LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Geometric Brownian motion is a stochastic differential equation, and its solution is a formula for the price at a later time. Apply Ito's lemma, the chain rule for a function of a random process, to the log of the price. The log's differential is a constant drift plus a constant times the Brownian shock. Integrating and exponentiating produces the price: the starting price times an exponential of a drift term and a Brownian increment. That expression is what a simulator samples when it builds a path.
Why it matters
Path-dependent contracts need a model of the underlying before any payoff can be averaged. The lesson's example is an exotic option: choose a model, simulate paths to the end, score the payoff, and discount. Geometric Brownian motion is the model being solved here, and the lesson is frank that other models track market behavior more closely. It uses this one because the algebra is the one to learn first. A 2021 John Hull paper on neural nets trained across several processes is mentioned, with prices not forced to be arbitrage-free, and then set aside. The worked result on this page is the closed form.
The idea in plain words
An ordinary differential equation can be separated and integrated. This equation cannot, in the same way, because one term is a Brownian motion. Ito's lemma supplies the differential of a function y of time and of the random process. It looks like a Taylor expansion: the time partial times d t, the process partial times the process differential, and one half the second process partial times the squared process differential. Squares of d t, and higher, are dropped.
The function chosen is the log of the price. Its time partial is zero. Its first price partial is one over the price. Its second price partial is minus one over the price squared. The squared price increment, using the geometric-Brownian-motion equation and the multiplication rules, keeps only volatility squared times price squared times d t. Those rules are: d t squared is zero, d t times the Brownian increment is zero, and the squared Brownian increment equals d t.
Substitute. Every bare price cancels. The log's differential collapses to drift minus one half volatility squared, times d t, plus volatility times the Brownian increment. That is an equation with constant coefficients, so both sides integrate. From a start time to a horizon T, the change in the log is the grouped drift times the length of the interval, plus volatility times the change in the Brownian motion. The Brownian value at the start is usually zero, so the change is the Brownian value at T. Exponentiate, multiply by the starting price, and the price at T is determined.
How to use it
Write the price equation as a drift times price times d t, plus volatility times price times the Brownian increment. Set the log as the function to differentiate. Compute the three partials. Replace the squared increment by volatility squared times price squared times d t. Cancel the price. Integrate from the start date to T. Convert the difference of logs into a ratio of prices, then multiply through by the starting price.
A simulator samples that result. One normal draw, scaled by the square root of the gap, covers the whole increment when only the ending price is required. The lesson ties discounted payoff averages to a Feynman-Kac link and does not derive that theorem.
The lesson points to a book on mathematical modeling and computation in finance, with MATLAB and Python examples, for a longer treatment.
What usually goes wrong
Skipping the log leaves a differential still full of the price, which is the equation you already had. The log is chosen because its derivatives are powers of one over the price, and those powers cancel the price factors in the original equation.
Dropping the one-half volatility term is the usual algebra slip. It comes from one half, times the second derivative, times the squared increment. The second derivative is negative, so the drift that multiplies d t is the original drift minus half the variance rate. A simulator that uses the raw drift in the exponential is sampling a different process.
The lesson's caution is that a better-fitting model may replace this one for a live exotic. The steps stay: transform, apply the lemma, integrate.
Questions people ask
What function is differentiated?
The logarithm of the price. Its time partial is zero, its first price partial is one over the price, and its second is minus one over the price squared.
What does the integrated log look like?
Drift minus half the volatility squared, times the time gap, plus volatility times the Brownian increment over that gap.
Why does the lesson solve this equation at all?
Because an ending price with a known formula can be simulated, a payoff can be scored on each ending, and that average can be discounted. That is the exotic-pricing sketch attached to the algebra.