A Workout in Computational Finance

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.

Table of Contents:
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 2.5.1 Ease of Implementation 12 2.5.2 Oscillations 12 2.5.3 Non-recombining Trees 14 2.5.4 Exotic Options and Trees 14 2.5.5 Greeks and Binomial Trees 15 2.5.6 Grid Adaptivity and Trees 15 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 3.6.1 Numerical Results 34 3.7 Appendix: Error Analysis 36 4 Mean Reversion and Trinomial Trees 39 4.1 Some Fixed Income Terms 39 4.1.1 Interest Rates and Compounding 39 4.1.2 Libor Rates and Vanilla Interest Rate Swaps 40 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 5.2.1 Model Equation 57 5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 63 5.3.1 Bond Details 64 5.3.2 Model Details 64 5.3.3 Numerical Method 65 5.3.4 An Algorithm in Pseudocode 68 5.3.5 Results 69 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 6.6.1 Double Barrier Options and Dirichlet Boundary Conditions 77 6.6.2 Artificial Boundary Conditions and the Neumann Case 78 7 Finite Element Methods 81 7.1 Introduction 81 7.1.1 Weighted Residual Methods 81 7.1.2 Basic Steps 82 7.2 Grid Generation 83 7.3 Elements 85 7.3.1 1D Elements 86 7.3.2 2D Elements 88 7.4 The Assembling Process 90 7.4.1 Element Matrices 93 7.4.2 Time Discretization 97 7.4.3 Global Matrices 98 7.4.4 Boundary Conditions 101 7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems 103 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105 7.6 Appendix: Higher Order Elements 107 7.6.1 3D Elements 109 7.6.2 Local and Natural Coordinates 111 8 Solving Systems of Linear Equations 117 8.1 Direct Methods 118 8.1.1 Gaussian Elimination 118 8.1.2 Thomas Algorithm 119 8.1.3 LU Decomposition 120 8.1.4 Cholesky Decomposition 121 8.2 Iterative Solvers 122 8.2.1 Matrix Decomposition 123 8.2.2 Krylov Methods 125 8.2.3 Multigrid Solvers 126 8.2.4 Preconditioning 129 9 Monte Carlo Simulation 133 9.1 The Principles of Monte Carlo Integration 133 9.2 Pricing Derivatives with Monte Carlo Methods 134 9.2.1 Discretizing the Stochastic Differential Equation 135 9.2.2 Pricing Formalism 137 9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model 137 9.3 An Introduction to the Libor Market Model 139 9.4 Random Number Generation 146 9.4.1 Properties of a Random Number Generator 147 9.4.2 Uniform Variates 148 9.4.3 Random Vectors 150 9.4.4 Recent Developments in Random Number Generation 151 9.4.5 Transforming Variables 152 9.4.6 Random Number Generation for Commonly Used Distributions 155 10 Advanced Monte Carlo Techniques 161 10.1 Variance Reduction Techniques 161 10.1.1 Antithetic Variates 161 10.1.2 Control Variates 163 10.1.3 Conditioning 166 10.1.4 Additional Techniques for Variance Reduction 168 10.2 Quasi Monte Carlo Method 169 10.2.1 Low-Discrepancy Sequences 169 10.2.2 Randomizing QMC 174 10.3 Brownian Bridge Technique 175 10.3.1 A Steepener under a Libor Market Model 177 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 11.2.1 Algorithm: Extended LSMC Method for Bermudan Options 182 11.2.2 Notes on Basis Functions and Regression 185 11.3 Examples 186 11.3.1 A Bermudan Callable Floater under Different Short-rate Models 186 11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model 188 11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework 189 12 Characteristic Function Methods for Option Pricing 193 12.1 Equity Models 194 12.1.1 Heston Model 196 12.1.2 Jump Diffusion Models 198 12.1.3 Infinite Activity Models 199 12.1.4 Bates Model 200 12.2 Fourier Techniques 201 12.2.1 Fast Fourier Transform Methods 201 12.2.2 Fourier-Cosine Expansion Methods 203 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 13.2.1 Discretization of the Spatial Domain 211 13.2.2 Discretization of the Time Domain 211 13.2.3 A European Option under the Kou Jump Diffusion Model 212 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214 14 Copulas and the Pitfalls of Correlation 217 14.1 Correlation 218 14.1.1 Pearson’s ρ 218 14.1.2 Spearman’s ρ 218 14.1.3 Kendall’s τ 220 14.1.4 Other Measures 221 14.2 Copulas 221 14.2.1 Basic Concepts 222 14.2.2 Important Copula Functions 222 14.2.3 Parameter estimation and sampling 229 14.2.4 Default Probabilities for Credit Derivatives 234 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 15.4.1 Dupire’s Inversion Formula 246 15.4.2 Identifying Local Volatility 246 15.4.3 Results 247 15.5 Identifying Parameters in Volatility Models 248 15.5.1 Model Calibration for the FTSE- 100 249 16 Optimization Techniques 253 16.1 Model Calibration and Optimization 255 16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems 256 16.2 Heuristically Inspired Algorithms 258 16.2.1 Simulated Annealing 259 16.2.2 Differential Evolution 260 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 17.1.1 Parametric VaR 270 17.1.2 Historical VaR 272 17.1.3 Monte Carlo VaR 273 17.1.4 Individual and Contribution VaR 274 17.2 Principal Component Analysis 276 17.2.1 Principal Component Analysis for Non-scalar Risk Factors 276 17.2.2 Principal Components for Fast Valuation 277 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 18.3.1 CUDA and OpenCL 289 18.3.2 Memory 289 18.4 Parallelization of Single Instrument Valuations using (Q)MC 290 18.5 Parallelization of Hybrid Calibration Algorithms 291 18.5.1 Implementation Details 292 18.5.2 Results 295 19 Building Large Software Systems for the Financial Industry 297 Bibliography 301 Index 307