A comprehensive introduction to various numerical methods used in computational finance today Quantitative skills are a prerequisite for anyone working in finance or beginning a career in the field, as well as risk managers. A thorough grounding in numerical methods is necessary, as is the ability to assess their quality, advantages, and limitations. This book offers a thorough introduction to each method, revealing the numerical traps that practitioners frequently fall into. Each method is referenced with practical, real-world examples in the areas of valuation, risk analysis, and calibration of specific financial instruments and models. It features a strong emphasis on robust schemes for the numerical treatment of problems within computational finance. Methods covered include PDE/PIDE using finite differences or finite elements, fast and stable solvers for sparse grid systems, stabilization and regularization techniques for inverse problems resulting from the calibration of financial models to market data, Monte Carlo and Quasi Monte Carlo techniques for simulating high dimensional systems, and local and global optimization tools to solve the minimization problem.
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
