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

✍ Scribed by Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 314
• Series: Springer Finance
• Edition: 2013
• Category: Library

✦ Synopsis

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