Part I. Financial Markets and Financial Time Series
1. Introduction ................................................. 3
1.1. Financial markets and financial time series ............. 3
1.2. Econometric modeling of asset returns ................... 4
1.3. Applications of non-Gaussian econometrics ............... 5
1.4. Option pricing with non-Gaussian distributions .......... 5
2. Statistical Properties of Financial Market Data .............. 7
2.1. Definitions of returns .................................. 7
2.1.1. Simple returns ................................... 8
2.1.2. Log-returns ...................................... 8
2.1.3. Stylized facts ................................... 9
2.2. Distribution of returns ................................ 10
2.2.1. Moments of a random variable .................... 10
2.2.2. Empirical moments ............................... 14
2.2.3. Testing for normality ........................... 16
2.3. Time dependency ........................................ 21
2.3.1. Serial correlation in returns ................... 22
2.3.2. Serial correlation in volatility ................ 23
2.3.3. Volatility asymmetry ............................ 25
2.3.4. Time-varying higher moments ..................... 26
2.4. Linear dependence across returns ....................... 26
2.4.1. Pearson's correlation coefficient ............... 27
2.4.2. Test for equality of two correlation
coefficients .................................... 28
2.4.3. Test for equality of two correlation matrices ... 30
2.5. Multivariate higher moments ............................ 31
2.5.1. Multivariate co-skewness and co-kurtosis ........ 31
2.5.2. Computing moments of portfolio returns .......... 32
3. Functioning of Financial Markets and Theoretical Models
for Returns ................................................. 33
3.1. Functioning of financial markets ....................... 34
3.1.1. Organization of financial markets ............... 34
3.1.2. Examples of orders .............................. 37
3.1.3. Components of the bid-ask spread ................ 39
3.2. Mandelbrot and the stable distribution ................. 39
3.2.1. A puzzling result ............................... 40
3.2.2. Stable distribution ............................. 41
3.3. Clark's subordination model ............................ 44
3.3.1. The idea of the model ........................... 44
3.3.2. The density of returns under subordination ...... 46
3.4. A bivariate mixture-of-distribution model for return and
volume ................................................. 48
3.4.1. A microstructure model for information
arrivals ........................................ 48
3.4.2. Implications of the mixture of distributions
hypothesis ...................................... 53
3.4.3. Testing the mixture of distribution
hypothesis ...................................... 57
3.4.4. Extensions ...................................... 61
3.5. A model of prices and quotes in a quote-driven
market ................................................. 62
3.5.1. A model based on the trade flow ................. 63
3.5.2. Estimating the parameters ....................... 66
3.5.3. The quote process ............................... 68
3.5.4. Extension to the liquidation of a large
portfolio ....................................... 73
Part II. Econometric Modeling of Asset Returns
4. Modeling Volatility ......................................... 79
4.1. Volatility at lower frequencies ........................ 79
4.2. ARCH model ............................................. 81
4.2.1. Forecasting ..................................... 81
4.2.2. Kurtosis of an ARCH model ....................... 82
4.2.3. Testing for ARCH effects ........................ 82
4.2.4. ARCH-in-mean model .............................. 83
4.2.5. Illustration .................................... 84
4.3. GARCH model ............................................ 84
4.3.1. Forecasting ..................................... 88
4.3.2. Integrated GARCH model .......................... 89
4.3.3. Estimation ...................................... 89
4.3.4. Testing for GARCH effects ....................... 92
4.3.5. Software to estimate ARCH and GARCH models ...... 92
4.3.6. Illustration .................................... 93
4.4. Asymmetric GARCH models ................................ 94
4.4.1. EGARCH model .................................... 94
4.4.2. TGARCH model .................................... 95
4.4.3. GJR model ....................................... 95
4.4.4. Cox-Box transform ............................... 95
4.4.5. News impact curve ............................... 96
4.4.6. Partially non-parametric estimation ............. 96
4.4.7. Testing for asymmetric effects .................. 97
4.4.8. Illustration .................................... 99
4.5. GARCH model with jumps ................................. 99
4.5.1. A model with time-varying jump intensity ....... 101
4.5.2. An empirical illustration ...................... 105
4.6. Aggregation of GARCH processes ........................ 108
4.6.1. Temporal aggregation ........................... 109
4.6.2. Cross-sectional aggregation .................... 113
4.6.3. Estimation of the weak GARCH process ........... 114
4.7. Stochastic volatility ................................. 115
4.7.1. From GARCH models to stochastic volatility
models ......................................... 115
4.7.2. Estimation of the discrete time SV model ....... 117
4.8. Realized volatility ................................... 118
4.8.1. The difficulty to disentangle jumps ............ 119
4.8.2. Quadratic variation ............................ 123
4.8.3. Power variation ................................ 124
4.8.4. Bipower variation .............................. 126
4.8.5. Estimation over finite time intervals .......... 128
4.8.6. Realized covariance ............................ 135
4.8.7. Further related results ........................ 141
5. Modeling Higher Moments .................................... 143
5.1. The general problem ................................... 144
5.1.1. Higher moments of a GARCH process .............. 145
5.1.2. Quasi Maximum Likelihood Estimation ............ 148
5.1.3. The existence of distribution with given
moments ........................................ 151
5.2. Distributions with higher moments ..................... 152
5.2.1. Semi-parametric approach ....................... 153
5.2.2. Series expansion about the normal
distribution ................................... 155
5.2.3. Skewed Student t distribution .................. 159
5.2.4. Generating asymmetric distributions ............ 166
5.2.5. Pearson IV distribution ........................ 169
5.2.6. Entropy distribution ........................... 172
5.3. Specification tests and inference ..................... 177
5.3.1. Moment specification tests ..................... 177
5.3.2. Adequacy tests based on density forecasts ...... 179
5.3.3. Adequacy tests based on interval forecasts ..... 180
5.4. Illustration .......................................... 182
5.5. Modeling conditional higher moments ................... 188
5.5.1. Tests for autoregressive conditional higher
moments ........................................ 189
5.5.2. Modeling higher moments directly ............... 189
5.5.3. Modeling the parameters of the distribution .... 191
6. Modeling Correlation ....................................... 195
6.1. Multivariate GARCH models ............................. 197
6.1.1. Vectorial and diagonal GARCH models ............ 198
6.1.2. Dealing with large-dimensional systems ......... 200
6.1.3. Modeling conditional correlation ............... 206
6.1.4. Estimation issues .............................. 210
6.1.5. Specification tests ............................ 212
6.1.6. Test of constant conditional correlation
matrix ......................................... 214
6.1.7. Illustration ................................... 217
6.2. Modeling the multivariate distribution ................ 223
6.2.1. Standard multivariate distributions ............ 225
6.2.2. Skewed elliptical distribution ................. 230
6.2.3. Skewed Student t distribution .................. 233
6.2.4. Estimation ..................................... 236
6.2.5. Adequacy tests ................................. 239
6.2.6. Illustration ................................... 240
6.3. Copula functions ...................................... 240
6.3.1. Definitions and properties ..................... 241
6.3.2. Measures of concordance ........................ 242
6.3.3. Non-parametric copulas ......................... 244
6.3.4. Review of some copula families ................. 245
6.3.5. Estimation ..................................... 254
6.3.6. Adequacy tests ................................. 258
6.3.7. Modeling the conditional dependency
parameter ...................................... 259
6.3.8. Illustration ................................... 261
7. Extreme Value Theory ....................................... 265
7.1. Univariate tail estimation ............................ 266
7.1.1. Distribution of extremes ....................... 266
7.1.2. Tail distribution .............................. 276
7.1.3. The case of weakly dependent data .............. 291
7.1.4. Estimation of high quantiles ................... 296
7.2. Multivariate dependence ............................... 300
7.2.1. Characterizing tail dependency ................. 303
7.2.2. Estimation and statistical inference
on χ and χ ..................................... 307
7.2.3. Modeling dependency ............................ 308
7.2.4. An illustration ................................ 309
7.2.5. Further investigations ......................... 311
Part III. Applications of Non-Gaussian Econometrics
8. Risk Management and VaR .................................... 315
8.1. Definitions and measures .............................. 316
8.1.1. Definitions .................................... 316
8.1.2. Models for portfolio returns ................... 320
8.2. Historical simulation ................................. 321
8.3. Semi-parametric approaches ............................ 322
8.3.1. Extreme Value Theory (EVT) ..................... 324
8.3.2. Quantile regression technique .................. 328
8.4. Parametric approaches ................................. 330
8.4.1. RiskMetrics - J.P. Morgan ...................... 331
8.4.2. The portfolio-level approach ................... 334
8.4.3. The asset-level approach ....................... 337
8.5. Non-linear models ..................................... 341
8.5.1. The "delta-only" method ........................ 341
8.5.2. The "delta-gamma" method ....................... 341
8.6. Comparison of VaR models .............................. 342
8.6.1. Evaluation of VaR models ....................... 343
8.6.2. Comparison of methods .......................... 343
8.6.3. 10-day VaR and scaling ......................... 344
8.6.4. Illustration ................................... 345
9. Portfolio Allocation ....................................... 349
9.1. Portfolio allocation under non-normality .............. 349
9.1.1 Direct maximization of expected utility ......... 350
9.1.2. An approximate solution based on moments ....... 353
9.2. Portfolio allocation under downside risk .............. 359
9.2.1. Definition ..................................... 360
9.2.2. Downside risk as an additional constraint ...... 360
9.2.3. Downside risk as an optimization criterion ..... 361
Part IV. Option Pricing with Non-Gaussian Returns
10. Fundamentals of Option Pricing ............................ 365
10.1. Notations ........................................... 366
10.2. The no-arbitrage approach to option pricing ......... 369
10.2.1. Choice of a stock price process ............. 369
10.2.2. The fundamental partial differential
equation .................................... 371
10.2.3. Solving the fundamental PDE ................. 373
10.2.4. The Black-Scholes-Merton formula ............ 375
10.3. Martingale measure and BSM formula .................. 377
10.3.1. Self-financing strategies and portfolio
construction ................................ 377
10.3.2. Change of numeraire ......................... 378
10.3.3. Change of Brownian motion ................... 378
10.3.4. Evolution of St under Q ..................... 379
10.3.5. The expected pay-off as a martingale ........ 379
10.3.6. The trading strategies ...................... 380
10.3.7. Equivalent martingale measure ............... 381
11. Non-structural Option Pricing ............................. 383
11.1. Difficulties with the standard BSM model ............ 384
11.2. Direct estimation of the risk-neutral density ....... 385
11.2.1. Expression for the RND ...................... 385
11.2.2. Estimating the parameters of the RND ........ 387
11.3. Parametric methods .................................. 389
11.3.1. Mixture of log-normal distributions ......... 389
11.3.2. Mixtures of hypergeometric functions ........ 394
11.3.3. Generalized beta distribution ............... 395
11.4. Semi-parametric methods ............................. 395
11.4.1. Edgeworth expansions ........................ 395
11.4.2. Hermite polynomials ......................... 399
11.5. Non-parametric methods .............................. 402
11.5.1. Spline methods .............................. 402
11.5.2. Tree-based methods .......................... 406
11.5.3. Maximum entropy principle ................... 407
11.5.4. Kernel regression ........................... 408
11.6. Comparison of various methods ....................... 409
11.7. Relationship with real probability .................. 414
11.7.1. The link between RNDs and objective
densities ................................... 414
11.7.2. Empirical findings .......................... 416
12. Structural Option Pricing ................................. 417
12.1. Stochastic volatility model ......................... 417
12.1.1. The square root process ..................... 418
12.1.2. Solving the PDE based on characteristic
function .................................... 419
12.1.3. A new partial differential equation ......... 422
12.2. Option pricing with stochastic volatility ........... 425
12.2.1. Hull and White (1987, 1988) ................. 425
12.2.2. Heston (1993) ............................... 426
12.2.3. Characteristic function of the SV model ..... 428
12.2.4. Further insights ............................ 429
12.3. Models with jumps ................................... 432
12.3.1. Stochastic process with jumps ............... 432
12.3.2. Diffusion with double exponential jumps ..... 434
12.3.3. Combining stochastic volatility with
jumps ....................................... 436
12.3.4. Jumpy affine models ......................... 440
12.4. Models with even wilder jumps: Levy option
pricing ............................................. 441
12.4.1. Commonly used Levy processes ................ 443
12.4.2. Choice of the time-changing process ......... 444
12.4.3. Option pricing .............................. 445
12.4.4. Pricing options with risk-neutral
characteristic function ..................... 446
12.4.5. Empirical results ........................... 447
Part V. Appendices on Option Pricing Mathematics
13. Brownian Motion and Stochastic Calculus ................... 451
13.1. Law of large numbers and the central limit
theorem ............................................. 451
13.2. Random walks ........................................ 453
13.3. Construction of the Brownian motion ................. 453
13.4. Properties of the Brownian motion ................... 456
13.5. Stochastic integration .............................. 457
13.6. Stochastic differential equations ................... 459
13.7. Ito's lemma ......................................... 460
13.8. Multivariate extension of Ito's lemma ............... 462
13.9. Transition probabilities and partial differential
equations ........................................... 463
13.10. Kolmogorov backward and forward equations .......... 464
13.11. PDE associated with diffusions ..................... 466
13.12. Feynman-Kac formula ................................ 468
14. Martingale and Changing Measure ........................... 471
14.1. Martingales ......................................... 471
14.2. Changing probability of a normal distribution ....... 472
14.3. Radon-Nikodym derivative ............................ 473
14.4. Girsanov's theorem .................................. 474
14.5. Martingale representation theorem ................... 475
15. Characteristic Functions and Fourier Transforms ........... 477
15.1. Characteristic functions ............................ 477
15.1.1. Basic properties ............................ 478
15.1.2. Moments and the characteristic function ..... 478
15.1.3. Convolution theorem ......................... 479
15.1.4. Uniqueness .................................. 480
15.1.5. Inversion theorem ........................... 480
15.2. Fourier transform and characteristic function ....... 483
16. Jump Processes ............................................ 487
16.1. Counting and marked point process ................... 487
16.2. The Poisson process ................................. 489
16.2.1. Construction of the Poisson distribution .... 489
16.2.2. Properties of the Poisson distribution ...... 491
16.2.3. Moments of pure Poisson process ............. 492
16.2.4. Compound Poisson process .................... 493
16.3. The exponential distribution ........................ 494
16.3.1. Definition and properties ................... 494
16.3.2. Moments of the exponential variable ......... 495
16.3.3. Hazard and survivor functions ............... 496
16.4. Duration between Poisson jumps ...................... 497
16.5. Compensated Poisson processes ....................... 498
17. Levy Processes ............................................ 501
17.1. Construction of the Levy process .................... 501
17.2. Properties of Levy processes ........................ 505
References .................................................... 507
Index ......................................................... 535
|
