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Table of Contents

Preface xv

Contributors xix

Frequently Used Notation xxi

I Value at Risk 1

1 Approximating Value at Risk in Conditional Gaussian Models 3
Stefan R. Jaschke and Yuze Jiang

1.1 Introduction ~ 3
1.1.1 The Practical Need ~ 3
1.1.2 Statistical Modeling for VaR ~ 4
1.1.3 VaR Approximations ~ 6
1.1.4 Pros and Cons of Delta-Gamma Approximations ~ 7
1.2 General Properties of Delta-Gamma-Normal Models ~ 8
1.3 Cornish-Fisher Approximations ~ 12
1.3.1 Derivation ~ 12
1.3.2 Properties ~ 15
1.4 Fourier Inversion ~ 16
1.4.1 Error Analysis ~ 16
1.4.2 Tail Behavior ~ 20
1.4.3 Inversion of the cdf minus the Gaussian Approximation 21
1.5 Variance Reduction Techniques in Monte-Carlo Simulation ~ 24
1.5.1 Monte-Carlo Sampling Method ~ 24
1.5.2 Partial Monte-Carlo with Importance Sampling ~ 28
1.5.3 XploRe Examples ~ 30

2 Applications of Copulas for the Calculation of Value-at-Risk 35
Jorn Rank and Thomas Siegl

2.1 Copulas ~ 36
2.1.1 Definition ~ 36
2.1.2 Sklar's Theorem ~ 37
2.1.3 Examples of Copulas ~ 37
2.1.4 Further Important Properties of Copulas ~ 39
2.2 Computing Value-at-Risk with Copulas ~ 40
2.2.1 Selecting the Marginal Distributions ~ 40
2.2.2 Selecting a Copula ~ 41
2.2.3 Estimating the Copula Parameters ~ 41
2.2.4 Generating Scenarios - Monte Carlo Value-at-Risk ~ 43
2.3 Examples ~ 45
2.4 Results ~ 47

3 Quantification of Spread Risk by Means of Historical Simulation 51
Christoph Frisch and Germar Knochlein

3.1 Introduction ~ 51
3.2 Risk Categories { a Definition of Terms ~ 51
3.3 Descriptive Statistics of Yield Spread Time Series ~ 53
3.3.1 Data Analysis with XploRe ~ 54
3.3.2 Discussion of Results ~ 58
3.4 Historical Simulation and Value at Risk ~ 63
3.4.1 Risk Factor: Full Yield ~ 64
3.4.2 Risk Factor: Benchmark ~ 67
3.4.3 Risk Factor: Spread over Benchmark Yield ~ 68
3.4.4 Conservative Approach ~ 69
3.4.5 Simultaneous Simulation ~ 69
3.5 Mark-to-Model Backtesting ~ 70
3.6 VaR Estimation and Backtesting with XploRe ~ 70
3.7 P-P Plots ~ 73
3.8 Q-Q Plots ~ 74
3.9 Discussion of Simulation Results ~ 75
3.9.1 Risk Factor: Full Yield ~ 77
3.9.2 Risk Factor: Benchmark ~ 78
3.9.3 Risk Factor: Spread over Benchmark Yield ~ 78
3.9.4 Conservative Approach ~ 79
3.9.5 Simultaneous Simulation ~ 80
3.10 XploRe for Internal Risk Models ~ 81

II Credit Risk 85

4 Rating Migrations 87
Steffi Hose, Stefan Huschens and Robert Wania

4.1 Rating Transition Probabilities ~ 88
4.1.1 From Credit Events to Migration Counts ~ 88
4.1.2 Estimating Rating Transition Probabilities ~ 89
4.1.3 Dependent Migrations ~ 90
4.1.4 Computation and Quantlets ~ 93
4.2 Analyzing the Time-Stability of Transition Probabilities ~ 94
4.2.1 Aggregation over Periods ~ 94
4.2.2 Are the Transition Probabilities Stationary? ~ 95
4.2.3 Computation and Quantlets ~ 97
4.2.4 Examples with Graphical Presentation ~ 98
4.3 Multi-Period Transitions ~ 101
4.3.1 Time Homogeneous Markov Chain ~ 101
4.3.2 Bootstrapping Markov Chains ~ 102
4.3.3 Computation and Quantlets ~ 104
4.3.4 Rating Transitions of German Bank Borrowers ~ 106
4.3.5 Portfolio Migration ~ 106

5 Sensitivity analysis of credit portfolio models 111
Rudiger Kiesel and Torsten Kleinow

5.1 Introduction ~ 111
5.2 Construction of portfolio credit risk models ~ 113
5.3 Dependence modelling ~ 114
5.3.1 Factor modelling ~ 115
5.3.2 Copula modelling ~ 117
5.4 Simulations ~ 119
5.4.1 Random sample generation ~ 119
5.4.2 Portfolio results ~ 120

III Implied Volatility 125

6 The Analysis of Implied Volatilities 127
Matthias R. Fengler, Wolfgang Hardle and Peter Schmidt

6.1 Introduction ~ 128
6.2 The Implied Volatility Surface ~ 129
6.2.1 Calculating the Implied Volatility ~ 129
6.2.2 Surface smoothing ~ 131
6.3 Dynamic Analysis ~ 134
6.3.1 Data description ~ 134
6.3.2 PCA of ATM Implied Volatilities ~ 136
6.3.3 Common PCA of the Implied Volatility Surface ~ 137

7 How Precise Are Price Distributions Predicted by IBT? 145
Wolfgang Hardle and Jun Zheng

7.1 Implied Binomial Trees ~ 146
7.1.1 The Derman and Kani (D & K) algorithm ~ 147
7.1.2 Compensation ~ 151
7.1.3 Barle and Cakici (B & C) algorithm ~ 153
7.2 A Simulation and a Comparison of the SPDs ~ 154
7.2.1 Simulation using Derman and Kani algorithm ~ 154
7.2.2 Simulation using Barle and Cakici algorithm ~ 156
7.2.3 Comparison with Monte-Carlo Simulation ~ 158
7.3 Example { Analysis of DAX data ~ 162

8 Estimating State-Price Densities with Nonparametric Regression 171
Kim Huynh, Pierre Kervella and Jun Zheng

8.1 Introduction ~ 171
8.2 Extracting the SPD using Call-Options ~ 173
8.2.1 Black-Scholes SPD ~ 175
8.3 Semiparametric estimation of the SPD ~ 176
8.3.1 Estimating the call pricing function ~ 176
8.3.2 Further dimension reduction ~ 177
8.3.3 Local Polynomial Estimation ~ 181
8.4 An Example: Application to DAX data ~ 183
8.4.1 Data ~ 183
8.4.2 SPD, delta and gamma ~ 185
8.4.3 Bootstrap confidence bands ~ 187
8.4.4 Comparison to Implied Binomial Trees ~ 190

9 Trading on Deviations of Implied and Historical Densities 197
Oliver Jim Blaskowitz and Peter Schmidt

9.1 Introduction ~ 197
9.2 Estimation of the Option Implied SPD ~ 198
9.2.1 Application to DAX Data ~ 198
9.3 Estimation of the Historical SPD ~ 200
9.3.1 The Estimation Method ~ 201
9.3.2 Application to DAX Data ~ 202
9.4 Comparison of Implied and Historical SPD ~ 205
9.5 Skewness Trades ~ 207
9.5.1 Performance ~ 210
9.6 Kurtosis Trades ~ 212
9.6.1 Performance ~ 214
9.7 A Word of Caution ~ 216

IV Econometrics 219

10 Multivariate Volatility Models 221
Matthias R. Fengler and Helmut Herwart Z

10.1 Introduction ~ 221
10.1.1 Model specifications ~ 222
10.1.2 Estimation of the BEKK-model ~ 224
10.2 An empirical illustration ~ 225
10.2.1 Data description ~ 225
10.2.2 Estimating bivariate GARCH ~ 226
10.2.3 Estimating the (co)variance processes ~ 229
10.3 Forecasting exchange rate densities ~ 232

11 Statistical Process Control 237
Sven Knoth

11.1 Control Charts ~ 238
11.2 Chart characteristics ~ 243
11.2.1 Average Run Length and Critical Values ~ 247
11.2.2 Average Delay ~ 248
11.2.3 Probability Mass and Cumulative Distribution Function 248
11.3 Comparison with existing methods ~ 251
11.3.1 Two-sided EWMA and Lucas/Saccucci ~ 251
11.3.2 Two-sided CUSUM and Crosier ~ 251
11.4 Real data example { monitoring CAPM ~ 253

12 An Empirical Likelihood Goodness-of-Fit Test for Diffusions 259
Song Xi Chen, Wolfgang Hardle and Torsten Kleinow

12.1 Introduction ~ 259
12.2 Discrete Time Approximation of a Diffusion ~ 260
12.3 Hypothesis Testing ~ 261
12.4 Kernel Estimator ~ 263
12.5 The Empirical Likelihood concept ~ 264
12.5.1 Introduction into Empirical Likelihood ~ 264
12.5.2 Empirical Likelihood for Time Series Data ~ 265
12.6 Goodness-of-Fit Statistic ~ 268
12.7 Goodness-of-Fit test ~ 272
12.8 Application ~ 274
12.9 Simulation Study and Illustration ~ 276
12.10Appendix ~ 279

13 A simple state space model of house prices 283
Rainer Schulz and Axel Werwat Z

13.1 Introduction ~ 283
13.2 A Statistical Model of House Prices ~ 284
13.2.1 The Price Function ~ 284
13.2.2 State Space Form ~ 285
13.3 Estimation with Kalman Filter Techniques ~ 286
13.3.1 Kalman Filtering given all parameters ~ 286
13.3.2 Filtering and state smoothing ~ 287
13.3.3 Maximum likelihood estimation of the parameters ~ 288
13.3.4 Diagnostic checking ~ 289
13.4 The Data ~ 289
13.5 Estimating and filtering in XploRe ~ 293
13.5.1 Overview ~ 293
13.5.2 Setting the system matrices ~ 293
13.5.3 Kalman filter and maximized log likelihood ~ 295
13.5.4 Diagnostic checking with standardized residuals ~ 298
13.5.5 Calculating the Kalman smoother ~ 300
13.6 Appendix ~ 302
13.6.1 Procedure equivalence ~ 302
13.6.2 Smoothed constant state variables ~ 304

14 Long Memory Effects Trading Strategy 309
Oliver Jim Blaskowitz and Peter Schmidt

14.1 Introduction ~ 309
14.2 Hurst and Rescaled Range Analysis ~ 310
14.3 Stationary Long Memory Processes ~ 312
14.3.1 Fractional Brownian Motion and Noise ~ 313
14.4 Data Analysis ~ 315
14.5 Trading the Negative Persistence ~ 318

15 Locally time homogeneous time series modeling 323
Danilo Mercurio

15.1 Intervals of homogeneity ~ 323
15.1.1 The adaptive estimator ~ 326
15.1.2 A small simulation study ~ 327
15.2 Estimating the coefficients of an exchange rate basket ~ 329
15.2.1 The Thai Baht basket ~ 331
15.2.2 Estimation results ~ 335
15.3 Estimating the volatility of financial time series ~ 338
15.3.1 The standard approach ~ 339
15.3.2 The locally time homogeneous approach ~ 340
15.3.3 Modeling volatility via power transformation ~ 340
15.3.4 Adaptive estimation under local time-homogeneity ~ 341
15.4 Technical appendix ~ 344

16 Simulation based Option Pricing 349
Jens Lussem and Jurgen Schumacher

16.1 Simulation techniques for option pricing ~ 349
16.1.1 Introduction to simulation techniques ~ 349
16.1.2 Pricing path independent European options on one underlying ~ 350
16.1.3 Pricing path dependent European options on one underlying ~ 354
16.1.4 Pricing options on multiple underlyings ~ 355
16.2 Quasi Monte Carlo (QMC) techniques for option pricing ~ 356
16.2.1 Introduction to Quasi Monte Carlo techniques ~ 356
16.2.2 Error bounds ~ 356
16.2.3 Construction of the Halton sequence ~ 357
16.2.4 Experimental results ~ 359
16.3 Pricing options with simulation techniques - a guideline ~ 361
16.3.1 Construction of the payoff function ~ 362
16.3.2 Integration of the payoff function in the simulation framework ~ 362
16.3.3 Restrictions for the payoff functions ~ 365

17 Nonparametric Estimators of GARCH Processes 367
Jurgen Franke, Harriet Holzberger and Marlene Muller

17.1 Deconvolution density and regression estimates ~ 369
17.2 Nonparametric ARMA Estimates ~ 370
17.3 Nonparametric GARCH Estimates ~ 379

18 Net Based Spreadsheets in Quantitative Finance 385
Gokhan Aydinli

18.1 Introduction ~ 385
18.2 Client/Server based Statistical Computing ~ 386
18.3 Why Spreadsheets? ~ 387
18.4 Using MD*ReX ~ 388
18.5 Applications ~ 390
18.5.1 Value at Risk Calculations with Copulas ~ 391
18.5.2 Implied Volatility Measures ~ 393

Index 398