Book contents
- Frontmatter
- Contents
- Introduction and Preface
- 1 Probability
- 2 Normal Random Variables
- 3 Geometric Brownian Motion
- 4 Interest Rates and Present Value Analysis
- 5 Pricing Contracts via Arbitrage
- 6 The Arbitrage Theorem
- 7 The Black–Scholes Formula
- 8 Additional Results on Options
- 9 Valuing by Expected Utility
- 10 Optimization Models
- 11 Exotic Options
- 12 Beyond Geometric Brownian Motion Models
- 13 Autogressive Models and Mean Reversion
- Index
7 - The Black–Scholes Formula
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Introduction and Preface
- 1 Probability
- 2 Normal Random Variables
- 3 Geometric Brownian Motion
- 4 Interest Rates and Present Value Analysis
- 5 Pricing Contracts via Arbitrage
- 6 The Arbitrage Theorem
- 7 The Black–Scholes Formula
- 8 Additional Results on Options
- 9 Valuing by Expected Utility
- 10 Optimization Models
- 11 Exotic Options
- 12 Beyond Geometric Brownian Motion Models
- 13 Autogressive Models and Mean Reversion
- Index
Summary
Introduction
In this chapter we derive the celebrated Black–Scholes formula, which gives – under the assumption that the price of a security evolves according to a geometric Brownian motion – the unique no-arbitrage cost of a call option on this security. Section 7.2 gives the derivation of the no-arbitrage cost, which is a function of five variables, and Section 7.3 discusses some of the properties of this function. Section 7.4 gives the strategy that can, in theory, be used to obtain an arbitrage when the cost of the security is not as specified by the formula. Section 7.5, which is more theoretical than other sections of the text, presents simplified derivations of (1) the computational form of the Black–Scholes formula and (2) the partial derivatives of the no-arbitrage cost with respect to each of its five parameters.
The Black–Scholes Formula
Consider a call option having strike price K and expiration time t. That is, the option allows one to purchase a single unit of an underlying security at time t for the price K. Suppose further that the nominal interest rate is r, compounded continuously, and also that the price of the security follows a geometric Brownian motion with drift parameter μ and volatility parameter σ. Under these assumptions, we will find the unique cost of the option that does not give rise to an arbitrage.
To begin, let S(y) denote the price of the security at time y.
- Type
- Chapter
- Information
- An Elementary Introduction to Mathematical FinanceOptions and other Topics, pp. 95 - 117Publisher: Cambridge University PressPrint publication year: 2002