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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.
• What constitute “reasonable” assumptions
about stock prices
• Lognormal distribution as a reasonable
• Simulation of lognormal prices
Stock Price Characteristics
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
Wiggly lines
Lines are continuous solid with no jumps
Lines are positive
Average increases with time
Standard deviation increases with time
• 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
• The expected value is
• and the variance is
Lognormal distribution
Normal distribution pdf
Random number Generation
Simulating lognormal prices
• Requires VBA skills (optional)
• Also skip 15.3 Geometric diffuaions
• Calculating the parameters of the
lognormal distribution
Lognormal mean and sigma p.294