An Analysis of Option Pricing Models Charles Vu 2002-2003
Poster Outline
Title: An Analysis of Option Pricing Models

Abstract: The stock options market is a trillion d ollar industry and many models exist which attmept to price stock derivates. The Nobel-prize winning Black-Scholes model is the benchmark of option pricing models. However, whether or not the Black-Scholes model accurately predicts the intrinisic value of an option is still up to debate. Other widely accepted models exist, including the Cox, Ross & Rubenstein binomial model, Black's approximation, and the Barone-Adesi and Whaley quadratic approximation.

Introduction: The premise of a stock option is that someone sells the right to buy their shares of a particular stock at a certain price by a certain date. For example, if Microsoft (MSFT) is at $50 on October 1st, the price of the option to purchase a share of MSFT for $50 at anytime before October 15th might cost me $2. If MSFT is still at $50 on October 15, I will lose my $2. However, if MSFT is at $60 on October 15th, I can still purchase the shares at $50, and therefore make a 10-2 = $8 profit, a good profit since I only put $2 at risk. Further, if MSFT goes down to $40, or lower, I still only lose the $2 I spent for the option. If I had simply bought shares of MSFT, I would have been down $10. But the question remains, is the option I bought worth $2, or is it worth more than $2, or less? Many models have been proposed to answer this question.

Popular Pricing Models: The Black-Scholes Model was formulated in 1973 by Fischer Black and Myron Scholes. It won the Nobel Prize in economics in 1997. This is the premier option pricing model that most institutions use in pricing stock derivatives. In 1976 Fischer Black made some minor modifications to the Black Scholes model to adapt it for use in evaluating options on futures contracts. The Cox, Ross and Rubinstein model was developed using a similar approach to the Black-Scholes model, but assumes the stock moves in a binomial distribution. In 1987 Giovanni Barone-Adesi and Robert Whaley published a quadratic approximation method of valuing American options. Their Quadratic Approximation Model has become one of the more popular pricing algorithms.

Cox, Ross & Rubenstein Binomial Tree Model The binomial model breaks down the time to expiration into a number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price will either move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option. At the end of the tree -- at expiration of the option -- all the terminal option prices for each of the final possible stock prices are their intrinsic values -- stock price - strike price. One then calculates recursively the underlying option price, calculating the value of each parent node from its children. This process is continued until one reaches the root node -- and the value of the option.

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