Cox, Ross & Rubenstein (1979) Option Price Theory Option price is the expected discounted value of the cash flows from an option on a stock having the same variance as the stock on which the option is written and growing at the risk-free rate of interest. The cash flows are discounted continuously at the risk-free rate The price does not depend on the growth rate of the stock!

Modeling the Price of a Stock Most financial models of stock prices assume that the stock’s price follows a lognormal distribution. (The logarithm of the stock’s price is normally distributed) This implies the following relationship: P t = P 0 * exp[(μ-.5*σ 2 )*t + σ*Z*t.5 ]

Notation Definition –P 0 = Current price of stock –t = Number of years in future –P t = Price of stock at time t  Random Variable!! –Z = A standard normal random variable with mean 0 and standard deviation 1  Random Variable!! –μ = Mean percentage growth rate of stock per year expressed as a decimal –σ = Standard deviation of the growth rate of stock per year expressed as a decimal. Also referred to as the annual volatility.

Option Pricing Simulation Logic Simulate the stock price t years from now assuming that it grows at the risk-free rate r f. This implies the following relationship: P t = P 0 * exp[(r f -.5*σ 2 )*t + σ*Z*t.5 ] Compute the cash flows from the option at expiration t years from now. Discount the cash flow value back to time 0 by multiplying by e -rt to calculate the current value of the option. Select the current value of the option as the output variable and perform many iterations to quantify the expected value and distribution for the option.

Asian Options An option whose payoff depends in some way on the average price of the underlying asset over a period of time prior to option expiration To compute the value of these type of options, you must be able to compute possible price paths of the underlying asset

Example of an Asian Option –The option payoff is based on difference between the average price of the underlying asset and the strike price –Value at expiration = Max(Average underlying asset price – Strike Price, 0) –See example on AsianCallOption worksheet