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Relationship between Normal Distribution and Lognormal Distribution

Relationship between Normal Distribution and Lognormal Distribution

A variable \(X\) is said to have a lognormal distribution if \(Y = ln(X)\) is normally distributed, where “ln” denotes the natural logarithm. In other words, when the logarithms of values form a normal distribution, we say that the original values have a lognormal distribution.

Let’s consider this:

$$Y=e^X$$

 Where \(e\) is the exponential constant.

If we take natural logs on both sides, \(lnY = ln\ e^X\) which leads us to \(lnY = X\). Therefore, if \(X\) has a normal distribution, then \(Y\) has a lognormal distribution.

The lognormal distribution is positively skewed with many small values and just a few large values. Consequently, the mean is greater than the mode in most cases.

the-lognormal-distribution

Why the Lognormal Distribution is used to Model Stock Prices

Since the lognormal distribution is bound by zero on the lower side, it is perfect for modeling asset prices that cannot take negative values. On the other hand, the normal distribution cannot be used for the same purpose because it has a negative side.

When the returns on a stock (continuously compounded) follow a normal distribution, the stock prices follow a lognormal distribution. Note that even if returns do not follow a normal distribution, the lognormal distribution is still the most appropriate for stock prices.

The probability density function of the distribution is:

$$ f\left( x \right) =\frac { 1 }{ x\sqrt { 2\pi { \sigma }^{ 2 } } } { e }^{ -\frac { { \left( lnx-\mu \right) }^{ 2 } }{ \sqrt { 2{ \sigma }^{ 2 } } } } $$

The Black-Scholes-Merton model used to price options, which we will see in-depth in level II, uses the lognormal distribution as its foundation.

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