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 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

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)probability density function

Lognormal Distribution

lognormal The expected value isexpected value – and the variance isvariance –

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