## Real Stock Distributions

At this point it is natural to ask how well this theoretical model fits actual stock price behavior . Are real stock prices lognormal?

The answer is that, based on an analysis of past stock price records, the price distributions of most stocks are actually quite close to lognormal. To verify this, we select a nominal period length of, say, 1 week and record the differences In S{k 4-1) -In S(k) for many values of k; that is, we record the weekly changes in the logarithm of the prices for many weeks. We then construct a histogram of these values and compare it with that of a normal distribution of the same variance Typically, the measured distribution is quite close to being normal, except that the observed distribution often is slightly smaller near the mean and larger at extremely large values (either positive or negative large values), This slight change in shape is picturesquely termed fat tails. (See Figure 11 3 2) The observed distribution is larger in the tails than a normal

35 j- Number of samples 30

FIGURE 11.3 Observed distribution of the logarithm of return. The distribution has "fatter tails" than a normal distribution of the same variance

2The figure shows a histogram of American Airlines weekly log stock returns for the JO-year period of !982-i992 Shown superimposed is the norma) distribution with the same (sample) mean and standard deviation Along with fat tails there is invariably a "skinny middle "

distribution. This implies that large price changes tend to occur somewhat more frequently than would be predicted by a normal distribution of the same variance, For most applications (but not all) this slight discrepancy is not important

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