<P><BR></P>
<P>Contributors xix<BR>Frequently Used Notation xxi<BR>I Value at Risk 1<BR>1 Approximating Value at Risk in Conditional Gaussian Models 3<BR>Stefan R. Jaschke and Yuze Jiang<BR>1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<BR>1.1.1 The Practical Need . . . . . . . . . . . . . . . . . . . . . 3<BR>1.1.2 Statistical Modeling for VaR . . . . . . . . . . . . . . . 4<BR>1.1.3 VaR Approximations . . . . . . . . . . . . . . . . . . . . 6<BR>1.1.4 Pros and Cons of Delta-Gamma Approximations . . . . 7<BR>1.2 General Properties of Delta-Gamma-Normal Models . . . . . . 8<BR>1.3 Cornish-Fisher Approximations . . . . . . . . . . . . . . . . . . 12<BR>1.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 12<BR>1.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 15<BR>1.4 Fourier Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 16<BR>1.4.1 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . 16<BR>1.4.2 Tail Behavior . . . . . . . . . . . . . . . . . . . . . . . . 20<BR>1.4.3 Inversion of the cdf minus the Gaussian Approximation 21<BR>1.5 Variance Reduction Techniques in Monte-Carlo Simulation . . . 24<BR>1.5.1 Monte-Carlo Sampling Method . . . . . . . . . . . . . . 24<BR>1.5.2 Partial Monte-Carlo with Importance Sampling . . . . . 28<BR>1.5.3 XploRe Examples . . . . . . . . . . . . . . . . . . . . . 30<BR>2 Applications of Copulas for the Calculation of Value-at-Risk 35<BR>Jorn Rank and Thomas Siegl<BR>2.1 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<BR>2.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . 36<BR>2.1.2 Sklar's Theorem . . . . . . . . . . . . . . . . . . . . . . 37<BR>2.1.3 Examples of Copulas . . . . . . . . . . . . . . . . . . . . 37<BR>2.1.4 Further Important Properties of Copulas . . . . . . . . 39<BR>2.2 Computing Value-at-Risk with Copulas . . . . . . . . . . . . . 40<BR>2.2.1 Selecting the Marginal Distributions . . . . . . . . . . . 40<BR>2.2.2 Selecting a Copula . . . . . . . . . . . . . . . . . . . . . 41<BR>2.2.3 Estimating the Copula Parameters . . . . . . . . . . . . 41<BR>2.2.4 Generating Scenarios - Monte Carlo Value-at-Risk . . . 43<BR>2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45<BR>2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47<BR>3 Quantication of Spread Risk by Means of Historical Simulation 51<BR>Christoph Frisch and Germar Knochlein<BR>3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<BR>3.2 Risk Categories { a Denition of Terms . . . . . . . . . . . . . 51<BR>3.3 Descriptive Statistics of Yield Spread Time Series . . . . . . . . 53<BR>3.3.1 Data Analysis with XploRe . . . . . . . . . . . . . . . . 54<BR>3.3.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . 58<BR>3.4 Historical Simulation and Value at Risk . . . . . . . . . . . . . 63<BR>3.4.1 Risk Factor: Full Yield . . . . . . . . . . . . . . . . . . . 64<BR>3.4.2 Risk Factor: Benchmark . . . . . . . . . . . . . . . . . . 67<BR>3.4.3 Risk Factor: Spread over Benchmark Yield . . . . . . . 68<BR>3.4.4 Conservative Approach . . . . . . . . . . . . . . . . . . 69<BR>3.4.5 Simultaneous Simulation . . . . . . . . . . . . . . . . . . 69<BR>3.5 Mark-to-Model Backtesting . . . . . . . . . . . . . . . . . . . . 70<BR>3.6 VaR Estimation and Backtesting with XploRe . . . . . . . . . . 70<BR>3.7 P-P Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73<BR>3.8 Q-Q Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74<BR>3.9 Discussion of Simulation Results . . . . . . . . . . . . . . . . . 75<BR>3.9.1 Risk Factor: Full Yield . . . . . . . . . . . . . . . . . . . 77<BR>3.9.2 Risk Factor: Benchmark . . . . . . . . . . . . . . . . . . 78<BR>3.9.3 Risk Factor: Spread over Benchmark Yield . . . . . . . 78<BR>3.9.4 Conservative Approach . . . . . . . . . . . . . . . . . . 79<BR>3.9.5 Simultaneous Simulation . . . . . . . . . . . . . . . . . . 80<BR>3.10 XploRe for Internal Risk Models . . . . . . . . . . . . . . . . . 81<BR>II Credit Risk 85<BR>4 Rating Migrations 87<BR>Ste Hose, Stefan Huschens and Robert Wania<BR>4.1 Rating Transition Probabilities . . . . . . . . . . . . . . . . . . 88<BR>4.1.1 From Credit Events to Migration Counts . . . . . . . . 88<BR>4.1.2 Estimating Rating Transition Probabilities . . . . . . . 89<BR>4.1.3 Dependent Migrations . . . . . . . . . . . . . . . . . . . 90<BR>4.1.4 Computation and Quantlets . . . . . . . . . . . . . . . . 93<BR>4.2 Analyzing the Time-Stability of Transition Probabilities . . . . 94<BR>4.2.1 Aggregation over Periods . . . . . . . . . . . . . . . . . 94<BR>4.2.2 Are the Transition Probabilities Stationary? . . . . . . . 95<BR>4.2.3 Computation and Quantlets . . . . . . . . . . . . . . . . 97<BR>4.2.4 Examples with Graphical Presentation . . . . . . . . . . 98<BR>4.3 Multi-Period Transitions . . . . . . . . . . . . . . . . . . . . . . 101<BR>4.3.1 Time Homogeneous Markov Chain . . . . . . . . . . . . 101<BR>4.3.2 Bootstrapping Markov Chains . . . . . . . . . . . . . . 102<BR>4.3.3 Computation and Quantlets . . . . . . . . . . . . . . . . 104<BR>4.3.4 Rating Transitions of German Bank Borrowers . . . . . 106<BR>4.3.5 Portfolio Migration . . . . . . . . . . . . . . . . . . . . . 106<BR>5 Sensitivity analysis of credit portfolio models 111<BR>Rudiger Kiesel and Torsten Kleinow<BR>5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111<BR>5.2 Construction of portfolio credit risk models . . . . . . . . . . . 113<BR>5.3 Dependence modelling . . . . . . . . . . . . . . . . . . . . . . . 114<BR>5.3.1 Factor modelling . . . . . . . . . . . . . . . . . . . . . . 115<BR>5.3.2 Copula modelling . . . . . . . . . . . . . . . . . . . . . . 117<BR>5.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119<BR>5.4.1 Random sample generation . . . . . . . . . . . . . . . . 119<BR>5.4.2 Portfolio results . . . . . . . . . . . . . . . . . . . . . . . 120<BR>III Implied Volatility 125<BR>6 The Analysis of Implied Volatilities 127<BR>Matthias R. Fengler, Wolfgang Hardle and Peter Schmidt<BR>6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128<BR>6.2 The Implied Volatility Surface . . . . . . . . . . . . . . . . . . . 129<BR>6.2.1 Calculating the Implied Volatility . . . . . . . . . . . . . 129<BR>6.2.2 Surface smoothing . . . . . . . . . . . . . . . . . . . . . 131<BR>6.3 Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 134<BR>6.3.1 Data description . . . . . . . . . . . . . . . . . . . . . . 134<BR>6.3.2 PCA of ATM Implied Volatilities . . . . . . . . . . . . . 136<BR>6.3.3 Common PCA of the Implied Volatility Surface . . . . . 137<BR>7 How Precise Are Price Distributions Predicted by IBT? 145<BR>Wolfgang Hardle and Jun Zheng<BR>7.1 Implied Binomial Trees . . . . . . . . . . . . . . . . . . . . . . 146<BR>7.1.1 The Derman and Kani (D &amp; K) algorithm . . . . . . . . 147<BR>7.1.2 Compensation . . . . . . . . . . . . . . . . . . . . . . . 151<BR>7.1.3 Barle and Cakici (B &amp; C) algorithm . . . . . . . . . . . 153<BR>7.2 A Simulation and a Comparison of the SPDs . . . . . . . . . . 154<BR>7.2.1 Simulation using Derman and Kani algorithm . . . . . . 154<BR>7.2.2 Simulation using Barle and Cakici algorithm . . . . . . 156<BR>7.2.3 Comparison with Monte-Carlo Simulation . . . . . . . . 158<BR>7.3 Example { Analysis of DAX data . . . . . . . . . . . . . . . . . 162<BR>8 Estimating State-Price Densities with Nonparametric Regression 171<BR>Kim Huynh, Pierre Kervella and Jun Zheng<BR>8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171<BR>8.2 Extracting the SPD using Call-Options . . . . . . . . . . . . . 173<BR>8.2.1 Black-Scholes SPD . . . . . . . . . . . . . . . . . . . . . 175<BR>8.3 Semiparametric estimation of the SPD . . . . . . . . . . . . . . 176<BR>8.3.1 Estimating the call pricing function . . . . . . . . . . . 176<BR>8.3.2 Further dimension reduction . . . . . . . . . . . . . . . 177<BR>8.3.3 Local Polynomial Estimation . . . . . . . . . . . . . . . 181<BR>8.4 An Example: Application to DAX data . . . . . . . . . . . . . 183<BR>8.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183<BR>8.4.2 SPD, delta and gamma . . . . . . . . . . . . . . . . . . 185<BR>8.4.3 Bootstrap condence bands . . . . . . . . . . . . . . . . 187<BR>8.4.4 Comparison to Implied Binomial Trees . . . . . . . . . . 190<BR>9 Trading on Deviations of Implied and Historical Densities 197<BR>Oliver Jim Blaskowitz and Peter Schmidt<BR>9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197<BR>9.2 Estimation of the Option Implied SPD . . . . . . . . . . . . . . 198<BR>9.2.1 Application to DAX Data . . . . . . . . . . . . . . . . . 198<BR>9.3 Estimation of the Historical SPD . . . . . . . . . . . . . . . . . 200<BR>9.3.1 The Estimation Method . . . . . . . . . . . . . . . . . . 201<BR>9.3.2 Application to DAX Data . . . . . . . . . . . . . . . . . 202<BR>9.4 Comparison of Implied and Historical SPD . . . . . . . . . . . 205<BR>9.5 Skewness Trades . . . . . . . . . . . . . . . . . . . . . . . . . . 207<BR>9.5.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . 210<BR>9.6 Kurtosis Trades . . . . . . . . . . . . . . . . . . . . . . . . . . . 212<BR>9.6.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . 214<BR>9.7 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . 216<BR>IV Econometrics 219<BR>10 Multivariate Volatility Models 221<BR>Matthias R. Fengler and Helmut Herwartz<BR>10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221<BR>10.1.1 Model specications . . . . . . . . . . . . . . . . . . . . 222<BR>10.1.2 Estimation of the BEKK-model . . . . . . . . . . . . . . 224<BR>10.2 An empirical illustration . . . . . . . . . . . . . . . . . . . . . . 225<BR>10.2.1 Data description . . . . . . . . . . . . . . . . . . . . . . 225<BR>10.2.2 Estimating bivariate GARCH . . . . . . . . . . . . . . . 226<BR>10.2.3 Estimating the (co)variance processes . . . . . . . . . . 229<BR>10.3 Forecasting exchange rate densities . . . . . . . . . . . . . . . . 232<BR>11 Statistical Process Control 237<BR>Sven Knoth<BR>11.1 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . 238<BR>11.2 Chart characteristics . . . . . . . . . . . . . . . . . . . . . . . . 243<BR>11.2.1 Average Run Length and Critical Values . . . . . . . . . 247<BR>11.2.2 Average Delay . . . . . . . . . . . . . . . . . . . . . . . 248<BR>11.2.3 Probability Mass and Cumulative Distribution Function 248<BR>11.3 Comparison with existing methods . . . . . . . . . . . . . . . . 251<BR>11.3.1 Two-sided EWMA and Lucas/Saccucci . . . . . . . . . 251<BR>11.3.2 Two-sided CUSUM and Crosier . . . . . . . . . . . . . . 251<BR>11.4 Real data example { monitoring CAPM . . . . . . . . . . . . . 253<BR>12 An Empirical Likelihood Goodness-of-Fit Test for Diusions 259<BR>Song Xi Chen, Wolfgang Hardle and Torsten Kleinow<BR>12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259<BR>12.2 Discrete Time Approximation of a Diusion . . . . . . . . . . . 260<BR>12.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . 261<BR>12.4 Kernel Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 263<BR>12.5 The Empirical Likelihood concept . . . . . . . . . . . . . . . . . 264<BR>12.5.1 Introduction into Empirical Likelihood . . . . . . . . . . 264<BR>12.5.2 Empirical Likelihood for Time Series Data . . . . . . . . 265<BR>12.6 Goodness-of-Fit Statistic . . . . . . . . . . . . . . . . . . . . . . 268<BR>12.7 Goodness-of-Fit test . . . . . . . . . . . . . . . . . . . . . . . . 272<BR>12.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274<BR>12.9 Simulation Study and Illustration . . . . . . . . . . . . . . . . . 276<BR>12.10Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279<BR>13 A simple state space model of house prices 283<BR>Rainer Schulz and Axel Werwatz<BR>13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283<BR>13.2 A Statistical Model of House Prices . . . . . . . . . . . . . . . . 284<BR>13.2.1 The Price Function . . . . . . . . . . . . . . . . . . . . . 284<BR>13.2.2 State Space Form . . . . . . . . . . . . . . . . . . . . . . 285<BR>13.3 Estimation with Kalman Filter Techniques . . . . . . . . . . . 286<BR>13.3.1 Kalman Filtering given all parameters . . . . . . . . . . 286<BR>13.3.2 Filtering and state smoothing . . . . . . . . . . . . . . . 287<BR>13.3.3 Maximum likelihood estimation of the parameters . . . 288<BR>13.3.4 Diagnostic checking . . . . . . . . . . . . . . . . . . . . 289<BR>13.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289<BR>13.5 Estimating and ltering in XploRe . . . . . . . . . . . . . . . . 293<BR>13.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 293<BR>13.5.2 Setting the system matrices . . . . . . . . . . . . . . . . 293<BR>13.5.3 Kalman lter and maximized log likelihood . . . . . . . 295<BR>13.5.4 Diagnostic checking with standardized residuals . . . . . 298<BR>13.5.5 Calculating the Kalman smoother . . . . . . . . . . . . 300<BR>13.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302<BR>13.6.1 Procedure equivalence . . . . . . . . . . . . . . . . . . . 302<BR>13.6.2 Smoothed constant state variables . . . . . . . . . . . . 304<BR>14 Long Memory Eects Trading Strategy 309<BR>Oliver Jim Blaskowitz and Peter Schmidt<BR>14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309<BR>14.2 Hurst and Rescaled Range Analysis . . . . . . . . . . . . . . . . 310<BR>14.3 Stationary Long Memory Processes . . . . . . . . . . . . . . . . 312<BR>14.3.1 Fractional Brownian Motion and Noise . . . . . . . . . . 313<BR>14.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315<BR>14.5 Trading the Negative Persistence . . . . . . . . . . . . . . . . . 318<BR>15 Locally time homogeneous time series modeling 323<BR>Danilo Mercurio<BR>15.1 Intervals of homogeneity . . . . . . . . . . . . . . . . . . . . . . 323<BR>15.1.1 The adaptive estimator . . . . . . . . . . . . . . . . . . 326<BR>15.1.2 A small simulation study . . . . . . . . . . . . . . . . . 327<BR>15.2 Estimating the coecients of an exchange rate basket . . . . . 329<BR>15.2.1 The Thai Baht basket . . . . . . . . . . . . . . . . . . . 331<BR>15.2.2 Estimation results . . . . . . . . . . . . . . . . . . . . . 335<BR>15.3 Estimating the volatility of nancial time series . . . . . . . . . 338<BR>15.3.1 The standard approach . . . . . . . . . . . . . . . . . . 339<BR>15.3.2 The locally time homogeneous approach . . . . . . . . . 340<BR>15.3.3 Modeling volatility via power transformation . . . . . . 340<BR>15.3.4 Adaptive estimation under local time-homogeneity . . . 341<BR>15.4 Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . 344<BR>16 Simulation based Option Pricing 349<BR>Jens Lussem and Jurgen Schumacher<BR>16.1 Simulation techniques for option pricing . . . . . . . . . . . . . 349<BR>16.1.1 Introduction to simulation techniques . . . . . . . . . . 349<BR>16.1.2 Pricing path independent European options on one underlying<BR>. . . . . . . . . . . . . . . . . . . . . . . . . . . 350<BR>16.1.3 Pricing path dependent European options on one underlying<BR>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354<BR>16.1.4 Pricing options on multiple underlyings . . . . . . . . . 355<BR>16.2 Quasi Monte Carlo (QMC) techniques for option pricing . . . . 356<BR>16.2.1 Introduction to Quasi Monte Carlo techniques . . . . . 356<BR>16.2.2 Error bounds . . . . . . . . . . . . . . . . . . . . . . . . 356<BR>16.2.3 Construction of the Halton sequence . . . . . . . . . . . 357<BR>16.2.4 Experimental results . . . . . . . . . . . . . . . . . . . . 359<BR>16.3 Pricing options with simulation techniques - a guideline . . . . 361<BR>16.3.1 Construction of the payo function . . . . . . . . . . . . 362<BR>16.3.2 Integration of the payo function in the simulation framework<BR>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362<BR>16.3.3 Restrictions for the payo functions . . . . . . . . . . . 365<BR>17 Nonparametric Estimators of GARCH Processes 367<BR>Jurgen Franke, Harriet Holzberger and Marlene Muller<BR>17.1 Deconvolution density and regression estimates . . . . . . . . . 369<BR>17.2 Nonparametric ARMA Estimates . . . . . . . . . . . . . . . . . 370<BR>17.3 Nonparametric GARCH Estimates . . . . . . . . . . . . . . . . 379<BR>18 Net Based Spreadsheets in Quantitative Finance 385<BR>Gokhan Aydnl<BR>18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385<BR>18.2 Client/Server based Statistical Computing . . . . . . . . . . . . 386<BR>18.3 Why Spreadsheets? . . . . . . . . . . . . . . . . . . . . . . . . . 387<BR>18.4 Using MD*ReX . . . . . . . . . . . . . . . . . . . . . . . . . . . 388<BR>18.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390<BR>18.5.1 Value at Risk Calculations with Copulas . . . . . . . . . 391<BR>18.5.2 Implied Volatility Measures . . . . . . . . . . . . . . . . 393<BR>Index 398<BR></P>