Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

Cover......Page 1
About......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Introduction......Page 12
1.1 Market Model......Page 16
1.1.1 Brownian Motion......Page 17
1.1.2 Stochastic Integrals......Page 18
1.1.3 Risky Asset Price Model......Page 19
1.1.4 Ito’s Formula......Page 21
1.1.6 Ornstein–Uhlenbeck Process......Page 23
1.2.1 European Call and Put Options......Page 25
1.2.2 American Options......Page 26
1.2.3 Other Exotic Options......Page 27
1.3 Replicating Strategies......Page 28
1.3.1 Replicating Self-Financing Portfolios......Page 29
1.3.2 The Black–Scholes Partial Differential Equation......Page 30
1.3.3 Pricing to Hedge......Page 31
1.3.4 The Black–Scholes Formula......Page 32
1.3.5 The Greeks......Page 34
1.4 Risk-Neutral Pricing......Page 36
1.4.1 Equivalent Martingale Measure......Page 37
1.4.2 Self-Financing Portfolios......Page 38
1.4.3 Risk-Neutral Valuation......Page 39
1.4.4 Using the Markov Property......Page 40
1.5 Risk-Neutral Expectations and Partial Differential Equations......Page 42
1.5.1 Infinitesimal Generators and Associated Martingales......Page 43
1.5.2 Conditional Expectations and Parabolic Partial Differential Equations......Page 44
1.5.3 Kolmogorov Equations......Page 45
1.5.4 Application to the Black–Scholes Partial Differential Equation......Page 46
1.6.1 Optimal Stopping......Page 47
1.6.2 Free Boundary Value Problems......Page 49
1.7.1 Barrier Options......Page 51
1.7.2 Lookback Options......Page 54
1.7.4 Compound Options......Page 56
1.7.5 Asian Options......Page 57
1.8.1 Merton’s Approach......Page 58
1.8.2 First-Passage Model......Page 59
1.9.1 Multidimensional Ito’s Formula......Page 61
1.9.2 Girsanov’s Theorem......Page 62
1.9.3 The Feynman–Kac Formula......Page 63
1.10 Complete Market......Page 64
Notes......Page 65
2 
Introduction to Stochastic Volatility Models......Page 66
Remarks......Page 67
2.1.1 Interpretation of the Skew......Page 70
2.1.2 What Data to Use......Page 71
2.2 Local Volatility......Page 72
2.2.1 Time-Dependent Volatility......Page 73
2.2.2 Level-Dependent Volatility and Dupire’s Formula......Page 74
2.3.1 One-Factor Stochastic Volatility Models......Page 77
2.3.2 Stock Price Distribution under Stochastic Volatility......Page 79
2.4 Derivative Pricing......Page 80
2.4.1 Hedging Argument......Page 81
2.4.2 Pricing with Equivalent Martingale Measures......Page 84
2.4.3 Market Price of Volatility Risk and Data......Page 86
2.4.4 Short-Time Tight Fit vs. Long-Time Approximate Fit: The
(K,T, t)-Problem......Page 88
2.5.1 Implied Volatility as a Function of Moneyness......Page 89
2.5.2 Uncorrelated Volatility, Hull–White Formula, and Renault–Touzi Theorem......Page 90
2.5.3 Correlated Stochastic Volatility......Page 93
2.5.4 Multifactor Stochastic Volatility Models......Page 95
2.6 Summary and Conclusions......Page 98
Notes......Page 99
3.1 A Simple Picture of Fast and Slow Time Scales......Page 101
3.2 Ergodicity and Mean-Reversion......Page 103
3.3.1 Example: Markov Chain......Page 110
3.3.2 Example: Another Jump Process......Page 112
3.3.3 Example: The Ornstein–Uhlenbeck Process......Page 115
3.3.4 Example: The Cox–Ingersoll–Ross Process......Page 119
3.3.5 Other One-Dimensional Diffusion Processes......Page 121
3.4.1 The Returns Time Series......Page 125
3.4.2 Returns Process with Jump Volatility......Page 126
3.4.3 Returns Process with OU Volatility......Page 127
3.5.1 Short-Term Volatility Statistics of the S&P 500......Page 129
3.5.2 Variogram Analysis......Page 130
3.5.3 Effect of Longer Scales......Page 131
3.5.5 Spectrum......Page 132
3.6 Multiscale Models......Page 133
Notes......Page 135
4.1 Option Pricing under Multiscale Stochastic Volatility......Page 136
4.1.1 Models under Risk-Neutral Measures......Page 137
4.1.2 Pricing Partial Differential Equations......Page 139
4.2 Formal Regular and Singular Perturbation Analysis......Page 140
4.2.1 Zeroth-Order Approximation P0......Page 141
4.2.2 Fast Time Scale Correction Pε
1,0......Page 143
4.2.3 Explicit Formula for Pε
1,0 for European Options......Page 146
4.2.4 Slow Time Scale Correction Pδ
0,1......Page 147
4.2.5 Explicit Formula for Pδ
0,1 for European Options......Page 149
4.3 Parameter Reduction......Page 150
4.4 First-Order Approximation: Summary and Discussion......Page 152
4.5 Accuracy of First-Order Approximation......Page 153
5 
Implied Volatility Formulas and Calibration......Page 163
5.1 Approximate Call Prices and Implied Volatilities......Page 164
5.2 Calibration Procedure......Page 169
5.3 Illustration with S&P 500 Data......Page 170
5.3.1 Data Cleaning......Page 171
5.3.2 Fast Volatility Factor Only......Page 172
5.3.4 Fast and Slow Volatility Factors......Page 173
5.3.5 Parameter Stability......Page 174
5.4 Maturity Cycles......Page 178
5.4.1 Perturbation with Time-Varying Coefficients......Page 179
5.4.2 Calendar Time......Page 182
5.4.3 Calibration with Time-Dependent Parameters......Page 184
5.4.4 Practical Fitting with Time-Dependent Parameters......Page 186
5.4.5 Maturity Cycles for S&P 500 Data......Page 187
5.5 Higher-Order Corrections......Page 189
5.5.1 Second-Order Fitting......Page 190
5.5.2 Capturing the S&P 500 Wings......Page 192
Notes......Page 193
6.1.1 Approximation Formula......Page 194
6.1.2 Accuracy of the Approximation......Page 195
6.2.1 First-Order Correction......Page 196
6.2.2 Explicit Expressions......Page 199
6.3 Asian Options......Page 200
Notes......Page 203
7.1 American Options Valuation under Stochastic Volatility......Page 204
7.2 Stochastic Volatility Correction for American Put......Page 205
7.3 Parameter Reduction......Page 211
7.4 Summary......Page 212
8
 Hedging Strategies......Page 214
8.2 The Strategy and its Cost......Page 215
8.2.1 Averaging Effect......Page 217
8.2.2 Small Noise Effect and Summary of the Cost......Page 220
8.3 Mean Self-Financing Hedging Strategy......Page 221
8.4.1 The Black–Scholes Strategy with Frozen Volatility......Page 224
8.4.2 The Small Noise Effect......Page 226
8.4.3 Bias Reduction......Page 229
8.5.1 Strategy Based on Dynamically Calibrated Parameters......Page 232
8.5.2 Staying Close to the Price......Page 234
8.6.1 Main Argument......Page 235
8.6.2 Decomposition Result......Page 237
8.6.3 Comparison with the PDE Approach......Page 240
8.7 Non-Markovian Models of Volatility......Page 241
8.7.1 Setting: An Example......Page 242
8.7.2 Asymptotics in the Non-Markovian Case......Page 243
9.1.1 Dividends......Page 247
9.1.2 Varying Interest Rates......Page 249
9.2 Probabilistic Representation of the Approximate Prices......Page 252
9.3 Second-Order Correction from Fast Scale......Page 253
9.3.1 Expansion and Successive Equations......Page 254
9.3.2 Computation of the Second Correction......Page 256
9.3.3 Accuracy and Parameter Reduction......Page 257
9.3.4 Correction to the Skew......Page 260
9.4 Second-Order Corrections from Slow and Fast Scales......Page 262
9.5 Periodic Day Effect......Page 264
9.6 Markovian Jump Volatility Models......Page 266
9.7 Multidimensional Models......Page 269
10 
Around the Heston Model......Page 274
10.1.1 European Derivatives under the Heston Model......Page 275
10.1.2 Numerical Evaluation of Call Options......Page 278
10.2.1 Fast Mean-Reverting Heston Model......Page 280
10.2.2 Second-Order Approximation for a Fast Mean-Reverting Heston Model......Page 282
10.2.3 Slowly Varying Heston Model......Page 283
10.3.1 A Multiscale Generalized Heston Model......Page 286
10.3.2 Pricing Equation......Page 287
10.3.3 Asymptotic Analysis......Page 288
10.3.4 The Multiscale Implied Volatility Surface......Page 290
10.4 Large Deviations and Short Maturity Asymptotics......Page 291
10.4.2 Moment-Generating Function and its Asymptotic......Page 293
10.4.3 Applications to Pricing and Implied Volatilities......Page 296
11.1.1 Importance Sampling......Page 298
11.1.2 Control Variates......Page 301
11.2.1 Constant Volatility Merton Problem......Page 302
11.2.2 Stochastic Volatility Merton Problem......Page 304
11.2.3 A Practical Solution......Page 308
11.3.1 Discrete-Time CAPM and Forward-Looking Betas......Page 311
11.3.2 Continuous-Time CAPM with Stochastic Volatility......Page 313
11.3.3 Pricing Risk-Neutral Measure......Page 315
11.3.4 Market and Asset Option Prices......Page 316
11.3.6 Examples......Page 319
12.1 The Vasicek Model......Page 322
12.1.1 Bond Pricing in the Vasicek Model......Page 324
12.1.2 Yield Curve......Page 326
12.1.4 Forward Rates......Page 328
12.2 The Bond Price and its Expansion......Page 330
12.2.1 The Full Bond-Pricing Problem......Page 331
12.2.2 Bond Price Expansion......Page 332
12.2.3 The Fast Scale Correction......Page 334
12.2.4 The Slow Scale Correction......Page 335
12.2.5 Combined Corrections......Page 336
12.2.6 Group Parameter Reduction......Page 337
12.2.7 Bond Approximation Formulas......Page 338
12.2.8 Yield Correction......Page 340
12.3 The Quadratic Model......Page 342
12.4 The CIR Model......Page 345
12.5.1 Pricing via Partial Differential Equations......Page 350
12.5.2 Pricing via Forward Measures......Page 351
12.5.3 Bond Option Pricing under the Vasicek Model......Page 353
12.5.4 Stochastic Volatility Correction Factors......Page 354
13.1 Single-Name Credit Derivatives......Page 357
13.1.1 Black–Cox Model for Defaultable Bond Price with Stochastic Volatility......Page 360
13.1.2 Effect of Stochastic Volatility on Yield Spreads......Page 363
13.1.3 Accuracy of Approximation......Page 365
13.1.4 Calibration......Page 366
13.2.1 Model Setup......Page 368
13.2.2 Approximated Joint Survival Probabilities......Page 371
13.2.3 Loss Distribution......Page 380
13.2.4 Models with Name–Name Correlation......Page 386
14.1.1 Poisson Process......Page 392
14.1.2 Inhomogeneous Poisson Process......Page 394
14.1.3 Doubly Stochastic or Cox Processes......Page 395
14.1.4 Connection with Interest Rate Derivatives......Page 396
14.1.5 Credit Default Swap......Page 398
14.2 Multiname Credit Derivatives......Page 400
14.2.1 The CDO Contract......Page 402
14.3 Symmetric Vasicek Model......Page 403
14.3.1 Vasicek Intensities......Page 404
14.3.3 The Loss Distribution......Page 406
14.3.4 Stochastic Volatility Effects......Page 409
14.4 Homogeneous Group Structure......Page 417
14.4.1 Factor Models......Page 418
14.4.2 Homogeneous Group Structure......Page 421
14.4.3 Stochastic Volatility in the Default Intensity Process......Page 423
14.4.4 CDO Mechanics......Page 428
14.4.5 Dow Jones CDX Fitting......Page 431
15
 Epilogue......Page 439
Sideways......Page 442
References......Page 445
Index......Page 454