 164
12.4 Volatility Swap Hedging 167
12.5 Numerical Results 169
12.6 Summary 171
Bibliography 171
13. Pricing of Variance and Volatility Swaps with Semi-Markov Volatilities 173
13.1 Introduction 173
13.2 Martingale Characterization of Semi-Markov Processes 173
13.2.1 Markov Renewal and Semi-Markov Processes 173
13.2.2 Jump Measure for Semi-Markov Process 175
13.2.3 Martingale Characterization of Semi-Markov Processes 175
13.3 Minimal Risk-Neutral (Martingale) Measure for Stock Price with Semi-Markov Stochastic Volatility 176
13.3.1 Current Life Stochastic Volatility Driven by Semi-Markov Process (Current Life Semi-Markov Volatility) 176
13.3.2 Minimal Martingale Measure 176
13.4 Pricing of Variance Swaps for Stochastic Volatility Driven by a Semi-Markov Process 177
13.5 Example of Variance Swap for Stochastic Volatility Driven by Two-State Continuous-Time Markov Chain 179
13.6 Pricing of Volatility Swaps for Stochastic Volatility Driven by a Semi-Markov Process 179
13.6.1 Volatility Swap 179
13.6.2 Pricing of Volatility Swap 181
13.7 Discussions of Some Extensions 182
13.7.1 Local Current Stochastic Volatility Driven by a Semi-Markov Process (Local Current Semi-Markov Volatility) 182
13.7.2 Local Stochastic Volatility Driven by a Semi-Markov Process (Local Semi-Markov Volatility) 183
13.7.3 Dupire Formula for Semi-Markov Local Volatility 183
13.7.4 Risk-Minimizing Strategies (or Portfolios) and Residual Risk 184
13.8 Summary 186
Bibliography 186
14. Covariance and Correlation Swaps for Markov-Modulated Volatilities 189
14.1 Introduction 189
14.2 Martingale Representation of Markov Processes 191
14.3 Variance and Volatility Swaps for Financial Markets with Markov-Modulated Stochastic Volatilities 194
14.3.1 Pricing Variance Swaps 195
14.3.2 Pricing Volatility Swaps 196
14.4 Covariance and Correlation Swaps for a Two Risky Assets for Financial Markets with Markov-Modulated Stochastic Volatilities 198
14.4.1 Pricing Covariance Swaps 198
14.4.2 Pricing Correlation Swaps 200
14.4.3 Correlation Swap Made Simple 200
14.5 Example: Variance, Volatility, Covariance and Correlation Swaps for Stochastic Volatility Driven by Two-State Continuous Markov Chain 202
14.6 Numerical Example 203
14.6.1 S&P500: Variance and Volatility Swaps 203
14.6.2 S&P500 and NASDAQ-100: Covariance and Correlation Swaps 205
14.7 Correlation Swaps: First Order Correction 206
14.8 Summary 209
Bibliography 209
15. Volatility and Variance Swaps for the COGARCH(1,1) Model 211
15.1 Introduction 211
15.2 Lévy Processes 212
15.3 The COGARCH Process of Klüppelberg et al 213
15.3.1 The COGARCH(1,1) Equations 213
15.3.2 Informal Derivation of COGARCH(1,1) Equation 213
15.3.3 The Second Order Properties of the Volatility Process σt 214
15.4 Pricing Variance and Volatility Swaps under the COGARCH(1,1) Model 214
15.4.1 Variance Swaps 215
15.4.2 Volatility Swaps 217
15.5 Formula for ξ1 and ξ2 220
15.6 Summary 223 Bibliography 223
16. Variance and Volatility Swaps for Volatilities Driven by Fractional Brownian Motion 225
16.1 Introduction 225
16.2 Variance and Volatility Swaps 226
16.3 Fractional Brownian Motion and Financial Markets with Long-Range Dependence 227
16.3.1 Definition and Some Properties of Fractional Brownian Motion 227
16.3.2 How to Model Long-Range Dependence on Financial Market 228
16.4 Modeling of Financial Markets with Stochastic Volatilities Driven by Fractional Brownian Motion (fBm) 229
16.4.1 Markets with Stochastic Volatility Driven by Fractional Ornstein-Uhlenbeck Process 230
16.4.2 Markets with Stochastic Volatility Driven by Fractional Vasićek Process 230
16.4.3 Markets with Stochastic Volatility Driven by Geometric Fractional Brownian Motion 231
16.4.4 Markets with Stochastic Volatility Driven by Fractional Continuous-Time GARCH Process 231
16.5 Pricing of Variance Swaps 231
16.5.1 Variance Swaps for Markets with Stochastic Volatility Driven by Fractional Ornstein-Uhlenbeck Process 232
16.5.2 Variance Swaps for Markets with Stochastic Volatility Driven by Fractional Vasićek Process 232
16.5.3 Variance Swaps for Markets with Stochastic Volatility Driven by Geometric fBm 233
16.5.4 Variance Swaps for Markets with Stochastic Volatility Driven by Fractional Continuous-Time GARCH Process 233
16.6 Pricing of Volatility Swaps 234
16.6.1 Volatility Swaps for Markets with Stochastic Volatility Driven by Fractional Ornstein-Uhlenbeck Process 235
16.6.2 Volatility Swaps for Markets with Stochastic Volatility Driven by Fractional Vasićek Process 236
16.6.3 Volatility Swaps for Markets with Stochastic Volatility Driven by Geometric fBm 236
16.6.4 Volatility Swaps for Markets with Stochastic Volatility Driven by Fractional Continuous-Time GARCH Process 237
16.7 Discussion: Asymptotic Results for the Pricing of Variance Swaps with Zero Risk-Free Rate when the Expiration Date Increases 238
16.8 Summary 239
Bibliography 239
17. Variance and Volatility Swaps in Energy Markets 241
17.1 Introduction 241
17.2 Mean-Reverting Stochastic Volatility Model (MRSVM) 243
17.2.1 Explicit Solution of MRSVM 244
17.2.2 Some Properties of the Process W~(øt-1) 244
17.2.3 Explicit Expression for the Process W~(øt-1) 245
17.2.4 Some Properties of the Mean-Reverting Stochastic Volatility σ2(t) : First Two Moments, Variance and Covariation 246
17.3 Variance Swap for MRSVM 247
17.4 Volatility Swap for MRSVM 247
17.5 Mean-Reverting Risk-Neutral Stochastic Volatility Model 249
17.5.1 Risk-Neutral Stochastic Volatility Model (SVM) 249
17.5.2 Variance and Volatility Swaps for Risk-Neutral SVM 250
17.5.3 Numerical Example: AECO Natural GAS Index (1 May 1998-30 April 1999) 250
17.6 Summary 252
Bibliography 252
18. Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets 255
18.1 Introduction 255
18.2 Mean-Reverting Asset Model (MRAM) 256
18.3 Explicit Option Pricing Formula for European Call Option for MRAM under Physical Measure 256
18.3.1 Explicit Solution of MRAM 256
18.3.2 Properties of the Process W~(øt-1) 257
18.3.3 Explicit Expression for the Process W~(øt-1) 258
18.3.4 Some Properties of the Mean-Reverting Asset St 259
18.3.5 Explicit Option Pricing Formula for European Call Option for MRAM under Physical Measure 260
18.4 Mean-Reverting Risk-Neutral Asset Model (MRRNAM) 263
18.5 Explicit Option Pricing Formula for European Call Option for MRRNAM 264
18.5.1 Explicit Solution for the Mean-Reverting Risk-Neutral Asset Model 264
18.5.2 Some Properties of the Process W~*((ø*t)-1) 265
18.5.3 Explicit Expression for the Process W~*(øt-1) 265
18.5.4 Some Properties of the Mean-Reverting Risk-Neutral Asset St 267
18.5.5 Explicit Option Pricing Formula for European Call Option for MRAM under Risk-Neutral Measure 268
18.5.6 Black-Scholes Formula Follows: L* = 0 and α = r 268
18.6 Numerical Example: AECO Natural GAS Index (1 May 1998-30 April 1999) 269
18.7 Summary 271
Bibliography 271
19. Forward and Futures in Energy Markets: Multi-Factor Lévy Models 273
19.1 Introduction 273
19.2 α-Stable Lévy Processes and Their Properties 274
19.2.1 Lévy Processes 274
19.2.2 Lévy-Khintchine Formula and Lévy-Ito Decomposition for Lévy Processes L(t) 274
19.2.3 α-Stable Distributions and Lévy Processes 275
19.3 Stochastic Differential Equations Driven by α-Stable Lévy Processes 277
19.3.1 One-Factor α-Stable Lévy Models 277
19.3.2 Multi-Factor α-Stable Lévy Models 277
19.4 Change of Time Method (CTM) for SDEs Driven by Lévy Processes 278
19.4.1 Solutions of One-Factor Lévy Models using the CTM 278
19.4.2 Solution of Multi-Factor Lévy Models using CTM 279
19.5 Applications in Energy Markets 280
19.5.1 Energy Forwards and Futures 280
19.5.2 Gaussian- and Lévy-Based SABR/LIBOR Market Models 282
19.6 Summary 282
Bibliography 283
20. Generalization of Black-76 Formula: Markov-Modulated Volatility 285
20.1 Introduction 285
20.2 Generalization of Black-76 Formula with Markov-Modulated Volatility 286
20.2.1 Black-76 Formula 286
20.2.2 Pricing Options for Markov-Modulated Markets 287
20.2.3 Proof of Theorem 20.3 292
20.2.4 Proof of Theorem 20.5 293
20.3 Numerical Results for Synthetic Data 293
20.3.1 Case Without Jumps 293
20.3.2 Case with Jumps 293
20.4 Applications: Data from Nordpool 296
20.5 Summary 298
Bibliography 299
Index 301
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