Probability

Lognormal Distribution

A distribution whose logarithm is normally distributed, used to model asset prices.

If log X is normal with parameters mu and sigma, X is lognormal. Positive and right-skewed.

Prices are modelled this way because returns compound multiplicatively: many small log-returns add, so by the central limit theorem the log of the price is normal and the price itself is lognormal. It also enforces positivity, which prices require.

The trap. E[X] = exp(mu + sigma^2 / 2), not exp(mu). The mean exceeds the median, and that gap grows with volatility - which is the source of the difference between arithmetic and geometric average returns, and a favourite follow-up.

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