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