A Workout in Computational 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

2.6 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

3.7 Appendix: Error Analysis 36

4 Mean Reversion and Trinomial Trees 39

4.1 Some Fixed Income Terms 39

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

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

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

7 Finite Element Methods 81

7.1 Introduction 81

7.2 Grid Generation 83

7.3 Elements 85

7.4 The Assembling Process 90

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

7.6 Appendix: Higher Order Elements 107

8 Solving Systems of Linear Equations 117

8.1 Direct Methods 118

8.2 Iterative Solvers 122

9 Monte Carlo Simulation 133

9.1 The Principles of Monte Carlo Integration 133

9.2 Pricing Derivatives with Monte Carlo Methods 134

9.3 An Introduction to the Libor Market Model 139

9.4 Random Number Generation 146

10 Advanced Monte Carlo Techniques 161

10.1 Variance Reduction Techniques 161

10.2 Quasi Monte Carlo Method 169

10.3 Brownian Bridge Technique 175

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

11.3 Examples 186

12 Characteristic Function Methods for Option Pricing 193

12.1 Equity Models 194

12.2 Fourier Techniques 201

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

13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214

14 Copulas and the Pitfalls of Correlation 217

14.1 Correlation 218

14.2 Copulas 221

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

15.5 Identifying Parameters in Volatility Models 248

16 Optimization Techniques 253

16.1 Model Calibration and Optimization 255

16.2 Heuristically Inspired Algorithms 258

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

17.2 Principal Component Analysis 276

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

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

18.5 Parallelization of Hybrid Calibration Algorithms 291

19 Building Large Software Systems for the Financial Industry 297

Bibliography 301

Index 307