Book contents
- Frontmatter
- Contents
- Preface
- Part One Basic Option Theory
- 1 An Introduction to Options and Markets
- 2 Asset Price Random Walks
- 3 The Black–Scholes Model
- 4 Partial Differential Equations
- 5 The Black–Scholes Formulæ
- 6 Variations on the Black–Scholes Model
- 7 American Options
- Part Two Numerical Methods
- Part Three Further Option Theory
- Part Four Interest Rate Derivative Products
- Hints to Selected Exercises
- Bibliography
- Index
4 - Partial Differential Equations
from Part One - Basic Option Theory
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Part One Basic Option Theory
- 1 An Introduction to Options and Markets
- 2 Asset Price Random Walks
- 3 The Black–Scholes Model
- 4 Partial Differential Equations
- 5 The Black–Scholes Formulæ
- 6 Variations on the Black–Scholes Model
- 7 American Options
- Part Two Numerical Methods
- Part Three Further Option Theory
- Part Four Interest Rate Derivative Products
- Hints to Selected Exercises
- Bibliography
- Index
Summary
Introduction
The modelling of Chapter 3 culminates in the formulation of the pricing problem for a derivative product as a partial differential equation. We now take a break from the financial modelling to discuss, in this and the next chapter, some of the theory behind such differential equations. In this chapter we describe the elementary theory and the nature of boundary and initial conditions. In Chapter 5 we derive some explicit solutions, including the original Black–Scholes formulæ. Later, in Chapter 7, we describe in detail the special problems arising when there are free boundaries. This chapter is of particular importance when considering the valuation of American options.
The study of partial differential equations in complete generality is a vast undertaking. Fortunately, however, almost all the partial differential equations encountered in financial applications belong to a much more manageable subset of the whole: second order linear parabolic equations. These technical terms are discussed below; more detailed treatments of the areas beyond the scope of this text are given in some of the references at the end of the chapter.
We begin this chapter with a review of second order linear parabolic equations: their physical interpretation, mathematical properties of their solutions, and techniques for obtaining explicit solutions to specific problems. Then, we exploit this knowledge in the context of financial models to derive explicit solutions to some option valuation problems, and we set the scene for the numerical methods of Chapters 8 and 9.
- Type
- Chapter
- Information
- The Mathematics of Financial DerivativesA Student Introduction, pp. 58 - 70Publisher: Cambridge University PressPrint publication year: 1995
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