The Lognormal Distribution
MGT 4850
Spring 2008
University of Lethbridge
Binomial Option Pricing
• Computational, not analytic
• closed-form solution – solution can be
expressed analytically in terms of certain
"well-known" functions (e.g. BSOPM)
• To develop a formula we need
assumptions in this case about the
statistical properties of the underlying
stock prices.
Overview
• What constitute “reasonable” assumptions
about stock prices
• Lognormal distribution as a reasonable
distribution
• Simulation of lognormal prices
Stock Price Characteristics
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The Stock Price is uncertain
Changes are continuous
The stock price is never 0 or negative
The average return tends to increase
Uncertainty increases with time
Stock Price Paths
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Wiggly lines
Lines are continuous solid with no jumps
Lines are positive
Average increases with time
Standard deviation increases with time
examples
Definition
• the log-normal distribution is the
probability distribution of any random
variable whose logarithm is normally
distributed. If X is a random variable with a
normal distribution, then exp(X) or e X has
a log-normal distribution; likewise, if Y is
log-normally distributed, then log(Y) is
normally distributed.
Lognormal Distribution
• probability density function (pdf)
Lognormal Distribution
lognormal
• The expected value is
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• and the variance is
–
Lognormal distribution
Normal distribution pdf
Random number Generation
Simulating lognormal prices
• Requires VBA skills (optional)
• Also skip 18.3 Geometric diffusions
• Calculating the parameters of the
lognormal distribution
Lognormal mean and sigma