Geometric Brownian Motion and Its Limitations

Geometric Brownian motion assumes constant volatility and continuous paths, and both assumptions break down in ways candidates must name.

Geometric Brownian motion models an asset price as a process whose percentage changes, rather than its absolute changes, follow a normal distribution with constant drift and constant volatility, and it is the process underlying the Black-Scholes-Merton model. Its limitations, chiefly constant volatility and continuous paths with no jumps, are exactly what interviewers probe once a candidate has demonstrated they understand the model itself.

The stochastic differential equation is: the differential of S equals mu times S times dt plus sigma times S times dW, where mu is the constant drift and sigma is the constant volatility. Applying Ito's lemma to the logarithm of S, rather than to S directly, removes the state-dependent coefficients. The differential of the logarithm of S equals mu minus one half sigma squared, all times dt, plus sigma times dW. The minus one half sigma squared term is the Ito correction, arising directly from the second-order term in Ito's lemma discussed in Ito's Lemma and Stochastic Calculus Derivation, and it is the single detail candidates most often forget when reproducing this derivation under…

Integrating both sides gives S at time t equal to S at time zero, times the exponential of mu minus one half sigma squared, times t, plus sigma times W at t. This is why geometric Brownian motion produces log-normally distributed prices. The logarithm of the price is normally distributed, so the price itself, being an exponential of a normal random variable, is log-normal and always strictly positive.

Log-normal prices never go negative, which ordinary Brownian motion applied directly to the price would allow with positive probability over any sufficiently long horizon. Percentage returns being normally distributed also matches, to a first approximation, how traders naturally think about performance, quoting returns as a percentage rather than an absolute currency move. These two properties, combined with the tractability that comes from a closed-form solution and a closed-form option pricing formula, are why the model became the default starting point rather than a more general alternative.

Real markets exhibit volatility that changes over time and clusters, meaning large moves tend to be followed by other large moves rather than being spread evenly across the calendar. Geometric Brownian motion assumes sigma is fixed, which the observed shape of implied volatility across strikes, known as the volatility smile, directly contradicts. If the model were correct, every option on the same underlying with the same maturity would trade at the same implied volatility regardless of strike, which is not what markets show. This mismatch and the models built to address it are covered in Volatility Smile, Skew, and Implied Volatility.