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
8 - Additional Results on Options
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 look at some extensions of the basic call option model. In Section 8.2 we consider European call options on dividend-paying securities under three different scenarios for how the dividend is paid. In Section 8.2.1 we suppose that the dividend for each share owned is paid continuously in time at a rate equal to a fixed fraction of the price of the security. In Sections 8.2.2 and 8.2.3 we suppose that the dividend is to be paid at a specified time, with the amount paid equal to a fixed fraction of the price of the security (Section 8.2.2) or to a fixed amount (Section 8.2.3). In Section 8.3 we show how to determine the no-arbitrage price of an American put option. In Section 8.4 we introduce a model that allows for the possibilities of jumps in the price of a security. This model supposes that the security's price changes according to a geometric Brownian motion, with the exception that at random times the price is assumed to change by a random multiplicative factor. In Section 8.4.1 we derive an exact formula for the no-arbitrage cost of a call option when the multiplicative jumps have a lognormal probability distribution. In Section 8.4.2 we suppose that the multiplicative jumps have an arbitrary probability distribution; we show that the no-arbitrage cost is always at least as large as the Black–Scholes formula when there are no jumps, and we then present an approximation for the no-arbitrage cost.
- Type
- Chapter
- Information
- An Elementary Introduction to Mathematical FinanceOptions and other Topics, pp. 118 - 151Publisher: Cambridge University PressPrint publication year: 2002