A Workout in Computational Finance -- Contents -- Acknowledgements -- About the Authors -- 1 Introduction and Reading Guide -- 2 Binomial Trees -- 2.1 Equities and Basic Options -- 2.2 The One Period Model -- 2.3 The Multiperiod Binomial Model -- 2.4 Black-Scholes and Trees -- 2.5 Strengths and Weaknesses of Binomial Trees -- 2.5.1 Ease of Implementation -- 2.5.2 Oscillations -- 2.5.3 Non-recombining Trees -- 2.5.4 Exotic Options and Trees -- 2.5.5 Greeks and Binomial Trees -- 2.5.6 Grid Adaptivity and Trees -- 2.6 Conclusion -- 3 Finite Differences and the Black-Scholes PDE -- 3.1 A Continuous Time Model for Equity Prices -- 3.2 Black-Scholes Model: From the SDE to the PDE -- 3.3 Finite Differences -- 3.4 Time Discretization -- 3.5 Stability Considerations -- 3.6 Finite Differences and the Heat Equation -- 3.6.1 Numerical Results -- 3.7 Appendix: Error Analysis -- 4 Mean Reversion and Trinomial Trees -- 4.1 Some Fixed Income Terms -- 4.1.1 Interest Rates and Compounding -- 4.1.2 Libor Rates and Vanilla Interest Rate Swaps -- 4.2 Black76 for Caps and Swaptions -- 4.3 One-Factor Short Rate Models -- 4.3.1 Prominent Short Rate Models -- 4.4 The Hull-White Model in More Detail -- 4.5 Trinomial Trees -- 5 Upwinding Techniques for Short Rate Models -- 5.1 Derivation of a PDE for Short Rate Models -- 5.2 Upwind Schemes -- 5.2.1 Model Equation -- 5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model -- 5.3.1 Bond Details -- 5.3.2 Model Details -- 5.3.3 Numerical Method -- 5.3.4 An Algorithm in Pseudocode -- 5.3.5 Results -- 6 Boundary, Terminal and Interface Conditions and their Influence -- 6.1 Terminal Conditions for Equity Options -- 6.2 Terminal Conditions for Fixed Income Instruments -- 6.3 Callability and Bermudan Options -- 6.4 Dividends -- 6.5 Snowballs and TARNs -- 6.6 Boundary Conditions.

6.6.1 Double Barrier Options and Dirichlet Boundary Conditions -- 6.6.2 Artificial Boundary Conditions and the Neumann Case -- 7 Finite Element Methods -- 7.1 Introduction -- 7.1.1 Weighted Residual Methods -- 7.1.2 Basic Steps -- 7.2 Grid Generation -- 7.3 Elements -- 7.3.1 1D Elements -- 7.3.2 2D Elements -- 7.4 The Assembling Process -- 7.4.1 Element Matrices -- 7.4.2 Time Discretization -- 7.4.3 Global Matrices -- 7.4.4 Boundary Conditions -- 7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems -- 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model -- 7.6 Appendix: Higher Order Elements -- 7.6.1 3D Elements -- 7.6.2 Local and Natural Coordinates -- 8 Solving Systems of Linear Equations -- 8.1 Direct Methods -- 8.1.1 Gaussian Elimination -- 8.1.2 Thomas Algorithm -- 8.1.3 LU Decomposition -- 8.1.4 Cholesky Decomposition -- 8.2 Iterative Solvers -- 8.2.1 Matrix Decomposition -- 8.2.2 Krylov Methods -- 8.2.3 Multigrid Solvers -- 8.2.4 Preconditioning -- 9 Monte Carlo Simulation -- 9.1 The Principles of Monte Carlo Integration -- 9.2 Pricing Derivatives with Monte Carlo Methods -- 9.2.1 Discretizing the Stochastic Differential Equation -- 9.2.2 Pricing Formalism -- 9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model -- 9.3 An Introduction to the Libor Market Model -- 9.4 Random Number Generation -- 9.4.1 Properties of a Random Number Generator -- 9.4.2 Uniform Variates -- 9.4.3 Random Vectors -- 9.4.4 Recent Developments in Random Number Generation -- 9.4.5 Transforming Variables -- 9.4.6 Random Number Generation for Commonly Used Distributions -- 10 Advanced Monte Carlo Techniques -- 10.1 Variance Reduction Techniques -- 10.1.1 Antithetic Variates -- 10.1.2 Control Variates -- 10.1.3 Conditioning -- 10.1.4 Additional Techniques for Variance Reduction -- 10.2 Quasi Monte Carlo Method.

10.2.1 Low-Discrepancy Sequences -- 10.2.2 Randomizing QMC -- 10.3 Brownian Bridge Technique -- 10.3.1 A Steepener under a Libor Market Model -- 11 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks -- 11.1 Pricing American options using the Longstaff and Schwartz algorithm -- 11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments -- 11.2.1 Algorithm: Extended LSMC Method for Bermudan Options -- 11.2.2 Notes on Basis Functions and Regression -- 11.3 Examples -- 11.3.1 A Bermudan Callable Floater under Different Short-rate Models -- 11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model -- 11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework -- 12 Characteristic Function Methods for Option Pricing -- 12.1 Equity Models -- 12.1.1 Heston Model -- 12.1.2 Jump Diffusion Models -- 12.1.3 Infinite Activity Models -- 12.1.4 Bates Model -- 12.2 Fourier Techniques -- 12.2.1 Fast Fourier Transform Methods -- 12.2.2 Fourier-Cosine Expansion Methods -- 13 Numerical Methods for the Solution of PIDEs -- 13.1 A PIDE for Jump Models -- 13.2 Numerical Solution of the PIDE -- 13.2.1 Discretization of the Spatial Domain -- 13.2.2 Discretization of the Time Domain -- 13.2.3 A European Option under the Kou Jump Diffusion Model -- 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae -- 14 Copulas and the Pitfalls of Correlation -- 14.1 Correlation -- 14.1.1 Pearson's ρ -- 14.1.2 Spearman's ρ -- 14.1.3 Kendall's ρ -- 14.1.4 Other Measures -- 14.2 Copulas -- 14.2.1 Basic Concepts -- 14.2.2 Important Copula Functions -- 14.2.3 Parameter estimation and sampling -- 14.2.4 Default Probabilities for Credit Derivatives -- 15 Parameter Calibration and Inverse Problems -- 15.1 Implied Black-Scholes Volatilities.

15.2 Calibration Problems for Yield Curves -- 15.3 Reversion Speed and Volatility -- 15.4 Local Volatility -- 15.4.1 Dupire's Inversion Formula -- 15.4.2 Identifying Local Volatility -- 15.4.3 Results -- 15.5 Identifying Parameters in Volatility Models -- 15.5.1 Model Calibration for the FTSE-100 -- 16 Optimization Techniques -- 16.1 Model Calibration and Optimization -- 16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems -- 16.2 Heuristically Inspired Algorithms -- 16.2.1 Simulated Annealing -- 16.2.2 Differential Evolution -- 16.3 A Hybrid Algorithm for Heston Model Calibration -- 16.4 Portfolio Optimization -- 17 Risk Management -- 17.1 Value at Risk and Expected Shortfall -- 17.1.1 Parametric VaR -- 17.1.2 Historical VaR -- 17.1.3 Monte Carlo VaR -- 17.1.4 Individual and Contribution VaR -- 17.2 Principal Component Analysis -- 17.2.1 Principal Component Analysis for Non-scalar Risk Factors -- 17.2.2 Principal Components for Fast Valuation -- 17.3 Extreme Value Theory -- 18 Quantitative Finance on Parallel Architectures -- 18.1 A Short Introduction to Parallel Computing -- 18.2 Different Levels of Parallelization -- 18.3 GPU Programming -- 18.3.1 CUDA and OpenCL -- 18.3.2 Memory -- 18.4 Parallelization of Single Instrument Valuations using (Q)MC -- 18.5 Parallelization of Hybrid Calibration Algorithms -- 18.5.1 Implementation Details -- 18.5.2 Results -- 19 Building Large Software Systems for the Financial Industry -- Bibliography -- Index.