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 xiiiAbout the Authors xv1 Introduction and Reading Guide 12 Binomial Trees 72.1 Equities and Basic Options 72.2 The One Period Model 82.3 The Multiperiod Binomial Model 92.4 Black-Scholes and Trees 102.5 Strengths and Weaknesses of Binomial Trees 122.5.1 Ease of Implementation 122.5.2 Oscillations 122.5.3 Non-recombining Trees 142.5.4 Exotic Options and Trees 142.5.5 Greeks and Binomial Trees 152.5.6 Grid Adaptivity and Trees 152.6 Conclusion 163 Finite Differences and the Black-Scholes PDE 173.1 A Continuous Time Model for Equity Prices 173.2 Black-Scholes Model: From the SDE to the PDE 193.3 Finite Differences 233.4 Time Discretization 273.5 Stability Considerations 303.6 Finite Differences and the Heat Equation 303.6.1 Numerical Results 343.7 Appendix: Error Analysis 364 Mean Reversion and Trinomial Trees 394.1 Some Fixed Income Terms 394.1.1 Interest Rates and Compounding 394.1.2 Libor Rates and Vanilla Interest Rate Swaps 404.2 Black76 for Caps and Swaptions 434.3 One-Factor Short Rate Models 454.3.1 Prominent Short Rate Models 454.4 The Hull-White Model in More Detail 464.5 Trinomial Trees 475 Upwinding Techniques for Short Rate Models 555.1 Derivation of a PDE for Short Rate Models 555.2 Upwind Schemes 565.2.1 Model Equation 575.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 635.3.1 Bond Details 645.3.2 Model Details 645.3.3 Numerical Method 655.3.4 An Algorithm in Pseudocode 685.3.5 Results 696 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 776.6.1 Double Barrier Options and Dirichlet Boundary Conditions 776.6.2 Artificial Boundary Conditions and the Neumann Case 787 Finite Element Methods 817.1 Introduction 817.1.1 Weighted Residual Methods 817.1.2 Basic Steps 827.2 Grid Generation 837.3 Elements 857.3.1 1D Elements 867.3.2 2D Elements 887.4 The Assembling Process 907.4.1 Element Matrices 937.4.2 Time Discretization 977.4.3 Global Matrices 987.4.4 Boundary Conditions 1017.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems 1037.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 1057.6 Appendix: Higher Order Elements 1077.6.1 3D Elements 1097.6.2 Local and Natural Coordinates 1118 Solving Systems of Linear Equations 1178.1 Direct Methods 1188.1.1 Gaussian Elimination 1188.1.2 Thomas Algorithm 1198.1.3 LU Decomposition 1208.1.4 Cholesky Decomposition 1218.2 Iterative Solvers 1228.2.1 Matrix Decomposition 1238.2.2 Krylov Methods 1258.2.3 Multigrid Solvers 1268.2.4 Preconditioning 1299 Monte Carlo Simulation 1339.1 The Principles of Monte Carlo Integration 1339.2 Pricing Derivatives with Monte Carlo Methods 1349.2.1 Discretizing the Stochastic Differential Equation 1359.2.2 Pricing Formalism 1379.2.3 Valuation of a Steepener under a Two Factor Hull-White Model 1379.3 An Introduction to the Libor Market Model 1399.4 Random Number Generation 1469.4.1 Properties of a Random Number Generator 1479.4.2 Uniform Variates 1489.4.3 Random Vectors 1509.4.4 Recent Developments in Random Number Generation 1519.4.5 Transforming Variables 1529.4.6 Random Number Generation for Commonly Used Distributions 15510 Advanced Monte Carlo Techniques 16110.1 Variance Reduction Techniques 16110.1.1 Antithetic Variates 16110.1.2 Control Variates 16310.1.3 Conditioning 16610.1.4 Additional Techniques for Variance Reduction 16810.2 Quasi Monte Carlo Method 16910.2.1 Low-Discrepancy Sequences 16910.2.2 Randomizing QMC 17410.3 Brownian Bridge Technique 17510.3.1 A Steepener under a Libor Market Model 17711 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 18111.2.1 Algorithm: Extended LSMC Method for Bermudan Options 18211.2.2 Notes on Basis Functions and Regression 18511.3 Examples 18611.3.1 A Bermudan Callable Floater under Different Short-rate Models 18611.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model 18811.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework 18912 Characteristic Function Methods for Option Pricing 19312.1 Equity Models 19412.1.1 Heston Model 19612.1.2 Jump Diffusion Models 19812.1.3 Infinite Activity Models 19912.1.4 Bates Model 20012.2 Fourier Techniques 20112.2.1 Fast Fourier Transform Methods 20112.2.2 Fourier-Cosine Expansion Methods 20313 Numerical Methods for the Solution of PIDEs 20913.1 A PIDE for Jump Models 20913.2 Numerical Solution of the PIDE 21013.2.1 Discretization of the Spatial Domain 21113.2.2 Discretization of the Time Domain 21113.2.3 A European Option under the Kou Jump Diffusion Model 21213.3 Appendix: Numerical Integration via Newton-Cotes Formulae 21414 Copulas and the Pitfalls of Correlation 21714.1 Correlation 21814.1.1 Pearson’s ρ 21814.1.2 Spearman’s ρ 21814.1.3 Kendall’s τ 22014.1.4 Other Measures 22114.2 Copulas 22114.2.1 Basic Concepts 22214.2.2 Important Copula Functions 22214.2.3 Parameter estimation and sampling 22914.2.4 Default Probabilities for Credit Derivatives 23415 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 24515.4.1 Dupire’s Inversion Formula 24615.4.2 Identifying Local Volatility 24615.4.3 Results 24715.5 Identifying Parameters in Volatility Models 24815.5.1 Model Calibration for the FTSE- 100 24916 Optimization Techniques 25316.1 Model Calibration and Optimization 25516.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems 25616.2 Heuristically Inspired Algorithms 25816.2.1 Simulated Annealing 25916.2.2 Differential Evolution 26016.3 A Hybrid Algorithm for Heston Model Calibration 26116.4 Portfolio Optimization 26517 Risk Management 26917.1 Value at Risk and Expected Shortfall 26917.1.1 Parametric VaR 27017.1.2 Historical VaR 27217.1.3 Monte Carlo VaR 27317.1.4 Individual and Contribution VaR 27417.2 Principal Component Analysis 27617.2.1 Principal Component Analysis for Non-scalar Risk Factors 27617.2.2 Principal Components for Fast Valuation 27717.3 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 28818.3.1 CUDA and OpenCL 28918.3.2 Memory 28918.4 Parallelization of Single Instrument Valuations using (Q)MC 29018.5 Parallelization of Hybrid Calibration Algorithms 29118.5.1 Implementation Details 29218.5.2 Results 29519 Building Large Software Systems for the Financial Industry 297Bibliography 301Index 307