Lognormal Distribution

The construction runs in one direction only: start with a normally distributed variable X with mean mu and variance sigma squared, then define Y as e raised to the power of X. Y is lognormal. Equivalently, taking the natural log of Y recovers a normal variable. Because the exponential function maps the entire real line onto the positive half-line, Y can never be zero or negative, however far into the left tail X wanders.

The parameters mu and sigma belong to the underlying normal, not to the lognormal itself, and the two sets of moments are not interchangeable. The median of Y is e to the mu. Its mean is e raised to mu plus half of sigma squared, which always exceeds the median whenever volatility is non-zero. Its variance is e raised to twice mu plus sigma squared, multiplied by the quantity e raised to sigma squared minus one. A lognormal distribution is therefore fully described by two parameters, but they are the mean and variance of the logarithm.

Because the exponential transformation stretches the upper half of the normal distribution and compresses the lower half, the resulting shape is bounded at zero, has a single peak to the left of centre, and trails off to the right without limit. The ordering mode, then median, then mean holds strictly, and the gaps widen as sigma rises. At very low volatility the lognormal is nearly symmetric and closely resembles the normal; at high volatility it becomes severely skewed.

This geometry has a direct financial reading. The gap between the arithmetic mean and the median terminal price is the compounding penalty imposed by volatility: a small number of very large outcomes pull the average above the typical outcome. Investors who plan around expected terminal wealth rather than median terminal wealth systematically overstate what a representative path will deliver.

The link to markets runs through the continuously compounded return, defined as the natural log of the ending price divided by the beginning price. Rearranged, the ending price equals the beginning price multiplied by e raised to that return. Log returns add across time, so the return over T periods is the sum of T single-period returns. If those single-period returns are independent and identically distributed normals, the T-period return is normal with mean mu times T and variance sigma squared times T, so the standard deviation scales with the square root of T.

Exponentiating that sum makes the T-period price ratio lognormal, with dispersion widening as the horizon lengthens. Note the asymmetry in aggregation: log returns aggregate cleanly through time, but a weighted sum of lognormal asset prices is not itself lognormal, which is why the model applies naturally to a single asset path and only approximately to a portfolio.

Expect to move fluently between holding period return and continuously compounded return: take the natural log of one plus the holding period return to get the log return, and subtract one from the exponential of the log return to reverse the step. Expect to annualise a log return by multiplying the mean by the number of periods and the variance by the same factor, never the standard deviation directly.

A standard task builds a confidence interval for a future price. Construct the interval for the continuously compounded return using the normal distribution, then exponentiate each bound and apply it to the current price. The resulting price interval is asymmetric around the current level, which is the expected result rather than an error. Finally, be ready to state the model's weaknesses: real return series exhibit fatter tails, volatility clustering and serial dependence, so the independent-and-normal assumption understates the frequency of extreme moves.