Itô's lemma on the log: for GBM, d ln S = (μ − ½σ²)dt + σ dW. The corrected drift sits below the naive μ.
The path is built from 50 increments. Each ΔW is normal with mean 0 and variance Δt = 0.020; adding them up gives W(t).
On one Brownian path: the left side is f(W_t) − f(0); the right side is the Itô sum Σ f′(W)ΔW (evaluated at the left point) plus the correction ½Σ f″(W)Δt. Increase n: the corrected sum sticks to the left side, the uncorrected one does not.
Drag the time cursor (or focus it and use the arrow keys) to read the values at t.
The corrected error falls as n grows; the uncorrected one levels off at the size of the correction.
Itô's lemma is the chain rule of stochastic calculus. Applied to the logarithm of geometric Brownian motion, it shows that log prices drift at μ − ½σ², not μ. This visualization makes that −½σ² correction visible and explains where it comes from: the quadratic variation of Brownian motion.
Brownian increments are of size √dt, so (dW)² behaves like dt and does not vanish. The second-order Taylor term ½f″σ²dt therefore survives, unlike in ordinary calculus.
It is the gap between the arithmetic mean return μ and the growth rate of a typical path. Higher volatility lowers the long-run growth of a single investment even when the mean return is unchanged.