Lognormal Distribution

A variable is lognormally distributed if its natural logarithm follows a normal distribution. In finance, the lognormal distribution is the standard simple model for asset prices: if log returns are normal, prices are lognormal. This captures two real features of prices: they cannot fall below zero, and their potential upside is open ended while the downside is limited to losing everything. The lognormal assumption underpins the Black Scholes model and many simulations of price paths.

Simple vs log returns#

simple return R = (P_t / P_(t-1)) - 1
log return r = ln(P_t / P_(t-1)) = ln(1 + R)

For small returns, the two are nearly equal: a 1% simple return is a 0.995% log return.

Why prices are modelled as lognormal#

If each period's log return is normal and independent, the log of the price after many periods is the sum of many normal variables, which is normal. So the price itself is lognormal:

ln(P_T) ~ Normal(ln(P_0) + (μ - σ²/2) × T, σ² × T)

The shape of a lognormal distribution#

• Always positive.
• Right skewed: a long tail to the upside.
• Mean > median > mode.

Volatility drag#

The difference between the mean and median growth rate is about σ²/2 per year, often called volatility drag. High volatility lowers the typical compound return even if the average simple return stays the same.

median growth rate ≈ arithmetic mean return - σ² / 2

With an arithmetic mean return of 10% and volatility of 40%, the median growth rate is about 10% minus 8% = 2% a year. This explains why highly volatile assets and leveraged products can disappoint over time. See Compounding and Geometric vs Arithmetic Returns.

Lognormal and option pricing#

The Black Scholes model assumes the underlying price follows geometric Brownian motion, which makes future prices lognormal. This is why its formulas involve ln(S/K) and normal distribution functions. See Black-Scholes Model and Monte Carlo Option Pricing.

Where the lognormal model falls short#

The volatility smile in option markets is partly a sign that real price distributions differ from lognormal. See Volatility Smile and Skew.

Practical tips#

Use log returns when adding returns over time or building time series models, and simple returns when combining assets in a portfolio. When simulating prices, generate normal log returns and exponentiate them, which keeps prices positive. When reporting performance, convert back to simple returns, since that is what investors experience. See Measuring Returns and CAGR.

Frequently asked questions#

What is a lognormal distribution?#

A distribution of a variable whose natural logarithm is normally distributed; it is always positive and skewed to the right.

Why are stock prices modelled as lognormal?#

Because prices cannot go below zero and compound over time, and assuming normal log returns produces lognormal prices.

What is the difference between simple and log returns?#

Simple returns measure percentage change; log returns are the natural log of the price ratio, which add up neatly over time.

Next, learn a fat tailed alternative in Student's t-Distribution.