Lognormal Distribution and 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.

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