A Workout in Computational Finance.

By: Binder, AndreasContributor(s): Aichinger, MichaelSeries: The Wiley Finance SerPublisher: Somerset : John Wiley & Sons, Incorporated, 2013Copyright date: ©2013Edition: 1st edDescription: 1 online resource (338 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781119973485Subject(s): Finance -- Mathematical modelsGenre/Form: Electronic books. Additional physical formats: Print version:: A Workout in Computational FinanceDDC classification: 332.015195 LOC classification: HG106 -- .A387 2013ebOnline resources: Click to View
Contents:
Intro -- 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.
Summary: 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.
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Intro -- 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.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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