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.

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

2.6 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 555.2 Upwind Schemes 565.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 636. Boundary, Terminal and Interface Conditions and their Influence 716.1 Terminal Conditions for Equity Options 716.2 Terminal Conditions for Fixed Income Instruments 726.3 Callability and Bermudan Options 746.4 Dividends 746.5 Snowballs and TARNs 756.6 Boundary Conditions 777 Finite Element Methods 817.1 Introduction 817.2 Grid Generation 837.3 Elements 857.4 The Assembling Process 907.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 1057.6 Appendix: Higher Order Elements 1078 Solving Systems of Linear Equations 1178.1 Direct Methods 1188.2 Iterative Solvers 1229 Monte Carlo Simulation 1339.1 The Principles of Monte Carlo Integration 1339.2 Pricing Derivatives with Monte Carlo Methods 1349.3 An Introduction to the Libor Market Model 1399.4 Random Number Generation 14610 Advanced Monte Carlo Techniques 16110.1 Variance Reduction Techniques 16110.2 Quasi Monte Carlo Method 16910.3 Brownian Bridge Technique 17511 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks 17911.1 Pricing American options using the Longstaff and Schwartz algorithm 17911.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments 18111.3 Examples 18612 Characteristic Function Methods for Option Pricing 19312.1 Equity Models 19412.2 Fourier Techniques 20113 Numerical Methods for the Solution of PIDEs 20913.1 A PIDE for Jump Models 20913.2 Numerical Solution of the PIDE 21013.3 Appendix: Numerical Integration via Newton-Cotes Formulae 21414 Copulas and the Pitfalls of Correlation 21714.1 Correlation 21814.2 Copulas 22115 Parameter Calibration and Inverse Problems 23915.1 Implied Black-Scholes Volatilities 23915.2 Calibration Problems for Yield Curves 24015.3 Reversion Speed and Volatility 24515.4 Local Volatility 24515.5 Identifying Parameters in Volatility Models 24816 Optimization Techniques 25316.1 Model Calibration and Optimization 25516.2 Heuristically Inspired Algorithms 25816.3 A Hybrid Algorithm for Heston Model Calibration 26116.4 Portfolio Optimization 26517 Risk Management 26917.1 Value at Risk and Expected Shortfall 26917.2 Principal Component Analysis 27617.3 Extreme Value Theory 27818 Quantitative Finance on Parallel Architectures 28518.1 A Short Introduction to Parallel Computing 28518.2 Different Levels of Parallelization 28818.3 GPU Programming 28818.4 Parallelization of Single Instrument Valuations using (Q)MC 29018.5 Parallelization of Hybrid Calibration Algorithms 29119 Building Large Software Systems for the Financial Industry 297Bibliography 301Index 307