Statistics of Financial Markets: An Introduction

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Contents
Preface to the Third Edition
Preface to the Second Edition
I Option Pricing 1
1 Derivatives 3
1.1 Recommended Literature . . . . . . . . . . . . . . . . . . . . 10
1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Introduction to Option Management 13
2.1 Arbitrage Relations . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Portfolio Insurance . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Binary One-Period Model . . . . . . . . . . . . . . . . . . . . 32
2.4 Recommended Literature . . . . . . . . . . . . . . . . . . . . 37
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Basic Concepts of Probability Theory 43
3.1 Real Valued Random Variables . . . . . . . . . . . . . . . . . 43
3.2 Expectation and Variance . . . . . . . . . . . . . . . . . . . . 46
3.3 Skewness and Kurtosis . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Random Vectors, Dependence, Correlation . . . . . . . . . . . 48
3.5 Conditional Probabilities and Expectations . . . . . . . . . . 49
3.6 Recommended Literature . . . . . . . . . . . . . . . . . . . .
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Stochastic Processes in Discrete Time 55
4.1 Binomial Processes . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Trinomial Processes . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 General Random Walks . . . . . . . . . . . . . . . . . . . . . 61
4.4 Geometric Random Walks . . . . . . . . . . . . . . . . . . . . 62
4.5 Binomial Models with State Dependent Increments . . . . . . 63
4.6 Recommended Literature . . . . . . . . . . . . . . . . . . . . 64
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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5 Stochastic Integrals and Differential Equations 67
5.1 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . 73
5.4 The Stock Price as a Stochastic Process . . . . . . . . . . . . 76
5.5 Itˆo’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 Recommended Literature . . . . . . . . . . . . . . . . . . . . 82
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Black–Scholes Option Pricing Model 85
6.1 Black–Scholes Differential Equation . . . . . . . . . . . . . . . 85
6.2 Black–Scholes Formula for European Options . . . . . . . . . 92
6.2.1 Numerical Approximation . . . . . . . . . . . . . . . . 96
6.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.1 Linear Congruential Generator . . . . . . . . . . . . . 100
6.3.2 Fibonacci Generators . . . . . . . . . . . . . . . . . . 104
6.3.3 Inversion Method . . . . . . . . . . . . . . . . . . . . . 106
6.3.4 Box-Muller Method . . . . . . . . . . . . . . . . . . . 107
6.3.5 Marsaglia Method . . . . . . . . . . . . . . . . . . . . 109
6.4 Risk Management and Hedging . . . . . . . . . . . . . . . . . 110
6.4.1 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . 113
6.4.2 Gamma and Theta . . . . . . . . . . . . . . . . . . . . 116
6.4.3 Rho and Vega . . . . . . . . . . . . . . . . . . . . . . .
6.4.4 Volga and Vanna . . . . . . . . . . . . . . . . . . . . . 121
6.4.5 Historical and Implied Volatility . . . . . . . . . . . .
6.4.6 Realised Volatility . . . . . . . . . . . . . . . . . . . .
6.5 Recommended Literature . . . . . . . . . . . . . . . . . . . . 126
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7 Binomial Model for European Options 133
7.1 Cox–Ross–Rubinstein Approach to Option Pricing . . . . . . 134
7.2 Discrete Dividends . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Dividends as a Percentage of the Stock Price . . . . . 139
7.2.2 Dividends as a Fixed Amount of Money . . . . . . . . 140
7.3 Recommended Literature . . . . . . . . . . . . . . . . . . . . 143
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 American Options 145
8.1 Arbitrage Relations for American Options . . . . . . . . . . . 145
8.2 The Trinomial Model for American Options . . . . . . . . . . 153
8.3 Recommended Literature . . . . . . . . . . . . . . . . . . . . 157
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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9 Exotic Options 159
9.1 Compound Options, Option on Option . . . . . . . . . . . . . 159
9.2 Chooser Options or “As You Wish” Options . . . . . . . . . . 161
9.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.4 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.5 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . 167
9.6 Cliquet Options . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.7 Basket Options . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.8 Recommended Literature . . . . . . . . . . . . . . . . . . . . 170
9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10 Interest Rates and Interest Rate Derivatives 173
10.1 Interest Rates and Prices . . . . . . . . . . . . . . . . . . . . 173
10.1.1 Money Market Account . . . . . . . . . . . . . . . . . 176
10.1.2 Forward Rate Agreement . . . . . . . . . . . . . . . . 176
10.1.3 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . 177
10.2 Risk Neutral Valuation and Numeraire Measures . . . . . . . 179
10.2.1 Principles of Risk Neutral Valuation . . . . . . . . . . 180
10.2.2 Change of Numeraire . . . . . . . . . . . . . . . . . . 181
10.2.3 Equivalent Martingale Measure . . . . . . . . . . . . . 182
10.2.4 Traditional Risk Neutral Numeraire . . . . . . . . . . 183
10.2.5 Other Choices of Numeraire . . . . . . . . . . . . . . . 184
10.3 Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . 185
10.3.1 The Black Model . . . . . . . . . . . . . . . . . . . . . 186
10.3.2 Bond Option . . . . . . . . . . . . . . . . . . . . . . . 186
10.3.3 Caps and Floors . . . . . . . . . . . . . . . . . . . . . 187
10.3.4 Swaption . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.4 Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . 190
10.4.1 One-Factor Short-Rate Models . . . . . . . . . . . . . 191
10.4.2 Two-Factor Short-Rate Models . . . . . . . . . . . . . 193
10.5 Heath Jarrow Morton Framework . . . . . . . . . . . . . . . . 195
10.5.1 HJM Approach . . . . . . . . . . . . . . . . . . . . . . 195
10.5.2 Short Rate Process in the HJM Framework . . . . . . 197
10.6 LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . 197
10.6.1 Dynamics in the LMM . . . . . . . . . . . . . . . . . . 198
10.6.2 The Numeraire Measure . . . . . . . . . . . . . . . . . 198
10.7 Bond Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.7.1 The Bond Valuation Equation . . . . . . . . . . . . . 199
10.7.2 Solving the Zero Bond Valuation . . . . . . . . . . . . 200
10.8 Calibrating Short-Rate Models . . . . . . . . . . . . . . . . . 201
10.8.1 CIR Process Densities . . . . . . . . . . . . . . . . . . 202
10.8.2 Initial Estimates . . . . . . . . . . . . . . . . . . . . .
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Statistics of Financial Markets: An Introduction Statistics of Financial Markets，3rd Edition.pdf (17.31 MB)

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Statistics of Financial Markets: An Introduction

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关键词：introduction troduction Statistics financial statistic Insurance

II Statistical Models of Financial Time Series 213
11 Introduction: Definitions and Concepts 215
11.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.2 Statistical Analysis of German and British Stock Returns . . 223
11.3 Expectations and Efficient Markets . . . . . . . . . . . . . . . 226
11.4 Econometric Models: A Brief Summary . . . . . . . . . . . . 231
11.4.1 Stock Prices: the CAPM . . . . . . . . . . . . . . . . 231
11.4.2 Exchange Rate: Theory of the Interest Rate Parity . . 233
11.4.3 Term Structure: The Cox-Ingersoll-Ross Model . . . . 235
11.4.4 Options: The Black-Scholes Model . . . . . . . . . . . 237
11.4.5 The Market Price of Risk . . . . . . . . . . . . . . . . 239
11.5 The Random Walk Hypothesis . . . . . . . . . . . . . . . . . 242
11.6 Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 244
11.6.1 Dickey-Fuller Test . . . . . . . . . . . . . . . . . . . . 245
11.6.2 The KPSS Test . . . . . . . . . . . . . . . . . . . . . . 248
11.6.3 Variance Ratio Tests . . . . . . . . . . . . . . . . . . . 249
11.7 Recommended Literature . . . . . . . . . . . . . . . . . . . . 252
11.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
12 ARIMA Time Series Models 255
12.1 Moving Average Processes . . . . . . . . . . . . . . . . . . . . 256
12.2 Autoregressive Process . . . . . . . . . . . . . . . . . . . . . . 257
12.3 ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.4 Partial Autocorrelation . . . . . . . . . . . . . . . . . . . . . 263
12.5 Estimation of Moments . . . . . . . . . . . . . . . . . . . . . 267
12.5.1 Estimation of the Mean Function . . . . . . . . . . . . 267
12.5.2 Estimation of the Covariance Function . . . . . . . . . 269
12.5.3 Estimation of the ACF . . . . . . . . . . . . . . . . . . 270
12.6 Portmanteau Statistics . . . . . . . . . . . . . . . . . . . . . . 271
12.7 Estimation of AR(p) Models . . . . . . . . . . . . . . . . . . . 272
12.8 Estimation of MA(q) and ARMA(p, q) Models . . . . . . . . . 273
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Contents
12.9 Recommended Literature . . . . . . . . . . . . . . . . . . . . 278
12.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
13 Time Series with Stochastic Volatility 283
13.1 ARCH and GARCH Models . . . . . . . . . . . . . . . . . . . 285
13.1.1 ARCH(1): Definition and Properties . . . . . . . . . . 286
13.1.2 Estimation of ARCH(1) Models . . . . . . . . . . . . . 295
13.1.3 ARCH(q): Definition and Properties . . . . . . . . . . 299
13.1.4 Estimation of an ARCH(q) Model . . . . . . . . . . . 301
13.1.5 Generalised ARCH (GARCH) . . . . . . . . . . . . . . 302
13.1.6 Estimation of GARCH(p, q) Models . . . . . . . . . . 305
13.2 Extensions of the GARCH Model . . . . . . . . . . . . . . . .
13.2.1 Exponential GARCH . . . . . . . . . . . . . . . . . . . 308
13.2.2 Threshold ARCH Models . . . . . . . . . . . . . . . . 310
13.2.3 Risk and Returns . . . . . . . . . . . . . . . . . . . . . 311
13.2.4 Estimation Results for DAX and FTSE 100 Returns . 312
13.3 Shortfalls of GARCH . . . . . . . . . . . . . . . . . . . . . . . 314
13.3.1 Recent Challenges to GARCH Models . . . . . . . . . 314
13.3.2 Volatility Forecasting for DAX and FTSE 100 Returns 321
13.4 Multivariate GARCH Models . . . . . . . . . . . . . . . . . . 323
13.4.1 The Vec Specification . . . . . . . . . . . . . . . . . . 324
13.4.2 The BEKK Specification . . . . . . . . . . . . . . . . .
13.4.3 The CCC Model . . . . . . . . . . . . . . . . . . . . . 328
13.4.4 The DCC Model . . . . . . . . . . . . . . . . . 328
13.4.5 An Empirical Illustration . . . . . . . . . . . . . . . . 329
13.5 Continuous-Time GARCH Models . . . . . . . . . . . . . . . 333
13.5.1 COGARCH(1,1): Definition and Properties . . . . . . 334
13.5.2 Relation between GARCH and COGARCH . . . . . . 335
13.5.3 Estimation of the COGARCH(1,1) Model . . . . . . . 336
13.5.4 Extensions of the COGARCH Model . . . . . . . . . . 337
13.6 Recommended Literature . . . . . . . . . . . . . . . . . . . .
13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Long Memory Time Series 343
14.1 Definition of long range dependence . . . . . . . . . . . . . . 343
14.2 Fractional Integration and Long-Memory . . . . . . . . . . . 344
14.3 Long Memory and Self-Similar Processes . . . . . . . . . . . . 347
14.4 Detection of the Long Memory . . . . . . . . . . . . . . . . . 350
14.4.1 Rescaled Variance Test . . . . . . . . . . . . . . . . . 351
14.4.2 Semiparametric test . . . . . . . . . . . . . . . . . . . 352
14.4.3 Tests for Spurious Long Memory . . . . . . . . . . . . 353
14.5 Estimation of the Long Memory parameter . . . . . . . . . . 354
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14.5.1 Exact Maximum Likelihood Estimator . . . . . . . . . 354
14.5.2 Regression on the Periodogram . . . . . . . . . . . . . 355
14.5.3 Gaussian Semiparametric Estimator . . . . . . . . . . 355
14.6 Long Memory Models . . . . . . . . . . . . . . . . . . . . . . 357
14.6.1 ARFIMA Model . . . . . . . . . . . . . . . . . . . . . 357
14.6.2 GARCH Long Memory Models . . . . . . . . . . . . . 358
14.6.3 FIAPARCH Model . . . . . . . . . . . . . . . . . . . . 360
14.6.4 HYGARCH Model . . . . . . . . . . . . . . . . . . . . 361
14.7 An empirical illustration . . . . . . . . . . . . . . . . . . . . . 361
14.8 Recommended Literature . . . . . . . . . . . . . . . . . . . . 364
15 Non-Parametric and Flexible Time Series Estimators 367
15.1 Nonparametric Regression . . . . . . . . . . . . . . . . . . . . 368
15.2 Construction of the Estimator . . . . . . . . . . . . . . . . . . 370
15.3 Empirical illustration . . . . . . . . . . . . . . . . . . . . . . . 372
15.4 Flexible Volatility Estimators . . . . . . . . . . . . . . . . . . 373
15.5 Pricing Options with ARCH-Models . . . . . . . . . . . . . . 374
15.6 Application to the Valuation of DAX Calls . . . . . . . . . . 381
15.7 Recommended Literature . . . . . . . . . . . . . . . . . . . . 384
III Selected Financial Applications 387
16 Value at Risk and Backtesting 389
16.1 Forecast and VaR Models . . . . . . . . . . . . . . . . . . . . 391
16.2 Backtesting with Expected Shortfall . . . . . . . . . . . . . . 393
16.3 Backtesting in Action . . . . . . . . . . . . . . . . . . . . . . 395
16.4 Recommended Literature . . . . . . . . . . . . . . . . . . . . 400
16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
17 Copulae and Value at Risk 405
17.1 Copulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
17.2 Copula Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 409
17.2.1 Simplest Copulae . . . . . . . . . . . . . . . . . . . . . 409
17.2.2 Elliptical Copulae . . . . . . . . . . . . . . . . . . 412
17.2.3 Archimedean Copulae . . . . . . . . . . . . . . . . . . 415
17.2.4 Hierarchical Archimedean Copulae . . . . . . . . . . . 418
17.2.5 Generalisations . . . . . . . . . . . . . . . . . . . . . . 419
17.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 420
17.3.1 Conditional Inverse Method . . . . . . . . . . . . . . . 420
17.3.2 Marshal-Olkin Method . . . . . . . . . . . . . . . . . .
17.4 Copula Estimation . . . . . . . . . . . . . . . . . . . . . . . .
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17.4.1 FML – Full Maximum Likelihood Estimation . . . . . 426
17.4.2 IFM – Inference for Margins . . . . . . . . . . . . . . 426
17.4.3 CML – Canonical Maximum Likelihood . . . . . . . . 427
17.4.4 Gaussian Copula Estimation . . . . . . . . . . . . . .
17.4.5 t-Copula Estimation . . . . . . . . . . . . . . . . . . . 428
17.5 Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 429
17.6 Value-at-Risk of the Portfolio Returns . . . . . . . . . . . . . 430
17.6.1 VaR of the P&L . . . . . . . . . . . . . . . . . . . . .
17.6.2 3-dimensional Portfolio . . . . . . . . . . . . . . . . .
17.7 Recommended Literature . . . . . . . . . . . . . . . . . . . . 445
17.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Statistics of Extreme Risks 447
18.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 447
18.2 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . 449
18.3 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . 452
18.3.1 The Block Maxima Method . . . . . . . . . . . . . . . 453
18.3.2 The Peaks-Over-Threshold (POT) Method . . . . . . 463
18.4 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
18.5 Extreme Value Theory for Time Series . . . . . . . . . . . . . 478
18.6 Recommended Literature . . . . . . . . . . . . . . . . . . . . 483
18.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
19 Neural Networks 489
19.1 . . . . . . . . . 490
19.2 Back Propagation . . . . . . . . . . . . . . . . . . . . . . . 499
19.3 Neural Networks in Non-parametric Regression Analysis . . . 501
19.4 Forecasts of Financial Time Series with Neural Networks . . . 508
19.5 . . . . . . . . . . . . 512
19.6 Recommended Literature . . . . . . . . . . . 516

20 Volatility Risk of Option Portfolios
20.1 Description of the Data . . . . . . . . . . . . . . . . . . . . .
20.2 Principal Component Analysis of the VDAX’s Dynamics . . .
20.3 Stability Analysis of the VDAX’s Dynamics . . . . . . . . . .
20.4 . . . . . . . . . . . .
20.5 Recommended Literature . . . . . . . . . . . . . . . . . . . .
20.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 Nonparametric Estimators for the Probability of Default
21.1 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . .
21.2 Semi-parametric Model for Credit Rating . . . . . . . . . . .
21.3 Credit Ratings with Neural Networks . . . . . . . . . . . . . .
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Quantifying Risk with Neural Networks
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Measure of the Implied Volatility’s Risk 528
. . . . .
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From Perceptron to Non-linear Neuron . . . .
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Exercises
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Contents
22 Credit Risk Management
22.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .
22.2 The Bernoulli Model . . . . . . . . . . . . . . . . . . . . . . .
22.3 . . . . . . . . . . . . . . . . . . .
22.4 The Industrial Models . . . . . . . . . . . . . . . . . . . .
22.5 One Factor Models . . . . . . . . . . . . . . . . . . .
22.6 Copulae and Loss Distributions . . . . . .
22.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Technical Appendix
A.1 . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . .
Frequently Used Notations

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