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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. 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) Lognormal Distribution lognormal • 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