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
5 - Pricing Contracts via Arbitrage
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
An Example in Options Pricing
Suppose that the nominal interest rate is r, and consider the following model for pricing an option to purchase a stock at a future time at a fixed price. Let the present price (in dollars) of the stock be 100 per share, and suppose we know that, after one time period, its price will be either 200 or 50 (see Figure 5.1). Suppose further that, for any y, at a cost of Cy you can purchase at time 0 the option to buy y shares of the stock at time 1 at a price of 150 per share. Thus, for instance, if you purchase this option and the stock rises to 200, then you would exercise the option at time 1 and realize a gain of 200 − 150 = 50 for each of the y options purchased. On the other hand, if the price of the stock at time 1 is 50 then the option would be worthless. In addition to the options, you may also purchase x shares of the stock at time 0 at a cost of 100 x, and each share would be worth either 200 or 50 at time 1.
We will suppose that both x and y can be positive, negative, or zero. That is, you can either buy or sell both the stock and the option.
- Type
- Chapter
- Information
- An Elementary Introduction to Mathematical FinanceOptions and other Topics, pp. 63 - 80Publisher: Cambridge University PressPrint publication year: 2002