A variable follows a lognormal distribution if its natural logarithm is normally distributed.
Properties of lognormal distribution:
- It is bounded below by 0.
- It is skewed to the right i.e. the distribution has a long right tail.
Since the asset prices cannot be negative and thus are bounded below by zero, the lognormal distribution is ideal for modeling the asset prices. The returns on the assets can be negative as well. So, a normal distribution is ideal for the return distribution.
Two parameters completely describe the lognormal distribution: mean and standard deviation of the associated normal distribution: the mean and standard deviation of lnY where Y is lognormal.
If the mean and standard deviation of the associated normal distribution are µ and σ, then:
Mean(µL) of a lognormal random variable = exp(µ + 0.50σ2)
Variance (σL2) of a longnormal random variable = exp(2µ + σ2)*[exp(σ2) - 1]
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