Stochastic Differential Equations & Geometric Brownian Motion

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This is the 2nd post in a series of five (first post here) where we build up to the Black-Scholes-Merton model for pricing European put and call options.

1. Brownian Motion as a Symmetric Random Walk Limit
2. Stochastic Differential Equations & Geometric Brownian Motion (you are here)
3. Ito's Lemma
4. The Black-Scholes Partial Differential Equation
5. (Coming soon) - tying the pieces together for the pricing formula

This track closely follows Stephen Blyth's An Introduction to Quantitative Finance with a few Python experiments sprinkled in. Expect:

• not-very-practical advice if you simply want to price an option,
• a heavy dose of math if you're allergic,
• and perhaps gaps if you're already deep into stochastic calculus.

Preamble

Last time we built Brownian motion by taking the limit of a symmetric random walk as the step size shrinks to zero. With that in hand, we can start to sketch the idea of a stochastic differential equation (SDE).

Difference equations

For $\Delta t > 0$ we can write

The first term $\mu$ is a deterministic drift; the second scales a Brownian increment. Holding one term fixed while varying the other gives intuition for their roles:

$\mu$ sets the slope-add or subtract a little every timestep and the path tilts accordingly. $\sigma$ dials how violently Brownian motion kicks the process around; larger $\sigma$ makes the path spikier unless a strong drift overpowers it.

Stochastic differential equations

Letting $\Delta t \to 0$ gives the differential form

or more generally

The discrete intuition ($\sqrt{\Delta t} W$) still helps, but $dW_t$ is subtle-it represents an infinitesimal Brownian increment.

Why SDEs show up everywhere

Stochastic differential equations take an ordinary differential equation and inject randomness. Use a normal draw, you get diffusions; use something else and you can model jumps, regime switches, and more.

• Short rates. The Cox-Ingersoll-Ross model adds a square-root diffusion term to produce mean-reverting short rates.

• Jump diffusion. Swap the Gaussian draw for a Poisson jump and you reach the Merton jump-diffusion model, capturing sudden moves from earnings releases or breaking news.

In both cases, parameter choices (e.g., the Poisson rate $\lambda$) change the texture of the path dramatically.

Choosing $\mu(S_t, t)$ and $\sigma(S_t, t)$ for equity prices

A simple, finance-flavored specification is:

• Drift is proportional to price: $\mu(S_t, t) = \mu S_t$. Without randomness this solves to $S_t = S_0 e^{\mu t}$.
• Volatility scales with price: for small $\Delta t$, assume $\operatorname{Var}(\Delta S_t / S_t) \approx \sigma^2 \Delta t$, so $\operatorname{Var}(\Delta S_t) \approx \sigma^2 S_t^2 \Delta t$.

That gives the familiar geometric Brownian motion (GBM):

Why GBM is beloved (and where it falls short)

GBM has the right vibe for equities-multiplicative growth, log-normal distributions, paths that "look" like prices. But it's not quite a duck:

• Volatility is fixed. Real markets exhibit volatility smiles and clustering. Allowing $\sigma$ to depend on $(S_t, t)$ leads to local volatility models; letting $\sigma$ follow its own SDE yields stochastic volatility models.
• No jumps. Earnings surprises, takeovers, and crises produce discontinuities. Jump-diffusion blends or Levy processes push past GBM's smoothness.

Next time

We'll need Ito's Lemma to manipulate functions of diffusions like GBM. That's next on the path toward Black-Scholes.

Extra notes

• Most surfaces look cooler in 3D. Here's a Brownian surface for eye candy:

• Why stochastic volatility matters. Black-Scholes assumes constant volatility; letting volatility evolve stochastically repairs that simplification and better fits observed option prices.