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### FRM 2006-Lognormal Property of Stock Prices

The Black-Scholes option pricing model assumes that a stock moves ina Geometric Brownian motion (GBM). Under GBM, the natural log of thestock price at future time T (ST) is normally distributed: And the ratio of tomorrow’s stock price divided by today’s stock price (ST/S0) is also normally distributed. This formula is important: In words, the natural log of the wealth ratio (ST/S0) has an expected mean that almost reachesthe expected return scaled (multiplied) by time (T) and a standarddeviation scaled (multiplied) by the square root of time. Remember thelognormal/normal distinction:
• The stock price and the wealth ratio (ST/S0) are lognormally distributed
• The natural log of the stock price [ln(ST)] and the natural log of the wealth ratio [ln(ST/S0)] are normally distributed
Ifa random variable is lognormally distributed (e.g., the stock price),that’s another way of saying the lognormal of the variable (e.g., thenatural log of the stock price) is normally distributed. Notice thatstock returns are normally distributed but stock prices are lognormally distributed.A periodic stock return can go up or down (e.g., plus or minus 10%) buta stock price cannot fall below zero, nor can the wealth ratio (ST/S0) fall below zero. As such, the stock price/level is captured by a nonzero distribution: There are two parameters in the normal distribution above: expected value (mean) and standard deviation. The standard deviation scales with the square root of time. You know this one by now: volatility scales with the square root of time. The mean looks curious: the expected mean is depressed by one-half the variance. The idea is that volatility depresses returns!
Considersix period returns as below. The geometric average (a.k.a., compoundannual growth rate) of this brief series is 105.4% and the arithmeticaverage is 105.8%: The standard deviation of the population [=STDEVP()] is 8.9% so that the variance is 0.8% [8.9%2].The geometric average (CAGR) is always lower than the arithmeticaverage (unless the series has zero volatility). Notice that