Brownian Motion and Random Walk Models › Limitations and Impact

Geometric Brownian Motion has been highly influential in quantitative finance, forming the foundation for the Black-Scholes option pricing framework. Yet its limitations are equally important to understand.

GBM assumes returns are normally distributed with constant volatility, but as we documented in the chapter on stylized facts, real returns exhibit fat tails and volatility clustering. Extreme market movements like the 1987 crash or the 2008 financial crisis are orders of magnitude more likely than GBM predicts. A daily return of $-20\%$ is essentially impossible under GBM with typical parameters ($\sigma = 0.16$ would make this a 25-standard-deviation event), yet such moves do occur. This misspecification matters enormously for risk management and option pricing, particularly for out-of-the-money options where the probability of extreme moves determines value.

The assumption of constant volatility is equally problematic. Real volatility varies over time and tends to cluster: periods of high volatility are followed by more high volatility. The VIX index, which measures implied volatility in S&P 500 options, swings from below 10 in calm markets to above 80 during crises. GBM has no mechanism to capture this variation. Models like GARCH (which we'll explore in later chapters on time series) and stochastic volatility models address this limitation by allowing volatility itself to evolve randomly.

GBM also assumes prices are continuous with no jumps. In reality, prices can gap overnight on earnings announcements or news events. Jump-diffusion models extend GBM by adding occasional discontinuous moves, better matching the observed behavior of individual stocks around events.

Despite these limitations, GBM remains foundational for several reasons. First, it provides closed-form solutions that build intuition. The Black-Scholes formula, which we'll derive in upcoming chapters, shows exactly how option prices depend on underlying price, volatility, and time. This transparency is invaluable for understanding risk sensitivities. Second, GBM is mathematically tractable and is a building block for more complex models. Once you understand GBM, you can add stochastic volatility, jumps, or other features to create more realistic models. Third, for many applications, GBM is "good enough." For at-the-money options with short maturities, GBM pricing errors are often within bid-ask spreads. The model's simplicity makes it practical for real-time trading applications where computational speed matters.

The practical lesson is to use GBM as a starting point, not an endpoint. Understand its assumptions, recognize when they fail, and know how to extend the framework when higher accuracy is needed. The options chapter on implied volatility and volatility smiles will show how practitioners implicitly correct for GBM's limitations through the prices they quote.