Applied quantitative finance

47 3 Quantification of Spread Risk by Means of Historical Simulation 51 Christoph Frisch and Germar Kn¨ochlein 3.1 Introduction... 145 Wolfgang H¨ardle and Jun Zheng 7.1 Implied Binomial

Applied Quantitative Finance

Wolfgang H¨ ardle Torsten Kleinow Gerhard Stahl

In cooperation withG¨okhan Aydınlı, Oliver Jim Blaskowitz, Song Xi Chen,Matthias Fengler, J¨urgen Franke, Christoph Frisch,Helmut Herwartz, Harriet Holzberger, Steffi H¨ose,Stefan Huschens, Kim Huynh, Stefan R Jaschke, Yuze JiangPierre Kervella, R¨udiger Kiesel, Germar Kn¨ochlein,Sven Knoth, Jens L¨ussem, Danilo Mercurio,

Marlene M¨uller, J¨orn Rank, Peter Schmidt,

Rainer Schulz, J¨urgen Schumacher, Thomas Siegl,Robert Wania, Axel Werwatz, Jun Zheng

1 Approximating Value at Risk in Conditional Gaussian Models 3

Stefan R Jaschkeand 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¨orn 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 Kn¨ochlein 3.1 Introduction 51

3.2 Risk Categories – a Definition 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 Steffi H¨ose, Stefan Huschens and Robert Wania 4.1 Rating Transition Probabilities 88

4.1.1 From Credit Events to Migration Counts 88

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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¨udiger Kiesel andTorsten 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

Contents vii

6 The Analysis of Implied Volatilities 127

Matthias R Fengler,Wolfgang H¨ardle andPeter 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 H¨ardle 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

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 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

Contents ix

Matthias R FenglerandHelmut Herwartz

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 H¨ardleandTorsten Kleinow 12.1 Introduction 259

x Contents

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 SchulzandAxel 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 filtering in XploRe 293

13.5.1 Overview 293

13.5.2 Setting the system matrices 293

Contents xi

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

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¨ussemandJ¨urgen 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 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 payoff function 362

16.3.2 Integration of the payoff function in the simulation frame-work 362

16.3.3 Restrictions for the payoff functions 365

17 Nonparametric Estimators of GARCH Processes 367 J¨urgen Franke, Harriet Holzberger andMarlene M¨uller 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¨okhan Aydınlı 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

This book is designed for students and researchers who want to develop fessional skill in modern quantitative applications in finance The Center forApplied Statistics and Economics (CASE) course at Humboldt-Universit¨at zuBerlin that forms the basis for this book is offered to interested students whohave had some experience with probability, statistics and software applicationsbut have not had advanced courses in mathematical finance Although thecourse assumes only a modest background it moves quickly between differentfields of applications and in the end, the reader can expect to have theoreticaland computational tools that are deep enough and rich enough to be relied onthroughout future professional careers

pro-The text is readable for the graduate student in financial engineering as well asfor the inexperienced newcomer to quantitative finance who wants to get a grip

on modern statistical tools in financial data analysis The experienced readerwith a bright knowledge of mathematical finance will probably skip some sec-tions but will hopefully enjoy the various computational tools of the presentedtechniques A graduate student might think that some of the econometrictechniques are well known The mathematics of risk management and volatil-ity dynamics will certainly introduce him into the rich realm of quantitativefinancial data analysis

The computer inexperienced user of this e-book is softly introduced into theinteractive book concept and will certainly enjoy the various practical exam-ples The e-book is designed as an interactive document: a stream of text andinformation with various hints and links to additional tools and features Oure-book design offers also a complete PDF and HTML file with links to worldwide computing servers The reader of this book may therefore without down-load or purchase of software use all the presented examples and methods viathe enclosed license code number with a local XploRe Quantlet Server (XQS).Such XQ Servers may also be installed in a department or addressed freely onthe 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 first part Jaschke andJiang treat the Approximation of the Value at Risk in conditional GaussianModels and Rank and Siegl show how the VaR can be calculated using copulas.The second part starts with an analysis of rating migration probabilities byH¨ose, Huschens and Wania Frisch and Kn¨ochlein quantify the risk of yieldspread changes via historical simulations This part is completed by an anal-ysis of the sensitivity of risk measures to changes in the dependency structurebetween single positions of a portfolio by Kiesel and Kleinow

The third part is devoted to the analysis of implied volatilities and their ics Fengler, H¨ardle and Schmidt start with an analysis of the implied volatilitysurface and show how common PCA can be applied to model the dynamics ofthe surface In the next two chapters the authors estimate the risk neutralstate price density from observed option prices and the corresponding impliedvolatilities While H¨ardle and Zheng apply implied binomial trees to estimatethe SPD, the method by Huynh, Kervella and Zheng is based on a local poly-nomial estimation of the implied volatility and its derivatives Blaskowitz andSchmidt use the proposed methods to develop trading strategies based on thecomparison of the historical SPD and the one implied by option prices.Recently developed econometric methods are presented in the last part of thebook Fengler and Herwartz introduce a multivariate volatility model and ap-ply it to exchange rates Methods used to monitor sequentially observed dataare treated by Knoth Chen, H¨ardle and Kleinow apply the empirical likeli-hood concept to develop a test about a parametric diffusion model Schulzand Werwatz estimate a state space model of Berlin house prices that can beused to construct a time series of the price of a standard house The influ-ence of long memory effects on financial time series is analyzed by Blaskowitzand Schmidt Mercurio propose a methodology to identify time intervals ofhomogeneity for time series The pricing of exotic options via a simulationapproach is introduced by L¨ussem and Schumacher The chapter by Franke,Holzberger and M¨uller is devoted to a nonparametric estimation approach ofGARCH models The book closes with a chapter of Aydınlı, who introduces

dynam-a technology to connect stdynam-anddynam-ard softwdynam-are with the XploRe server in order tohave access to quantlets developed in this book

We gratefully acknowledge the support of Deutsche Forschungsgemeinschaft,SFB 373 Quantifikation und Simulation ¨Okonomischer Prozesse A book of thiskind would not have been possible without the help of many friends, colleaguesand students For the technical production of the e-book platform we would