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 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Quanti cation of Spread Risk by Means of Historical Simulation 51
Christoph Frisch and Germar Knochlein
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Risk Categories { a De nition 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
Ste 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 con dence 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 Herwartz
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.1.1 Model speci cations . . . . . . . . . . . . . . . . . . . . 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 Di usions 259
Song Xi Chen, Wolfgang Hardle and Torsten Kleinow
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
12.2 Discrete Time Approximation of a Di usion . . . . . . . . . . . 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 Werwatz
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 ltering in XploRe . . . . . . . . . . . . . . . . 293
13.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 293
13.5.2 Setting the system matrices . . . . . . . . . . . . . . . . 293
13.5.3 Kalman lter 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 E ects 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 coecients 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 nancial 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 payo function . . . . . . . . . . . . 362
16.3.2 Integration of the payo function in the simulation framework
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
16.3.3 Restrictions for the payo 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 Aydnl
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 Measure