Why Stock Prices Have a Lognormal Distribution

Because the future price of a stock at time t cannot be predicted with certainty, we model it as a random variable, denoted here by S(t). Since random variables are characterised by their distribution functions it is useful to have a notation to express this concept. Definition 1.1 We use the symbol X D = Y to mean that the random variables X and Y have the same distribution,i.e., P (X ≤ t) = P (Y ≤ t), −∞ < t < ∞ Of the many distributions that the reader is likely to encounter in finance it is the normal and log normal distributions defined below, that are the most important. Definition 1.2 A random variable X with distribution function given by