Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing (Springer Finance)

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

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 301
• 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

Computational Methods for Quantitative Finance
Preface
Contents
Part I: Basic Techniques and Models
Chapter 1: Notions of Mathematical Finance
1.1 Financial Modelling
Stocks
Price Process
Derivative Securities
Options
Payoff
Modelling Assumptions
1.2 Stochastic Processes
1.3 Further Reading
Chapter 2: Elements of Numerical Methods for PDEs
2.1 Function Spaces
2.2 Partial Differential Equations
2.3 Numerical Methods for the Heat Equation
2.3.1 Finite Difference Method
2.3.2 Convergence of the Finite Difference Method
2.3.3 Finite Element Method
2.4 Further Reading
Chapter 3: 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
3.4.1 Elemental Forms and Assembly
3.4.2 Initial Data
3.5 Stability of the theta-Scheme
3.6 Error Estimates
3.6.1 Finite Element Interpolation
3.6.2 Convergence of the Finite Element Method
3.7 Further Reading
Chapter 4: European Options in BS Markets
4.1 Black-Scholes Equation
4.2 Variational Formulation
4.3 Localization
4.4 Discretization
4.4.1 Finite Difference Discretization
4.4.2 Finite Element Discretization
4.4.3 Non-smooth Initial Data
4.5 Extensions of the Black-Scholes Model
4.5.1 CEV Model
4.5.2 Local Volatility Models
4.6 Further Reading
Chapter 5: American Options
5.1 Optimal Stopping Problem
5.2 Variational Formulation
5.3 Discretization
5.3.1 Finite Difference Discretization
5.3.2 Finite Element Discretization
5.4 Numerical Solution of Linear Complementarity Problems
5.4.1 Projected Successive Overrelaxation Method
5.4.2 Primal-Dual Active Set Algorithm
5.5 Further Reading
Chapter 6: Exotic Options
6.1 Barrier Options
6.2 Asian Options
6.3 Compound Options
6.4 Swing Options
6.5 Further Reading
Chapter 7: Interest Rate Models
7.1 Pricing Equation
7.2 Interest Rate Derivatives
7.3 Further Reading
Chapter 8: Multi-asset Options
8.1 Pricing Equation
8.2 Variational Formulation
8.3 Localization
8.4 Discretization
8.4.1 Finite Difference Discretization
8.4.2 Finite Element Discretization
8.5 Further Reading
Chapter 9: Stochastic Volatility Models
9.1 Market Models
9.1.1 Heston Model
9.1.2 Multi-scale Model
9.2 Pricing Equation
9.3 Variational Formulation
9.4 Localization
9.5 Discretization
9.5.1 Finite Difference Discretization
9.5.2 Finite Element Discretization
9.6 American Options
9.7 Further Reading
Chapter 10: Lévy Models
10.1 Lévy Processes
10.2 Lévy Models
10.2.1 Jump-Diffusion Models
10.2.2 Pure Jump Models
10.2.3 Admissible Market Models
10.3 Pricing Equation
10.4 Variational Formulation
10.5 Localization
10.6 Discretization
10.6.1 Finite Difference Discretization
10.6.2 Finite Element Discretization
10.7 American Options Under Exponential Lévy Models
10.8 Further Reading
Chapter 11: Sensitivities and Greeks
11.1 Option Pricing
11.2 Sensitivity Analysis
11.2.1 Sensitivity with Respect to Model Parameters
11.2.2 Sensitivity with Respect to Solution Arguments
11.3 Numerical Examples
11.3.1 One-Dimensional Models
11.3.2 Multivariate Models
11.4 Further Reading
Part II: Advanced Techniques and Models
Chapter 12: Wavelet Methods
12.1 Spline Wavelets
12.1.1 Wavelet Transformation
12.1.2 Norm Equivalences
12.2 Wavelet Discretization
12.2.1 Space Discretization
12.2.2 Matrix Compression
12.2.3 Multilevel Preconditioning
12.3 Discontinuous Galerkin Time Discretization
12.3.1 Derivation of the Linear Systems
12.3.2 Solution Algorithm
12.4 Further Reading
Chapter 13: 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
13.4.1 Aggregated Black-Scholes Models
13.4.2 Stochastic Volatility Models
13.5 Numerical Examples
13.5.1 Full-Rank d-Dimensional Black-Scholes Model
13.5.2 Low-Rank d-Dimensional Black-Scholes
13.6 Further Reading
Chapter 14: Multidimensional Lévy Models
14.1 Lévy Processes
14.2 Lévy Copulas
14.3 Lévy Models
14.3.1 Subordinated Brownian Motion
14.3.2 Lévy Copula Models
14.3.3 Admissible Models
14.4 Pricing Equation
14.5 Variational Formulation
14.6 Wavelet Discretization
14.6.1 Wavelet Compression
14.6.2 Fully Discrete Scheme
14.7 Application: Impact of Approximations of Small Jumps
14.7.1 Gaussian Approximation
14.7.2 Basket Options
14.7.3 Barrier Options
14.8 Further Reading
Chapter 15: Stochastic Volatility Models with Jumps
15.1 Market Models
15.1.1 Bates Models
15.1.2 BNS Model
15.2 Pricing Equations
15.3 Variational Formulation
15.4 Wavelet Discretization
15.5 Further Reading
Chapter 16: 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
16.5.1 Sector Condition
16.5.2 Well-Posedness
16.6 Numerical Examples
16.7 Further Reading
Appendix A: Elliptic Variational Inequalities
A.1 Hilbert Spaces
A.2 Dual of a Hilbert Space
A.3 Theorems of Stampacchia and Lax-Milgram
Appendix B: Parabolic Variational Inequalities
B.1 Weak Formulation of PVI's
B.2 Existence
B.3 Proof of the Existence Result
References
Index