Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing

Many mathematical assumptions on which classical derivative pricing methods are based have come under scrutiny in recent years. The present volume offers an introduction to deterministic algorithms for the fast and accurate pricing of derivative contracts in modern finance. This unified, non-Monte-Carlo computational pricing methodology is capable of handling rather general classes of stochastic market models with jumps, including, in particular, all currently used Lévy and stochastic volatility models. It allows us e.g. to quantify model risk in computed prices on plain vanilla, as well as on various types of exotic contracts. The algorithms are developed in classical Black-Scholes markets, and then extended to market models based on multiscale stochastic volatility, to Lévy, additive and certain classes of Feller processes. This book is intended for graduate students and researchers, as well as for practitioners in the fields of quantitative finance and applied and computational mathematics with a solid background in mathematics, statistics or economics.​ Table of Contents Cover Computational Methods for Quantitative Finance - Finite Element Methods for Derivative Pricing ISBN 9783642354007 ISBN 9783642354014 Preface Contents Part I Basic Techniques and Models Notions of Mathematical Finance 1.1 Financial Modelling 1.2 Stochastic Processes 1.3 Further Reading Elements of Numerical Methods for PDEs 2.1 Function Spaces 2.2 Partial Differential Equations 2.3 Numerical Methods for the Heat Equation o 2.3.1 Finite Difference Method o 2.3.2 Convergence of the Finite Difference Method o 2.3.3 Finite Element Method 2.4 Further Reading Finite Element Methods for Parabolic Problems 3.1 Sobolev Spaces 3.2 Variational Parabolic Framework 3.3 Discretization 3.4 Implementation of the Matrix Form o 3.4.1 Elemental Forms and Assembly o 3.4.2 Initial Data 3.5 Stability of the .-Scheme 3.6 Error Estimates o 3.6.1 Finite Element Interpolation o 3.6.2 Convergence of the Finite Element Method 3.7 Further Reading European Options in BS Markets 4.1 Black-Scholes Equation 4.2 Variational Formulation 4.3 Localization 4.4 Discretization o 4.4.1 Finite Difference Discretization o 4.4.2 Finite Element Discretization o 4.4.3 Non-smooth Initial Data 4.5 Extensions of the Black-Scholes Model o 4.5.1 CEV Model o 4.5.2 Local Volatility Models 4.6 Further Reading American Options 5.1 Optimal Stopping Problem 5.2 Variational Formulation 5.3 Discretization o 5.3.1 Finite Difference Discretization o 5.3.2 Finite Element Discretization 5.4 Numerical Solution of Linear Complementarity Problems o 5.4.1 Projected Successive Overrelaxation Method o 5.4.2 Primal-Dual Active Set Algorithm 5.5 Further Reading Exotic Options 6.1 Barrier Options 6.2 Asian Options 6.3 Compound Options 6.4 Swing Options 6.5 Further Reading Interest Rate Models 7.1 Pricing Equation 7.2 Interest Rate Derivatives 7.3 Further Reading Multi-asset Options 8.1 Pricing Equation 8.2 Variational Formulation 8.3 Localization 8.4 Discretization o 8.4.1 Finite Difference Discretization o 8.4.2 Finite Element Discretization 8.5 Further Reading Stochastic Volatility Models 9.1 Market Models o 9.1.1 Heston Model o 9.1.2 Multi-scale Model 9.2 Pricing Equation 9.3 Variational Formulation 9.4 Localization 9.5 Discretization o 9.5.1 Finite Difference Discretization o 9.5.2 Finite Element Discretization 9.6 American Options 9.7 Further Reading L�vy Models 10.1 L�vy Processes 10.2 L�vy Models o 10.2.1 Jump-Diffusion Models o 10.2.2 Pure Jump Models o 10.2.3 Admissible Market Models 10.3 Pricing Equation 10.4 Variational Formulation 10.5 Localization 10.6 Discretization o 10.6.1 Finite Difference Discretization o 10.6.2 Finite Element Discretization 10.7 American Options Under Exponential L�vy Models 10.8 Further Reading Sensitivities and Greeks 11.1 Option Pricing 11.2 Sensitivity Analysis o 11.2.1 Sensitivity with Respect to Model Parameters o 11.2.2 Sensitivity with Respect to Solution Arguments 11.3 Numerical Examples o 11.3.1 One-Dimensional Models o 11.3.2 Multivariate Models 11.4 Further Reading Wavelet Methods 12.1 Spline Wavelets o 12.1.1 Wavelet Transformation o 12.1.2 Norm Equivalences 12.2 Wavelet Discretization o 12.2.1 Space Discretization o 12.2.2 Matrix Compression o 12.2.3 Multilevel Preconditioning 12.3 Discontinuous Galerkin Time Discretization o 12.3.1 Derivation of the Linear Systems o 12.3.2 Solution Algorithm 12.4 Further Reading Part II Advanced Techniques and Models Multidimensional Diffusion Models 13.1 Sparse Tensor Product Finite Element Spaces 13.2 Sparse Wavelet Discretization 13.3 Fully Discrete Scheme 13.4 Diffusion Models o 13.4.1 Aggregated Black-Scholes Models o 13.4.2 Stochastic Volatility Models 13.5 Numerical Examples o 13.5.1 Full-Rank d-Dimensional Black-Scholes Model o 13.5.2 Low-Rank d-Dimensional Black-Scholes 13.6 Further Reading Multidimensional L�vy Models 14.1 L�vy Processes 14.2 L�vy Copulas 14.3 L�vy Models o 14.3.1 Subordinated Brownian Motion o 14.3.2 L�vy Copula Models o 14.3.3 Admissible Models 14.4 Pricing Equation 14.5 Variational Formulation 14.6 Wavelet Discretization o 14.6.1 Wavelet Compression o 14.6.2 Fully Discrete Scheme 14.7 Application: Impact of Approximations of Small Jumps o 14.7.1 Gaussian Approximation o 14.7.2 Basket Options o 14.7.3 Barrier Options 14.8 Further Reading Stochastic Volatility Models with Jumps 15.1 Market Models o 15.1.1 Bates Models o 15.1.2 BNS Model 15.2 Pricing Equations 15.3 Variational Formulation 15.4 Wavelet Discretization 15.5 Further Reading Multidimensional Feller Processes 16.1 Pseudodifferential Operators 16.2 Variable Order Sobolev Spaces 16.3 Subordination 16.4 Admissible Market Models 16.5 Variational Formulation o 16.5.1 Sector Condition o 16.5.2 Well-Posedness 16.6 Numerical Examples 16.7 Further Reading Elliptic Variational Inequalities Parabolic Variational Inequalities Index