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
