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:

**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

And the ratio of tomorrow’s stock price divided by today’s stock price (ST/S0) is also normally distributed.

In words, the natural log of the wealth ratio (ST/S0) has an expected mean that

- The stock price and the wealth ratio (ST/S0) are
distributed**lognormally** - The
**natural log of**the stock price [ln(ST)] and the natural log of the wealth ratio [ln(ST/S0)] aredistributed**normally**

There are two parameters in the normal distribution above: expected value (mean) and standard deviation. The standard deviation

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