# Colloquium

## Introduction to the Pricing of Financial Options

### Prof. Richard Stockbridge

University of Wisoconsin-Milwaukee

**Friday, February 26, 2010, 2:00 PM, EMS E495A**

An option is a contract which gives the buyer the right to "exercise his option" and receive a certain payoff at time T if the value is in her/his favor, but the buyer does not have to exercise the option if it is against her/him. A typical option is a European call option which gives the option holder the right to buy 100 shares of stock XYZ for $50 per share in three months. If the actual share price is more than this, the option holder will invoke the contract to pay $50 per share and immediately sell the shares at the market price, making a profit. If the share price is below $50, the option holder will prefer not to buy the shares for the greater amount and will let the option expire.

The question of interest is "How much should the buyer pay for this contract?"

A key idea underlying the option pricing is the "no arbitrage principle" which prohibits a risk-free profit from being made. The essential idea is that a portfolio consisting of some number of shares of the stock and another number of units of a bond is formed that will be worth exactly the value of the option at its expiration time. Under the no arbitrage principle, the amount needed to form the portfolio and the price of the option must be the same.

Option pricing will be examined for two simple stochastic models that illustrate fundamental concepts in mathematical finance. The simplest model for the evolution of the stock price is the binomial model, for which closed formulas for any option price can be given. Further analysis indicates a key result in option pricing. A minor modification to the stock price process using a trinomial model leads to some dramatic differences and open questions. This talk will be accessible to undergraduates.

Refreshments will be served at 1:30pm in EMS E495B

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