The most comprehensive educational resources for finance

Why Lognormal Distribution is Used to Describe Stock Prices

The concept of lognormal distribution is very closely related to the concept of normal distribution.

Let’s say we have a random variable Y. This variable Y will have a lognormal distribution if the natural log of Y (ln Y) is normally distributed. So, we check if the natural logarithm of a random variable is normally distributed or not. If it is, then the random variable itself will have a lognormal distribution.

A lognormal distribution has two important characteristics:

  • It has a lower bound of zero.
  • The distribution is skewed to the right, i.e., it has a long right tail.

Note that this is in contrast with a normal distribution which has zero skew and can take both negative and positive values.

Just like a normal distribution, a lognormal distribution is also described by just two parameters, namely, m and s.

A lognormal distribution is commonly used to describe distributions of financial assets such as share prices. A lognormal distribution is more suitable for this purpose because asset prices cannot be negative. An important point to note is that when the continuously compounded returns of a stock follow normal distribution, then the stock prices follow a lognormal distribution. Even in cases where returns do not follow a normal distribution, stock prices are better described by a lognormal distribution.

Consider the expression Y = exp(X).

Exp(X) or ex is the opposite of taking logs. If we take log on both side, we will have ln y = X

So, if we assume that X has normal distribution, then Y has lognormal distribution (because ln Y is normally distributed).

We can compare this with how stock prices move. Let’s say that the initial stock price is S0 and the stock price after period t is St. If the rate of return r is continuously compounded then the future stock price can be expressed as:

St = S0*EXP(r)

S0 is a known quantity and is a constant. This expression is the same as Y = exp(X).

Therefore, if r is normally distributed, the stock price will be lognormally distributed.

Leave a Reply

Your email address will not be published. Required fields are marked *

Name *