Book Description and Chapter Summary
Finite Difference Methods in Financial Instrument Pricing - A Partial Differential Equation Approach
By Daniel J. Duffy, Datasim Component Technology BV Amsterdam
dduffy@datasim.nl
Plan: in progress, manuscript to be completed July 2004
© Datasim Component Technology BV
This book is a self-contained introduction on how to apply the Finite Difference Method (FDM) to approximate the partial differential equations that describe the Black Scholes model. The book is partitioned into seven parts where each part deals with a specific issue pertaining to Financial Engineering. Theory and practice are interleaved so that the book remains relevant and interesting for financial engineers and quantitative analysts. We discuss state-of-the art finite difference schemes that approximate one-factor and multi-factor models.
This section gives a thorough discussion of the mathematical foundations for those partial differential equations that describe the behaviour of financial derivatives. In particular, we examine second-order parabolic partial differential equations for the one-factor and multi-factor Black Scholes equation.
Chapter 1: PDE Classification
Chapter 2: Second-order Parabolic Equations
Chapter 3: The Convection-Diffusion Equation in one Dimension
Chapter 4: Multi-dimensional Parabolic Partial Differential Equations
Chapter 5: First-order Hyperbolic Equations
In this section we give a complete and self-contained introduction to the finite difference method for both ordinary and partial differential equations. The focus is on equations in two independent variables. We consider diffusion, convection and convection-diffusion equations and we approximate them using a number of one-step explicit and implicit difference schemes. This section lays the foundations for a study of FDM for the one-factor Black Scholes equation in Part III.
Chapter 6: Introduction to the Finite Difference Method
Chapter 7:
Finite Differences for the Heat Equation
Chapter 8: General Theory of
the Finite Difference Method
Chapter 9: Finite Difference Schemes for
First-Order Equations
Chapter 10: FDM for Convection-Diffusion Equations
Chapter 11: Monotone and Exponentially Fitted Schemes
In this section we focus exclusively on finite difference schemes for the one-factor Black Scholes equation. We consider most of main contenders that use three-point difference schemes in the underlying variable (for example, stock S or interest rate r) and explicit, implicit and Crank Nicolson in the time direction. We also pay special attention to devising accurate and non-oscillatory schemes that approximate option sensitivities.
Chapter 12: Benchmarking Traditional FDM with Exact Solutions
Chapter
13: A Critique of the Crank Nicolson Method
Chapter 14: Introduction to
Barrier Options with FDM
Chapter 15: Barrier Options with Discrete,
Exponential and Partial Barriers
Chapter 16: One-factor Bond
Approximation with Finite Differences
Chapter 17: An Introduction to FDM
for Real Options
We now discuss a number of finite difference schemes for partial differential equations in three independent variables. The main categories are:
We introduce the most important features of each method and we discuss their usefulness in practical applications. The ADI method is popular in the financial literature but we shall show that operator splitting is easier to apply and in some cases has better properties than ADI. We also pay attention to the approximation of cross (mixed) derivatives.
This section lays the foundations for a study of FDM for multi-factor Black Scholes equations in Part V.
Chapter 18: Direct Finite Difference Methods
Chapter 19: Alternating
Direction Implicit (ADI) Methods
Chapter 20: An Introduction to Operator Splitting Methods
Chapter 21: Advanced Splitting Methods
This section applies ADI and operator splitting methods to a number of important problems in financial engineering. In particular, we discuss FDM for several kinds of path-dependent options, bonds and other derivatives.
Chapter 22: Options with Stochastic Volatility
Chapter 23: Asian
Options
Chapter 24: Rainbow Options and Options with multiple Underlyings
Chapter 25: Two-factor Bonds and Finite Differences
In this section we show how to solve option-pricing problems that contain an early exercise feature. In particular, we examine how to solve American option problems. To this end, we must realise that these are free-boundary value problems and there is a wealth of information on this topic in the engineering and numerical analysis literature. We discuss the problem of finding the curve or surface that represent the boundary between exercise and no exercise and we introduce several approximation techniques that realise this goal.
Chapter 26: Background to Free Boundary Value Problems
Chapter 27: An
Overview of Numerical Methods
Chapter 28: Front-Fixing Methods and
American Style Options
Chapter 29: Penalty Methods and American Style
Options
Chapter 30: Free Boundaries and Convertible Bonds
The chapters in this section complement the material in the previous 6 sections. In some cases they can be used in their own right to approximate the PDEs that describe Black Scholes.
Chapter 31: Analytical Solutions of the one-dimensional Heat Equation
Chapter 32: The Method of Characteristics (MOC)
Chapter 33: The Method
of Lines (MOL)
Chapter 34: Numerical Approximation of Stochastic
Differential Equations (SDE) Chapter 35: An Introduction to the Finite
Element Method (FEM)
Chapter 36: An Introduction to Meshfree (Meshless)
Methods
Chapter 37: Parabolic Variational Inequalities for Free Boundary
Value Problems