A Workout in Computational Finance, with Website

MICHAEL AICHINGER obtained his Ph.D. in Theoretical Physics from the Johannes Kepler University Linz with a thesis on numerical methods in density functional theory and their application to 2D finite electron systems. A mobility grant led him to the Texas A&M University (2003) and to the Helsinki University of Technology (2004). In 2007 Michael Aichinger joined the Industrial Mathematics Competence Center where he has been working as a senior researcher and consultant in the field of quantitative finance for the last five years. He also works for the Austrian Academy of Sciences at the Radon Institute for Computational and Applied Mathematics where he is involved in several industrial mathematics and computational physics projects. Michael has (co-) authored around 20 journal articles in the fields of computational physics and quantitative finance. ANDREAS BINDER obtained his Ph.D. in Industrial Mathematics from the Johannes Kepler University Linz with a thesis on continuous casting of steel. A research grant led him to the Oxford Center for Industrial and Applied Mathematics, UK, in 1991, where he got in touch with mathematical finance for the first time. After some years being an assistant professor at the Industrial Mathematics Institute, in 1996, he left university and became managing director of MathConsult GmbH, where he heads also the Computational Finance Group. Andreas has authored two introductory books on mathematical finance and 25 journal articles in the fields of industrial mathematics and of mathematical finance.

Acknowledgements xiii

About the Authors xv

1 Introduction and Reading Guide 1

2 Binomial Trees 7

2.1 Equities and Basic Options 7

2.2 The One Period Model 8

2.3 The Multiperiod Binomial Model 9

2.4 Black-Scholes and Trees 10

2.5 Strengths and Weaknesses of Binomial Trees 12

2.5.1 Ease of Implementation 12

2.5.2 Oscillations 12

2.5.3 Non-recombining Trees 14

2.5.4 Exotic Options and Trees 14

2.5.5 Greeks and Binomial Trees 15

2.5.6 Grid Adaptivity and Trees 15

2.6 Conclusion 16

3 Finite Differences and the Black-Scholes PDE 17

3.1 A Continuous Time Model for Equity Prices 17

3.2 Black-Scholes Model: From the SDE to the PDE 19

3.3 Finite Differences 23

3.4 Time Discretization 27

3.5 Stability Considerations 30

3.6 Finite Differences and the Heat Equation 30

3.6.1 Numerical Results 34

3.7 Appendix: Error Analysis 36

4 Mean Reversion and Trinomial Trees 39

4.1 Some Fixed Income Terms 39

4.1.1 Interest Rates and Compounding 39

4.1.2 Libor Rates and Vanilla Interest Rate Swaps 40

4.2 Black76 for Caps and Swaptions 43

4.3 One-Factor Short Rate Models 45

4.3.1 Prominent Short Rate Models 45

4.4 The Hull-White Model in More Detail 46

4.5 Trinomial Trees 47

5 Upwinding Techniques for Short Rate Models 55

5.1 Derivation of a PDE for Short Rate Models 55

5.2 Upwind Schemes 56

5.2.1 Model Equation 57

5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 63

5.3.1 Bond Details 64

5.3.2 Model Details 64

5.3.3 Numerical Method 65

5.3.4 An Algorithm in Pseudocode 68

5.3.5 Results 69

6 Boundary, Terminal and Interface Conditions and their Influence 71

6.1 Terminal Conditions for Equity Options 71

6.2 Terminal Conditions for Fixed Income Instruments 72

6.3 Callability and Bermudan Options 74

6.4 Dividends 74

6.5 Snowballs and TARNs 75

6.6 Boundary Conditions 77

6.6.1 Double Barrier Options and Dirichlet Boundary Conditions 77

6.6.2 Artificial Boundary Conditions and the Neumann Case 78

7 Finite Element Methods 81

7.1 Introduction 81

7.1.1 Weighted Residual Methods 81

7.1.2 Basic Steps 82

7.2 Grid Generation 83

7.3 Elements 85

7.3.1 1D Elements 86

7.3.2 2D Elements 88

7.4 The Assembling Process 90

7.4.1 Element Matrices 93

7.4.2 Time Discretization 97

7.4.3 Global Matrices 98

7.4.4 Boundary Conditions 101

7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems 103

7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105

7.6 Appendix: Higher Order Elements 107

7.6.1 3D Elements 109

7.6.2 Local and Natural Coordinates 111

8 Solving Systems of Linear Equations 117

8.1 Direct Methods 118

8.1.1 Gaussian Elimination 118

8.1.2 Thomas Algorithm 119

8.1.3 LU Decomposition 120

8.1.4 Cholesky Decomposition 121

8.2 Iterative Solvers 122

8.2.1 Matrix Decomposition 123

8.2.2 Krylov Methods 125

8.2.3 Multigrid Solvers 126

8.2.4 Preconditioning 129

9 Monte Carlo Simulation 133

9.1 The Principles of Monte Carlo Integration 133

9.2 Pricing Derivatives with Monte Carlo Methods 134

9.2.1 Discretizing the Stochastic Differential Equation 135

9.2.2 Pricing Formalism 137

9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model 137

9.3 An Introduction to the Libor Market Model 139

9.4 Random Number Generation 146

9.4.1 Properties of a Random Number Generator 147

9.4.2 Uniform Variates 148

9.4.3 Random Vectors 150

9.4.4 Recent Developments in Random Number Generation 151

9.4.5 Transforming Variables 152

9.4.6 Random Number Generation for Commonly Used Distributions 155

10 Advanced Monte Carlo Techniques 161

10.1 Variance Reduction Techniques 161

10.1.1 Antithetic Variates 161

10.1.2 Control Variates 163

10.1.3 Conditioning 166

10.1.4 Additional Techniques for Variance Reduction 168

10.2 Quasi Monte Carlo Method 169

10.2.1 Low-Discrepancy Sequences 169

10.2.2 Randomizing QMC 174

10.3 Brownian Bridge Technique 175

10.3.1 A Steepener under a Libor Market Model 177

11 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks 179

11.1 Pricing American options using the Longstaff and Schwartz algorithm 179

11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments 181

11.2.1 Algorithm: Extended LSMC Method for Bermudan Options 182

11.2.2 Notes on Basis Functions and Regression 185

11.3 Examples 186

11.3.1 A Bermudan Callable Floater under Different Short-rate Models 186

11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model 188

11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework 189

12 Characteristic Function Methods for Option Pricing 193

12.1 Equity Models 194

12.1.1 Heston Model 196

12.1.2 Jump Diffusion Models 198

12.1.3 Infinite Activity Models 199

12.1.4 Bates Model 200

12.2 Fourier Techniques 201

12.2.1 Fast Fourier Transform Methods 201

12.2.2 Fourier-Cosine Expansion Methods 203

13 Numerical Methods for the Solution of PIDEs 209

13.1 A PIDE for Jump Models 209

13.2 Numerical Solution of the PIDE 210

13.2.1 Discretization of the Spatial Domain 211

13.2.2 Discretization of the Time Domain 211

13.2.3 A European Option under the Kou Jump Diffusion Model 212

13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214

14 Copulas and the Pitfalls of Correlation 217

14.1 Correlation 218

14.1.1 Pearson’s ρ 218

14.1.2 Spearman’s ρ 218

14.1.3 Kendall’s τ 220

14.1.4 Other Measures 221

14.2 Copulas 221

14.2.1 Basic Concepts 222

14.2.2 Important Copula Functions 222

14.2.3 Parameter estimation and sampling 229

14.2.4 Default Probabilities for Credit Derivatives 234

15 Parameter Calibration and Inverse Problems 239

15.1 Implied Black-Scholes Volatilities 239

15.2 Calibration Problems for Yield Curves 240

15.3 Reversion Speed and Volatility 245

15.4 Local Volatility 245

15.4.1 Dupire’s Inversion Formula 246

15.4.2 Identifying Local Volatility 246

15.4.3 Results 247

15.5 Identifying Parameters in Volatility Models 248

15.5.1 Model Calibration for the FTSE- 100 249

16 Optimization Techniques 253

16.1 Model Calibration and Optimization 255

16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems 256

16.2 Heuristically Inspired Algorithms 258

16.2.1 Simulated Annealing 259

16.2.2 Differential Evolution 260

16.3 A Hybrid Algorithm for Heston Model Calibration 261

16.4 Portfolio Optimization 265

17 Risk Management 269

17.1 Value at Risk and Expected Shortfall 269

17.1.1 Parametric VaR 270

17.1.2 Historical VaR 272

17.1.3 Monte Carlo VaR 273

17.1.4 Individual and Contribution VaR 274

17.2 Principal Component Analysis 276

17.2.1 Principal Component Analysis for Non-scalar Risk Factors 276

17.2.2 Principal Components for Fast Valuation 277

17.3 Extreme Value Theory 278

18 Quantitative Finance on Parallel Architectures 285

18.1 A Short Introduction to Parallel Computing 285

18.2 Different Levels of Parallelization 288

18.3 GPU Programming 288

18.3.1 CUDA and OpenCL 289

18.3.2 Memory 289

18.4 Parallelization of Single Instrument Valuations using (Q)MC 290

18.5 Parallelization of Hybrid Calibration Algorithms 291

18.5.1 Implementation Details 292

18.5.2 Results 295

19 Building Large Software Systems for the Financial Industry 297

Bibliography 301

Index 307