Lognormal Distribution

The lognormal distribution is the probability distribution of a quantity whose natural logarithm follows a normal distribution. A lognormal variable is always positive, and its distribution is skewed to the right: most values sit fairly close together, while a long tail stretches towards large values. It arises naturally when a quantity is the product of many small independent factors, in the same way that the normal distribution arises from sums.

In finance it is the standard model for the price of a share or other asset at a future date. Prices cannot fall below zero, and they grow by compounding one period's return on the last, which is a multiplicative process. The Black-Scholes-Merton model of option pricing rests on the assumption that the price of the underlying is lognormally distributed.

A random variable $X$ is lognormal with parameters $\mu$ and $\sigma$ if $\ln X$ is normal with mean $\mu$ and standard deviation $\sigma$. The two parameters describe the logarithm, not $X$ itself. The median of $X$ is:

The mean is:

The variance is:

Here $e$ is the base of natural logarithms. The mean is always above the median, and the gap widens as $\sigma$ grows, because a few very large values pull the average up. The mode, the single most likely value, lies below both.

Under the standard model, in which the logarithm of the price follows Brownian motion with drift, the price of an asset at a future time $T$ is:

• $S_0$ is the price today and $S_T$ is the price at time $T$, in years.
• $\mu$ is the expected annual return and $\sigma$ is the annual volatility.
• $Z$ is a draw from the standard normal distribution.

The model implies that the continuously compounded return, $\ln(S_T / S_0)$, is normal, while the price itself is lognormal. Suppose a share trades at 100, with an expected return of 8 percent and a volatility of 20 percent. After one year the median price is $100 \times e^{0.08 - 0.02} = 106.18$, and the mean price is $100 \times e^{0.08} = 108.33$. The share is as likely to finish below 106.18 as above it, even though the average outcome is 108.33. A one-standard-deviation range for the price runs from $106.18 \times e^{-0.20} = 86.93$ to $106.18 \times e^{0.20} = 129.69$, which is not symmetric around the median in price terms.

• No negative prices. A normal distribution for the price itself would give some probability to values below zero. The lognormal does not.
• Compounding. Returns over successive periods multiply. Taking logarithms turns the product into a sum, and by the central limit theorem a sum of many independent pieces tends towards a normal distribution.
• Scale. The model treats a move from 10 to 11 the same as a move from 100 to 110, which matches the way investors think about a return.
• Tractability. It gives closed-form prices for European options and simple formulas for the chance that a price ends above a given level, used in measures such as probability of profit.

The model is also the usual starting point for Monte Carlo simulation of price paths.

Observed returns depart from the model in consistent ways. Log returns over short horizons have fat tails and high kurtosis: very large moves happen far more often than a normal distribution of log returns allows. Equity returns are also negatively skewed, with crashes larger and more sudden than rallies, whereas the model's log returns are symmetric. Volatility is not constant, as the model assumes, but changes over time and tends to rise when prices fall.

Option markets show these departures directly. If prices were truly lognormal, every option on the same asset and expiry would have the same implied volatility. Instead implied volatility varies with the strike, forming the volatility smile or skew. Traders keep the lognormal framework as a quoting convention and adjust for its flaws through that pattern, or they move to models with changing volatility or jumps.

Some quantities do not suit the model at all. Interest rates and price spreads can be negative, and a difference between two lognormal prices is not itself lognormal. For these, normal models or other distributions are used.

Outside asset pricing, the lognormal distribution is used for quantities that are positive and right-skewed, such as insurance claim sizes, incomes, firm sizes and the time between trades. In risk management it appears in the simulation of future portfolio values and in models of loss severity. In each case the reasoning is the same: the quantity results from many multiplicative influences, so its logarithm behaves like a sum.