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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: Pt = P0 * exp[(μ-.5*σ2)*t + σ*Z*t.5] Notation Definition – – – – P0 = Current price of stock t = Number of years in future Pt = 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 rf. This implies the following relationship: Pt = P0 * exp[(rf-.5*σ2)*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