Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

Probability Distributions 67

FIGURE 2.14 The real shape of the Normal Distribution

presented in this chapter imply. The real shape of the Normal Distribution

is depicted in Figure 2.14.

Notice that here the scales of the two axes are the same, whereas in

the other graphic examples they differ so as to exaggerate the shape of the

distribution.

The Lognormal Distribution

Many of the real-world applications in trading require a small but crucial

modification to the Normal Distribution. This modification takes the Nor-

mal, and changes it to what is known as the Lognormal Distribution.

Consider that the price of any freely traded item has zero as a lower

limit.^2 Therefore, as the price of an item drops and approaches zero, it

(^2) This idea that the lowest an item can trade for is zero is not always entirely true.
For instance, during the stock market crash of 1929 and the ensuing bear market,
the shareholders of many failed banks were held liable to the depositors in those
banks. Persons who owned stock in such banks not only lost their full investment,
they also realized liabilitybeyondthe amount of their investment. The point here
isn’t to say that such an event can or cannot happen again. Rather, we cannot al-
ways say that zero is the absolute low end of what a freely traded item can be priced
at, although it usually is.