Wolfgang H¨rdle

a

Torsten Kleinow

Gerhard Stahl

In cooperation with

G¨khan Ayd±nl±, Oliver Jim Blaskowitz, Song Xi Chen,

o

Matthias Fengler, J¨rgen Franke, Christoph Frisch,

u

Helmut Herwartz, Harriet Holzberger, Ste¬ H¨se,

o

Stefan Huschens, Kim Huynh, Stefan R. Jaschke, Yuze Jiang

Pierre Kervella, R¨diger Kiesel, Germar Kn¨chlein,

u o

Sven Knoth, Jens L¨ssem, Danilo Mercurio,

u

Marlene M¨ller, J¨rn Rank, Peter Schmidt,

u o

Rainer Schulz, J¨rgen Schumacher, Thomas Siegl,

u

Robert Wania, Axel Werwatz, Jun Zheng

June 20, 2002

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

iv Contents

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

J¨rn Rank and Thomas Siegl

o

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 Kn¨chlein

o

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Risk Categories “ a De¬nition of Terms . . . . . . . . . . . . . 51

Contents v

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¬ H¨se, Stefan Huschens and Robert Wania

o

4.1 Rating Transition Probabilities . . . . . . . . . . . . . . . . . . 88

4.1.1 From Credit Events to Migration Counts . . . . . . . . 88

vi Contents

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

R¨diger Kiesel and Torsten Kleinow

u

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

Contents vii

III Implied Volatility 125

6 The Analysis of Implied Volatilities 127

Matthias R. Fengler, Wolfgang H¨rdle and Peter Schmidt

a

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 H¨rdle and Jun Zheng

a

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

viii Contents

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

Contents ix

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 H¨rdle and Torsten Kleinow

a

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

x Contents

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

Contents xi

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 coe¬cients 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

xii Contents

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 L¨ssem and J¨rgen Schumacher

u u

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 un-

derlying . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

16.1.3 Pricing path dependent European options on one under-

lying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 frame-

work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

16.3.3 Restrictions for the payo¬ functions . . . . . . . . . . . 365

17 Nonparametric Estimators of GARCH Processes 367

J¨rgen Franke, Harriet Holzberger and Marlene M¨ller

u u

17.1 Deconvolution density and regression estimates . . . . . . . . . 369

17.2 Nonparametric ARMA Estimates . . . . . . . . . . . . . . . . . 370

Contents xiii

17.3 Nonparametric GARCH Estimates . . . . . . . . . . . . . . . . 379

18 Net Based Spreadsheets in Quantitative Finance 385

G¨khan Ayd±nl±

o

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

Preface

This book is designed for students and researchers who want to develop pro-

fessional skill in modern quantitative applications in ¬nance. The Center for

Applied Statistics and Economics (CASE) course at Humboldt-Universit¨t zu a

Berlin that forms the basis for this book is o¬ered to interested students who

have had some experience with probability, statistics and software applications

but have not had advanced courses in mathematical ¬nance. Although the

course assumes only a modest background it moves quickly between di¬erent

¬elds of applications and in the end, the reader can expect to have theoretical

and computational tools that are deep enough and rich enough to be relied on

throughout future professional careers.

The text is readable for the graduate student in ¬nancial engineering as well as

for the inexperienced newcomer to quantitative ¬nance who wants to get a grip

on modern statistical tools in ¬nancial data analysis. The experienced reader

with a bright knowledge of mathematical ¬nance will probably skip some sec-

tions but will hopefully enjoy the various computational tools of the presented

techniques. A graduate student might think that some of the econometric

techniques are well known. The mathematics of risk management and volatil-

ity dynamics will certainly introduce him into the rich realm of quantitative

¬nancial data analysis.

The computer inexperienced user of this e-book is softly introduced into the

interactive book concept and will certainly enjoy the various practical exam-

ples. The e-book is designed as an interactive document: a stream of text and

information with various hints and links to additional tools and features. Our

e-book design o¬ers also a complete PDF and HTML ¬le with links to world

wide computing servers. The reader of this book may therefore without down-

load or purchase of software use all the presented examples and methods via

the enclosed license code number with a local XploRe Quantlet Server (XQS).

Such XQ Servers may also be installed in a department or addressed freely on

the web, click to www.xplore-stat.de and www.quantlet.com.

xvi Preface

”Applied Quantitative Finance” consists of four main parts: Value at Risk,

Credit Risk, Implied Volatility and Econometrics. In the ¬rst part Jaschke and

Jiang treat the Approximation of the Value at Risk in conditional Gaussian

Models and Rank and Siegl show how the VaR can be calculated using copulas.

The second part starts with an analysis of rating migration probabilities by

H¨se, Huschens and Wania. Frisch and Kn¨chlein quantify the risk of yield

o o

spread changes via historical simulations. This part is completed by an anal-

ysis of the sensitivity of risk measures to changes in the dependency structure

between single positions of a portfolio by Kiesel and Kleinow.

The third part is devoted to the analysis of implied volatilities and their dynam-

ics. Fengler, H¨rdle and Schmidt start with an analysis of the implied volatility

a

surface and show how common PCA can be applied to model the dynamics of

the surface. In the next two chapters the authors estimate the risk neutral

state price density from observed option prices and the corresponding implied

volatilities. While H¨rdle and Zheng apply implied binomial trees to estimate

a

the SPD, the method by Huynh, Kervella and Zheng is based on a local poly-

nomial estimation of the implied volatility and its derivatives. Blaskowitz and

Schmidt use the proposed methods to develop trading strategies based on the

comparison of the historical SPD and the one implied by option prices.

Recently developed econometric methods are presented in the last part of the

book. Fengler and Herwartz introduce a multivariate volatility model and ap-

ply it to exchange rates. Methods used to monitor sequentially observed data

are treated by Knoth. Chen, H¨rdle and Kleinow apply the empirical likeli-

a

hood concept to develop a test about a parametric di¬usion model. Schulz

and Werwatz estimate a state space model of Berlin house prices that can be

used to construct a time series of the price of a standard house. The in¬‚u-

ence of long memory e¬ects on ¬nancial time series is analyzed by Blaskowitz

and Schmidt. Mercurio propose a methodology to identify time intervals of

homogeneity for time series. The pricing of exotic options via a simulation

approach is introduced by L¨ssem and Schumacher The chapter by Franke,

u

Holzberger and M¨ller is devoted to a nonparametric estimation approach of

u

GARCH models. The book closes with a chapter of Ayd±nl±, who introduces

a technology to connect standard software with the XploRe server in order to

have access to quantlets developed in this book.

We gratefully acknowledge the support of Deutsche Forschungsgemeinschaft,

¨

SFB 373 Quanti¬kation und Simulation Okonomischer Prozesse. A book of this

kind would not have been possible without the help of many friends, colleagues

and students. For the technical production of the e-book platform we would

Preface xvii

like to thank J¨rg Feuerhake, Zdenˇk Hl´vka, Sigbert Klinke, Heiko Lehmann

o e a

and Rodrigo Witzel.

W. H¨rdle, T. Kleinow and G. Stahl

a

Berlin and Bonn, June 2002

Contributors

G¨khan Ayd±nl± Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

o a

Statistics and Economics

Oliver Jim Blaskowitz Humboldt-Universit¨t zu Berlin, CASE, Center for Ap-

a

plied Statistics and Economics

Song Xi Chen The National University of Singapore, Dept. of Statistics and

Applied Probability

Matthias R. Fengler Humboldt-Universit¨t zu Berlin, CASE, Center for Ap-

a

plied Statistics and Economics

J¨rgen Franke Universit¨t Kaiserslautern

u a

Christoph Frisch Landesbank Rheinland-Pfalz, Risiko¨berwachung

u

Wolfgang H¨rdle Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a a

Statistics and Economics

Helmut Herwartz Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

Harriet Holzberger IKB Deutsche Industriebank AG

Ste¬ H¨se Technische Universit¨t Dresden

o a

Stefan Huschens Technische Universit¨t Dresden

a

Kim Huynh Queen™s Economics Department, Queen™s University

Stefan R. Jaschke Weierstrass Institute for Applied Analysis and Stochastics

Yuze Jiang Queen™s School of Business, Queen™s University

xx Contributors

Pierre Kervella Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

R¨diger Kiesel London School of Economics, Department of Statistics

u

Torsten Kleinow Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

Germar Kn¨chlein Landesbank Rheinland-Pfalz, Risiko¨berwachung

o u

Sven Knoth European University Viadrina Frankfurt (Oder)

Jens L¨ssem Landesbank Kiel

u

Danilo Mercurio Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

Marlene M¨ller Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

u a

Statistics and Economics

J¨rn Rank Andersen, Financial and Commodity Risk Consulting

o

Peter Schmidt Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

Rainer Schulz Humboldt-Universit¨t zu Berlin, CASE, Center for Applied Statis-

a

tics and Economics

J¨rgen Schumacher University of Bonn, Department of Computer Science

u

Thomas Siegl BHF Bank

Robert Wania Technische Universit¨t Dresden

a

Axel Werwatz Humboldt-Universit¨t zu Berlin, CASE, Center for Applied

a

Statistics and Economics

Jun Zheng Department of Probability and Statistics, School of Mathematical

Sciences, Peking University, 100871, Beijing, P.R. China

Frequently Used Notation

def

x = . . . x is de¬ned as ...

R real numbers

def

R = R ∪ {∞, ∞}

A transpose of matrix A

X ∼ D the random variable X has distribution D

E[X] expected value of random variable X

Var(X) variance of random variable X

Std(X) standard deviation of random variable X

Cov(X, Y ) covariance of two random variables X and Y

N(µ, Σ) normal distribution with expectation µ and covariance matrix Σ, a

similar notation is used if Σ is the correlation matrix

cdf denotes the cumulative distribution function

pdf denotes the probability density function

P[A] or P(A) probability of a set A

1 indicator function

def

(F —¦ G)(x) = F {G(x)} for functions F and G

±n = O(βn ) i¬ ±n ’’ constant, as n ’’ ∞

n

β

±n = O(βn ) i¬ ±n ’’ 0, as n ’’ ∞

n

β

Ft is the information set generated by all information available at time t

Let An and Bn be sequences of random variables.

An = Op (Bn ) i¬ ∀µ > 0 ∃M, ∃N such that P[|An /Bn | > M ] < µ, ∀n > N .

An = Op (Bn ) i¬ ∀µ > 0 : limn’∞ P[|An /Bn | > µ] = 0.

Part I

Value at Risk

1 Approximating Value at Risk in

Conditional Gaussian Models

Stefan R. Jaschke and Yuze Jiang

1.1 Introduction

1.1.1 The Practical Need

Financial institutions are facing the important task of estimating and control-

ling their exposure to market risk, which is caused by changes in prices of

equities, commodities, exchange rates and interest rates. A new chapter of risk

management was opened when the Basel Committee on Banking Supervision

proposed that banks may use internal models for estimating their market risk

(Basel Committee on Banking Supervision, 1995). Its implementation into na-

tional laws around 1998 allowed banks to not only compete in the innovation

of ¬nancial products but also in the innovation of risk management methodol-

ogy. Measurement of market risk has focused on a metric called Value at Risk

(VaR). VaR quanti¬es the maximal amount that may be lost in a portfolio over

a given period of time, at a certain con¬dence level. Statistically speaking, the

VaR of a portfolio is the quantile of the distribution of that portfolio™s loss over

a speci¬ed time interval, at a given probability level.

The implementation of a ¬rm-wide risk management system is a tremendous

job. The biggest challenge for many institutions is to implement interfaces to

all the di¬erent front-o¬ce systems, back-o¬ce systems and databases (poten-

tially running on di¬erent operating systems and being distributed all over the

world), in order to get the portfolio positions and historical market data into a

centralized risk management framework. This is a software engineering prob-

lem. The second challenge is to use the computed VaR numbers to actually

4 1 Approximating Value at Risk in Conditional Gaussian Models

control risk and to build an atmosphere where the risk management system

is accepted by all participants. This is an organizational and social problem.

The methodological question how risk should be modeled and approximated

is “ in terms of the cost of implementation “ a smaller one. In terms of im-

portance, however, it is a crucial question. A non-adequate VaR-methodology

can jeopardize all the other e¬orts to build a risk management system. See

(Jorion, 2000) for more on the general aspects of risk management in ¬nancial

institutions.

1.1.2 Statistical Modeling for VaR

VaR methodologies can be classi¬ed in terms of statistical modeling decisions

and approximation decisions. Once the statistical model and the estimation

procedure is speci¬ed, it is a purely numerical problem to compute or approx-

imate the Value at Risk. The modeling decisions are:

1. Which risk factors to include. This mainly depends on a banks™ business

(portfolio). But it may also depend on the availability of historical data.

If data for a certain contract is not available or the quality is not su¬cient,

a related risk factor with better historical data may be used. For smaller

stock portfolios it is customary to include each stock itself as a risk factor.

For larger stock portfolios, only country or sector indexes are taken as

the risk factors (Longerstaey, 1996). Bonds and interest rate derivatives

are commonly assumed to depend on a ¬xed set of interest rates at key

maturities. The value of options is usually assumed to depend on implied

volatility (at certain key strikes and maturities) as well as on everything

the underlying depends on.

2. How to model security prices as functions of risk factors, which is usually

i

called “the mapping”. If Xt denotes the log return of stock i over the

i i i

time interval [t ’ 1, t], i.e., Xt = log(St ) ’ log(St’1 ), then the change in

the value of a portfolio containing one stock i is

i

∆St = St’1 (eXt ’ 1),

i i

i

where St denotes the price of stock i at time t. Bonds are ¬rst decomposed

into a portfolio of zero bonds. Zero bonds are assumed to depend on

the two key interest rates with the closest maturities. How to do the

interpolation is actually not as trivial as it may seem, as demonstrated

1.1 Introduction 5

by Mina and Ulmer (1999). Similar issues arise in the interpolation of

implied volatilities.

3. What stochastic properties to assume for the dynamics of the risk factors

Xt . The basic benchmark model for stocks is to assume that logarith-

mic stock returns are joint normal (cross-sectionally) and independent in

time. Similar assumptions for other risk factors are that changes in the

logarithm of zero-bond yields, changes in log exchange rates, and changes

in the logarithm of implied volatilities are all independent in time and

joint normally distributed.

4. How to estimate the model parameters from the historical data. The usual

statistical approach is to de¬ne the model and then look for estimators

that have certain optimality criteria. In the basic benchmark model the

minimal-variance unbiased estimator of the covariance matrix Σ of risk

factors Xt is the “rectangular moving average”

T

1

ˆ (Xt ’ µ)(Xt ’ µ)

Σ=

T ’1 t=1

def

(with µ = E[Xt ]). An alternative route is to ¬rst specify an estimator

and then look for a model in which that estimator has certain optimality

properties. The exponential moving average

T ’1

ˆ e’»(T ’t) (Xt ’ µ)(Xt ’ µ)

ΣT = (e» ’ 1)

t=’∞

can be interpreted as an e¬cient estimator of the conditional covariance

matrix ΣT of the vector of risk factors XT , given the information up to

time T ’ 1 in a very speci¬c GARCH model.

While there is a plethora of analyses of alternative statistical models for market

risks (see Barry Schachter™s Gloriamundi web site), mainly two classes of models

for market risk have been used in practice:

1. iid-models, i.e., the risk factors Xt are assumed to be independent in time,

but the distribution of Xt is not necessarily Gaussian. Apart from some

less common models involving hyperbolic distributions (Breckling, Eber-

lein and Kokic, 2000), most approaches either estimate the distribution

6 1 Approximating Value at Risk in Conditional Gaussian Models

of Xt completely non-parametrically and run under the name “histori-

cal simulation”, or they estimate the tail using generalized Pareto dis-

tributions (Embrechts, Kl¨ppelberg and Mikosch, 1997, “extreme value

u

theory”).

2. conditional Gaussian models, i.e., the risk factors Xt are assumed to be

joint normal, conditional on the information up to time t ’ 1.

Both model classes can account for unconditional “fat tails”.

1.1.3 VaR Approximations

In this paper we consider certain approximations of VaR in the conditional

Gaussian class of models. We assume that the conditional expectation of Xt ,

µt , is zero and its conditional covariance matrix Σt is estimated and given at

time t ’ 1. The change in the portfolio value over the time interval [t ’ 1, t] is

then

n

i

∆Vt (Xt ) = wi ∆St (Xt ),

i=1

i

where the wi are the portfolio weights and ∆St is the function that “maps” the

risk factor vector Xt to a change in the value of the i-th security value over the

time interval [t ’ 1, t], given all the information at time t ’ 1. These functions

are usually nonlinear, even for stocks (see above). In the following, we will

drop the time index and denote by ∆V the change in the portfolio™s value over

the next time interval and by X the corresponding vector of risk factors.

The only general method to compute quantiles of the distribution of ∆V is

Monte Carlo simulation. From discussion with practitioners “full valuation

Monte Carlo” appears to be practically infeasible for portfolios with securi-

ties whose mapping functions are ¬rst, extremely costly to compute “ like for

certain path-dependent options whose valuation itself relies on Monte-Carlo

simulation “ and second, computed inside complex closed-source front-o¬ce

systems, which cannot be easily substituted or adapted in their accuracy/speed

trade-o¬s. Quadratic approximations to the portfolio™s value as a function of

the risk factors

1

∆V (X) ≈ ∆ X + X “X, (1.1)

2

have become the industry standard since its use in RiskMetrics (Longerstaey,

1996). (∆ and “ are the aggregated ¬rst and second derivatives of the indi-

vidual mapping functions ∆S i w.r.t. the risk factors X. The ¬rst version of

1.1 Introduction 7

RiskMetrics in 1994 considered only the ¬rst derivative of the value function,

the “delta”. Without loss of generality, we assume that the constant term in

the Taylor expansion (1.1), the “theta”, is zero.)

1.1.4 Pros and Cons of Delta-Gamma Approximations

Both assumptions of the Delta-Gamma-Normal approach “ Gaussian innova-

tions and a reasonably good quadratic approximation of the value function V

“ have been questioned. Simple examples of portfolios with options can be

constructed to show that quadratic approximations to the value function can

lead to very large errors in the computation of VaR (Britton-Jones and Schae-

fer, 1999). The Taylor-approximation (1.1) holds only locally and is question-

able from the outset for the purpose of modeling extreme events. Moreover,

the conditional Gaussian framework does not allow to model joint extremal

events, as described by Embrechts, McNeil and Straumann (1999). The Gaus-

sian dependence structure, the copula, assigns too small probabilities to joint

extremal events compared to some empirical observations.

Despite these valid critiques of the Delta-Gamma-Normal model, there are good

reasons for banks to implement it alongside other models. (1) The statistical

assumption of conditional Gaussian risk factors can explain a wide range of

“stylized facts” about asset returns like unconditional fat tails and autocor-

relation in realized volatility. Parsimonious multivariate conditional Gaussian

models for dimensions like 500-2000 are challenging enough to be the subject of

ongoing statistical research, Engle (2000). (2) First and second derivatives of

¬nancial products w.r.t. underlying market variables (= deltas and gammas)

and other “sensitivities” are widely implemented in front o¬ce systems and

routinely used by traders. Derivatives w.r.t. possibly di¬erent risk factors used

by central risk management are easily computed by applying the chain rule

of di¬erentiation. So it is tempting to stay in the framework and language of

the trading desks and express portfolio value changes in terms of deltas and

gammas. (3) For many actual portfolios the delta-gamma approximation may

serve as a good control-variate within variance-reduced Monte-Carlo methods,

if it is not a su¬ciently good approximation itself. Finally (4), is it extremely

risky for a senior risk manager to ignore delta-gamma models if his friendly

consultant tells him that 99% of the competitors have it implemented.

Several methods have been proposed to compute a quantile of the distribution

de¬ned by the model (1.1), among them Monte Carlo simulation (Pritsker,

1996), Johnson transformations (Zangari, 1996a; Longerstaey, 1996), Cornish-

8 1 Approximating Value at Risk in Conditional Gaussian Models

Fisher expansions (Zangari, 1996b; Fallon, 1996), the Solomon-Stephens ap-

proximation (Britton-Jones and Schaefer, 1999), moment-based approxima-

tions motivated by the theory of estimating functions (Li, 1999), saddle-point

approximations (Rogers and Zane, 1999), and Fourier-inversion (Rouvinez,

1997; Albanese, Jackson and Wiberg, 2000). Pichler and Selitsch (1999) com-

pare ¬ve di¬erent VaR-methods: Johnson transformations, Delta-Normal, and

Cornish-Fisher-approximations up to the second, fourth and sixth moment.

The sixth-order Cornish-Fisher-approximation compares well against the other

techniques and is the ¬nal recommendation. Mina and Ulmer (1999) also com-

pare Johnson transformations, Fourier inversion, Cornish-Fisher approxima-

tions, and partial Monte Carlo. (If the true value function ∆V (X) in Monte

Carlo simulation is used, this is called “full Monte Carlo”. If its quadratic ap-

proximation is used, this is called “partial Monte Carlo”.) Johnson transforma-

tions are concluded to be “not a robust choice”. Cornish-Fisher is “extremely

fast” compared to partial Monte Carlo and Fourier inversion, but not as robust,

as it gives “unacceptable results” in one of the four sample portfolios.

The main three methods used in practice seem to be Cornish-Fisher expansions,

Fourier inversion, and partial Monte Carlo, whose implementation in XploRe

will be presented in this paper. What makes the Normal-Delta-Gamma model

especially tractable is that the characteristic function of the probability distri-

bution, i.e. the Fourier transform of the probability density, of the quadratic

form (1.1) is known analytically. Such general properties are presented in sec-

tion 1.2. Sections 1.3, 1.4, and 1.5 discuss the Cornish-Fisher, Fourier inversion,

and partial Monte Carlo techniques, respectively.

1.2 General Properties of Delta-Gamma-Normal

Models

The change in the portfolio value, ∆V , can be expressed as a sum of indepen-

dent random variables that are quadratic functions of standard normal random

variables Yi by means of the solution of the generalized eigenvalue problem

CC = Σ,

C “C = Λ.

1.2 General Properties of Delta-Gamma-Normal Models 9

This implies

m

1

(δi Yi + »i Yi2 )

∆V = (1.2)

2

i=1

m 2 2

1 δi δi

’

= »i + Yi

2 »i 2»i

i=1

with X = CY , δ = C ∆ and Λ = diag(»1 , . . . , »m ). Packages like LAPACK

(Anderson, Bai, Bischof, Blackford, Demmel, Dongarra, Croz, Greenbaum,

Hammarling, McKenney and Sorensen, 1999) contain routines directly for the

generalized eigenvalue problem. Otherwise C and Λ can be computed in two

steps:

1. Compute some matrix B with BB = Σ. If Σ is positive de¬nite, the

fastest method is Cholesky decomposition. Otherwise an eigenvalue de-

composition can be used.

2. Solve the (standard) symmetric eigenvalue problem for the matrix B “B:

Q B “BQ = Λ

def

with Q’1 = Q and set C = BQ.

The decomposition is implemented in the quantlet

npar= VaRDGdecomp(par)

uses a generalized eigen value decomposition to do a suitable co-

ordinate change. par is a list containing Delta, Gamma, Sigma on

input. npar is the same list, containing additionally B, delta,

and lambda on output.

The characteristic function of a non-central χ2 variate ((Z + a)2 , with standard

1

normal Z) is known analytically:

a2 it

2

Eeit(Z+a) = (1 ’ 2it)’1/2 exp .

1 ’ 2it

This implies the characteristic function for ∆V

1 12

Eeit∆V = exp{’ δj t2 /(1 ’ i»j t)},

√ (1.3)

2

1 ’ i»j t

j

10 1 Approximating Value at Risk in Conditional Gaussian Models

which can be re-expressed in terms of “ and B

1

Eeit∆V = det(I ’ itB “B)’1/2 exp{’ t2 ∆ B(I ’ itB “B)’1 B ∆}, (1.4)

2

or in terms of “ and Σ

1

Eeit∆V = det(I ’ it“Σ)’1/2 exp{’ t2 ∆ Σ(I ’ it“Σ)’1 ∆}. (1.5)

2

Numerical Fourier-inversion of (1.3) can be used to compute an approximation

to the cumulative distribution function (cdf) F of ∆V . (The ±-quantile is

computed by root-¬nding in F (x) = ±.) The cost of the Fourier-inversion is

O(N log N ), the cost of the function evaluations is O(mN ), and the cost of the

eigenvalue decomposition is O(m3 ). The cost of the eigenvalue decomposition

dominates the other two terms for accuracies of one or two decimal digits and

the usual number of risk factors of more than a hundred. Instead of a full

spectral decomposition, one can also just reduce B “B to tridiagonal form

B “B = QT Q . (T is tridiagonal and Q is orthogonal.) Then the evaluation

of the characteristic function in (1.4) involves the solution of a linear system

with the matrix I ’itT , which costs only O(m) operations. An alternative route

is to reduce “Σ to Hessenberg form “Σ = QHQ or do a Schur decomposition

“Σ = QRQ . (H is Hessenberg and Q is orthogonal. Since “Σ has the same

eigenvalues as B “B and they are all real, R is actually triangular instead of

quasi-triangular in the general case, Anderson et al. (1999). The evaluation of

(1.5) becomes O(m2 ), since it involves the solution of a linear system with the

matrix I ’ itH or I ’ itR, respectively. Reduction to tridiagonal, Hessenberg,

or Schur form is also O(m3 ), so the asymptotics in the number of risk factors

m remain the same in all cases. The critical N , above which the complete

spectral decomposition + fast evaluation via (1.3) is faster than the reduction

to tridiagonal or Hessenberg form + slower evaluation via (1.4) or (1.5) remains

to be determined empirically for given m on a speci¬c machine.

The computation of the cumulant generating function and the characteristic

function from the diagonalized form is implemented in the following quantlets:

1.2 General Properties of Delta-Gamma-Normal Models 11

z= VaRcgfDG(t,par)

Computes the cumulant generating function (cgf) for the class of

quadratic forms of Gaussian vectors.

z= VaRcharfDG(t,par)

Computes the characteristic function for the class of quadratic

forms of Gaussian vectors.

t is the complex argument and par the parameter list generated by

VaRDGdecomp.

The advantage of the Cornish-Fisher approximation is that it is based on the

cumulants, which can be computed without any matrix decomposition:

1 1

κ1 = »i = tr(“Σ),

2 2

i

1 1

{(r ’ 1)!»r + r!δi »r’2 } =

2

(r ’ 1)! tr((“Σ)r )

κr = i i

2 2

i

1

+ r!∆ Σ(“Σ)r’2 ∆

2

(r ≥ 2). Although the cost of computing the cumulants needed for the Cornish-

Fisher approximation is also O(m3 ), this method can be faster than the eigen-

value decomposition for small orders of approximation and relatively small

numbers of risk factors.

The computation of all cumulants up to a certain order directly from “Σ is im-

plemented in the quantlet VaRcumulantsDG, while the computation of a single

cumulant from the diagonal decomposition is provided by VaRcumulantDG:

vec= VaRcumulantsDG(n,par)

Computes the ¬rst n cumulants for the class of quadratic forms

of Gaussian vectors. The list par contains at least Gamma and

Sigma.

z= VaRcumulantDG(n,par)

Computes the n-th cumulant for the class of quadratic forms of

Gaussian vectors. The parameter list par is to be generated with

VaRDGdecomp.

12 1 Approximating Value at Risk in Conditional Gaussian Models

Partial Monte-Carlo (or partial Quasi-Monte-Carlo) costs O(m2 ) operations

per sample. (If “ is sparse, it may cost even less.) The number of samples

needed is a function of the desired accuracy. It is clear from the asymptotic

costs of the three methods that partial Monte Carlo will be preferable for

su¬ciently large m.

While Fourier-inversion and Partial Monte-Carlo can in principal achieve any

desired accuracy, the Cornish-Fisher approximations provide only a limited

accuracy, as shown in the next section.

1.3 Cornish-Fisher Approximations

1.3.1 Derivation

The Cornish-Fisher expansion can be derived in two steps. Let ¦ denote some

base distribution and φ its density function. The generalized Cornish-Fisher

expansion (Hill and Davis, 1968) aims to approximate an ±-quantile of F in

terms of the ±-quantile of ¦, i.e., the concatenated function F ’1 —¦ ¦. The key

to a series expansion of F ’1 —¦¦ in terms of derivatives of F and ¦ is Lagrange™s

inversion theorem. It states that if a function s ’ t is implicitly de¬ned by

t = c + s · h(t) (1.6)

and h is analytic in c, then an analytic function f (t) can be developed into a

power series in a neighborhood of s = 0 (t = c):

∞

sr r’1

D [f · hr ](c),

f (t) = f (c) + (1.7)

r!

r=1

where D denotes the di¬erentation operator. For a given probability c = ±,

f = ¦’1 , and h = (¦ ’ F ) —¦ ¦’1 this yields

∞

sr r’1

’1 ’1

D [((F ’ ¦)r /φ) —¦ ¦’1 ](±).

(’1)r

¦ (t) = ¦ (±) + (1.8)

r!

r=1

Setting s = 1 in (1.6) implies ¦’1 (t) = F ’1 (±) and with the notations x =

F ’1 (±), z = ¦’1 (±) (1.8) becomes the formal expansion

∞

1 r’1

D [((F ’ ¦)r /φ) —¦ ¦’1 ](¦(z)).

(’1)r

x=z+

r!

r=1

1.3 Cornish-Fisher Approximations 13

With a = (F ’ ¦)/φ this can be written as

∞

1

(’1)r D(r’1) [ar ](z)

x=z+ (1.9)

r!

r=1

with D(r) = (D+ φ )(D+2 φ ) . . . (D+r φ ) and D(0) being the identity operator.

φ φ φ

(1.9) is the generalized Cornish-Fisher expansion. The second step is to choose a

speci¬c base distribution ¦ and a series expansion for a. The classical Cornish-

Fisher expansion is recovered if ¦ is the standard normal distribution, a is

(formally) expanded into the Gram-Charlier series, and the terms are re-ordered

as described below.

The idea of the Gram-Charlier series is to develop the ratio of the moment

generating function of the considered random variable (M (t) = Eet∆V ) and

2

the moment generating function of the standard normal distribution (et /2 )

into a power series at 0:

∞

’t2 /2

ck tk .

M (t)e = (1.10)

k=0

(ck are the Gram-Charlier coe¬cients. They can be derived from the moments

by multiplying the power series for the two terms on the left hand side.) Com-

ponentwise Fourier inversion yields the corresponding series for the probability

density

∞

ck (’1)k φ(k) (x)

f (x) = (1.11)

k=0

and for the cumulative distribution function (cdf)

∞

ck (’1)k’1 φ(k’1) (x).

F (x) = ¦(x) ’ (1.12)

k=1

(φ und ¦ are now the standard normal density and cdf. The derivatives of

the standard normal density are (’1)k φ(k) (x) = φ(x)Hk (x), where the Her-

mite polynomials Hk form an orthogonal basis in the Hilbert space L2 (R, φ)

of the square integrable functions on R w.r.t. the weight function φ. The

Gram-Charlier coe¬cients can thus be interpreted as the Fourier coe¬cients

of the function f (x)/φ(x) in the Hilbert space L2 (R, φ) with the basis {Hk }

∞

f (x)/φ(x) = k=0 ck Hk (x).) Plugging (1.12) into (1.9) gives the formal Cornish-

Fisher expansion, which is re-grouped as motivated by the central limit theo-

rem.

14 1 Approximating Value at Risk in Conditional Gaussian Models

Assume that ∆V is already normalized (κ1 = 0, κ2 = 1) and consider the

normalized sum of independent random variables ∆Vi with the distribution F ,

n

1

Sn = √n i=1 ∆Vi . The moment generating function of the random variable

Sn is

∞

√n t2 /2

ck tk n’k/2 )n .

Mn (t) = M (t/ n) = e (

k=0

Multiplying out the last term shows that the k-th Gram-Charlier coe¬cient

ck (n) of Sn is a polynomial expression in n’1/2 , involving the coe¬cients ci up

to i = k. If the terms in the formal Cornish-Fisher expansion

r

∞ ∞

1

(’1)r D(r’1) ’

x=z+ ck (n)Hk’1 (z) (1.13)

r!

r=1 k=1

are sorted and grouped with respect to powers of n’1/2 , the classical Cornish-

Fisher series

∞

n’k/2 ξk (z)

x=z+ (1.14)

k=1

results. (The Cornish-Fisher approximation for ∆V results from setting n = 1

in the re-grouped series (1.14).)

It is a relatively tedious process to express the adjustment terms ξk correpond-

ing to a certain power n’k/2 in the Cornish-Fisher expansion (1.14) directly

in terms of the cumulants κr , see (Hill and Davis, 1968). Lee developed a

recurrence formula for the k-th adjustment term ξk in the Cornish-Fisher ex-

pansion, which is implemented in the algorithm AS269 (Lee and Lin, 1992; Lee

and Lin, 1993). (We write the recurrence formula here, because it is incorrect

in (Lee and Lin, 1992).)

k’1

j

—(k+1)

(ξk’j (H) ’ ξk’j ) — (ξj ’ aj H —(j+1) ) — H, (1.15)

’

ξk (H) = ak H

k

j=1

κk+2

with ak = (k+2)! . ξk (H) is a formal polynomial expression in H with the usual

algebraic relations between the summation “+” and the “multiplication” “—”.

Once ξk (H) is multiplied out in —-powers of H, each H —k is to be interpreted

as the Hermite polynomial Hk and then the whole term becomes a polynomial

in z with the “normal” multiplication “·”. ξk denotes the scalar that results

when the “normal” polynomial ξk (H) is evaluated at the ¬xed quantile z, while

ξk (H) denotes the expression in the (+, —)-algebra.

1.3 Cornish-Fisher Approximations 15

This formula is implemented by the quantlet

q = CornishFisher (z, n, cum)

Cornish-Fisher expansion for arbitrary orders for the standard

normal quantile z, order of approximation n, and the vector of

cumulants cum.

The following example prints the Cornish-Fisher approximation for increasing

orders for z=2.3 and cum=1:N:

XFGcofi.xpl

Contents of r

[1,] 2 4.2527

[2,] 3 5.3252

[3,] 4 5.0684

[4,] 5 5.2169

[5,] 6 5.1299

[6,] 7 5.1415

[7,] 8 5.255

1.3.2 Properties

The qualitative properties of the Cornish-Fisher expansion are:

+ If Fm is a sequence of distributions converging to the standard normal dis-

tribution ¦, the Edgeworth- and Cornish-Fisher approximations present

better approximations (asymptotically for m ’ ∞) than the normal ap-

proximation itself.

˜ ˜

’ The approximated functions F and F ’1 —¦¦ are not necessarily monotone.

˜

’ F has the “wrong tail behavior”, i.e., the Cornish-Fisher approximation

for ±-quantiles becomes less and less reliable for ± ’ 0 (or ± ’ 1).

’ The Edgeworth- and Cornish-Fisher approximations do not necessarily

improve (converge) for a ¬xed F and increasing order of approximation,

k.

16 1 Approximating Value at Risk in Conditional Gaussian Models

For more on the qualitative properties of the Cornish-Fisher approximation

see (Jaschke, 2001). It contains also an empirical analysis of the error of the

Cornish-Fisher approximation to the 99%-VaR in real-world examples as well

as its worst-case error on a certain class of one- and two-dimensional delta-

gamma-normal models:

+ The error for the 99%-VaR on the real-world examples - which turned

out to be remarkably close to normal - was about 10’6 σ, which is more

than su¬cient. (The error was normalized with respect to the portfolio™s

standard deviation, σ.)

’ The (lower bound on the) worst-case error for the one- and two-dimensional

problems was about 1.0σ, which corresponds to a relative error of up to

100%.

In summary, the Cornish-Fisher expansion can be a quick approximation with

su¬cient accuracy in many practical situations, but it should not be used

unchecked because of its bad worst-case behavior.

1.4 Fourier Inversion

1.4.1 Error Types in Approximating the Quantile through

Fourier Inversion

Let f denote a continuous, absolutely integrable function and φ(t) =

∞

eitx f (x)dx its Fourier transform. Then, the inversion formula

’∞

∞

1

e’itx φ(t)dt

f (x) = (1.16)

2π ’∞

holds.

The key to an error analysis of trapezoidal, equidistant approximations to the

integral (1.16)

∞

def ∆t

˜ φ(t + k∆t )e’i(t+k∆t )x

f (x, ∆t , t) = (1.17)

2π

k=’∞

1.4 Fourier Inversion 17

is the Poisson summation formula

∞

2π

˜ j)e2πitj/∆t ,

f (x, ∆t , t) = f (x + (1.18)

∆t

j=’∞

˜

see (Abate and Whitt, 1992, p.22). If f (x) is approximated by f (x, ∆t , 0), the

residual

2π

ea (x, ∆t , 0) = f (x + j) (1.19)

∆t

j=0

is called the aliasing error, since di¬erent “pieces” of f are aliased into the

window (’π/∆t , π/∆t ). Another suitable choice is t = ∆t /2:

∞

2π

˜ j)(’1)j .

f (x, ∆t , ∆t /2) = f (x + (1.20)

∆t

j=’∞

˜

If f is nonnegative, f (x, ∆t , 0) ≥ f (x). If f (x) is decreasing in |x| for |x| >

˜

π/∆t , then f (x, ∆t , ∆t /2) ¤ f (x) holds for |x| < π/∆t . The aliasing error

can be controlled by letting ∆t tend to 0. It decreases only slowly when f has

“heavy tails”, or equivalently, when φ has non-smooth features.

It is practical to ¬rst decide on ∆t to control the aliasing error and then decide

on the cut-o¬ in the sum (1.17):

∆t

˜

˜ φ(t + k∆t )e’i(t+k∆t )x .

f (x, T, ∆t , t) = (1.21)

2π

|t+k∆t |¤T

def ˜

˜ ˜

Call et (x, T, ∆t , t) = f (x, T, ∆t , t) ’ f (x, ∆t , t) the truncation error.

For practical purposes, the truncation error et (x, T, ∆t , t) essentially depends

only on (x, T ) and the decision on how to choose T and ∆t can be decoupled.

et (x, T, ∆t , t) converges to

T

def 1

e’itx φ(t)dt ’ f (x)

et (x, T ) = (1.22)

2π

’T

sin(πx) def

π

e’itx dt =

1

for ∆t “ 0. Using = sinc(x) and the convolution

’π

2π πx

theorem, one gets

π/∆x

∞

1

e’itx φ(t)dt = f (y∆x ) sinc(x/∆x ’ y)dy, (1.23)

2π ’∞

’π/∆x

18 1 Approximating Value at Risk in Conditional Gaussian Models

which provides an explicit expression for the truncation error et (x, T ) in terms

of f . It decreases only slowly with T ‘ ∞ (∆x “ 0) if f does not have in¬nitely

many derivatives, or equivalently, φ has “power tails”. The following lemma

leads to the asymptotics of the truncation error in this case.

∞

±(t)t’ν eit dt exists and is

LEMMA 1.1 If limt’∞ ±(t) = 1, ν > 0, and T

¬nite for some T , then

∞ ’ν+1

1

ν’1 T if x = 0

’ν itx

dt ∼

±(t)t e (1.24)

i ’ν ixT

xT e if x = 0

T

for T ’ ∞.

PROOF:

Under the given conditions, both the left and the right hand side converge to 0,

so l™Hospital™s rule is applicable to the ratio of the left and right hand sides.

THEOREM 1.1 If the asymptotic behavior of a Fourier transform φ of a

function f can be described as

φ(t) = w|t|’ν eib sign(t)+ix— t ±(t) (1.25)

with limt’∞ ±(t) = 1, then the truncation error (1.22)

∞

1

φ(t)e’itx dt

et (x, T ) = ’

π T

where denotes the real part, has the asymptotic behavior

wT ’ν+1

π(1’ν) cos(b) if x = x—

∼ (1.26)

wT ’ν π

’ π(x— ’x) cos(b + (x— ’ x)T )

+ if x = x—

2

T

for T ’ ∞ at all points x where 2π ’T φ(t)e’itx converges to f (x). (If in the

1

¬rst case cos(b) = 0, this shall mean that limT ’∞ et (x; T )T ν’1 = 0.)

1.4 Fourier Inversion 19

PROOF:

The previous lemma is applicable for all points x where the Fourier inversion

integral converges.

The theorem completely characterizes the truncation error for those cases,

where f has a “critical point of non-smoothness” and has a higher degree of

smoothness everywhere else. The truncation error decreases one power faster

away from the critical point than at the critical point. Its amplitude is inversely

proportional to the distance from the critical point.

˜

Let F be a (continuous) approximation to a (di¬erentiable) cdf F with f =

˜

F > 0. Denote by ≥ |F (x) ’ F (x)| a known error-bound for the cdf. Any

˜q

solution q (x) to F (˜(x)) = F (x) may be considered an approximation to the

˜

true F (x)-quantile x. Call eq (x) = q (x) ’ x the quantile error. Obviously, the

˜

quantile error can be bounded by

|eq (x)| ¤ , (1.27)

inf y∈U f (y)

where U is a suitable neighborhood of x. Given a sequence of approximations

˜ ˜

F with supx |F (x) ’ F (x)| = ’ 0,

˜

F (x) ’ F (x)

eq (x) ∼ ( ’ 0) (1.28)

f (x)

holds.

FFT-based Fourier inversion yields approximations for the cdf F on equidistant

∆x -spaced grids. Depending on the smoothness of F , linear or higher-order

interpolations may be used. Any monotone interpolation of {F (x0 + ∆x j)}j

yields a quantile approximation whose interpolation error can be bounded by

∆x . This bound can be improved if an upper bound on the density f in a

suitable neighborhood of the true quantile is known.

20 1 Approximating Value at Risk in Conditional Gaussian Models

1.4.2 Tail Behavior

22

If »j = 0 for some j, then |φ(t)| = O(e’δj t /2 ). In the following, we assume

that |»i | > 0 for all i. The norm of φ(t) has the form

m

δi t2 /2

2

(1 + »2 t2 )’1/4 exp ’

|φ(t)| = , (1.29)

i

1 + » 2 t2

i

i=1

|φ(t)| ∼ w— |t|’m/2 |t| ’ ∞ (1.30)

with

m

1

def

|»i |’1/2 exp ’ (δi /»i )2 .

w— = (1.31)

2

i=1

The arg has the form

m

1 12 »i t

arctan(»i t) ’ δi t2

arg φ(t) = θt + , (1.32)

1 + » 2 t2

2 2 i

i=1

m 2

π δi t

arg φ(t) ∼ θt + sign(»i t) ’ ) (1.33)

4 2»i