Transcript Risk Analysis - Loyola Marymount University

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