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Lognormal Returns and Compounding

Geometric Brownian motion
dPt=μPt dt+σPt dWtdP_t = \mu P_t\,dt + \sigma P_t\,dW_t

The workhorse price model, and the reason a price modelled this way can approach zero without ever reaching it.

Geometric Brownian motion: drift and volatility both proportional to the current price, so moves are multiplicative. A stock at $10 and one at $1,000 both move in percentage terms.

Two consequences follow, and both are correct descriptions of prices.

Prices stay positive. Multiplying by positive factors never reaches zero, which a model with additive normal shocks cannot promise.

Log returns are normal, so prices are lognormal.

The solution, and the term that surprises people

PT=P0exp⁡((μ−σ22)T+σWT)P_T = P_0\exp\left(\left(\mu - \tfrac{\sigma^2}{2}\right)T + \sigma W_T\right)

Note the −σ22-\frac{\sigma^2}{2}. It is not a typo and it is not a fudge; it comes out of Ito's lemma, and it has a clear meaning.

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