Lesson 6.2: The Language of Options Pricing: Geometric Brownian Motion (GBM)

The standard model for stock price paths used in the Black-Scholes formula.

Part 1: The Model for Stock Prices

The Random Walk model ( $P_t = P_{t-1} + \epsilon_t$ ) is a good start, but it has flaws. The size of the random step ( $\epsilon_t$ ) is constant, regardless of the stock price. A $1 step is a huge deal for a $10 stock but insignificant for a $1000 stock.

A more realistic model is one where the returns, not the price changes, are random. This leads to **Geometric Brownian Motion (GBM)**, the standard model for asset prices in the Black-Scholes world.

The Geometric Brownian Motion (GBM) Model

The SDE for a stock price $S_t$ under GBM is:

dS_t = \mu S_t dt + \sigma S_t dW_t

The change in the stock price ( $dS_t$ ) has two parts:

$\mu S_t dt$ : A deterministic **drift** term. The stock is expected to grow at a rate $\mu$ , proportional to its current price $S_t$ .
$\sigma S_t dW_t$ : A random **diffusion** term. The size of the random "jiggle" is also proportional to the current price. $\sigma$ is the volatility.

Part 2: Solving the SDE

The SDE $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a differential equation. Solving it gives us a formula for the stock price at a future time T, $S_T$ .

This requires Itô's Lemma. We can show that the solution is:

The Solution to the GBM SDE

S_T = S_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right)T + \sigma W_T \right)

Deconstructing the Solution

This solution shows that the future stock price $S_T$ follows a **log-normal distribution**. The natural logarithm of the price, $\ln(S_T)$ , is Normally distributed.

$\left(\mu - \frac{1}{2}\sigma^2\right)T$ : This is the mean of the log-return. Note the Itô correction term, $-\frac{1}{2}\sigma^2$ , which arises because of the convexity of the exponential function.
$\sigma W_T$ : This is the random component, where $W_T \sim \mathcal{N}(0, T)$ .

What's Next? Pricing Derivatives

Now that we have a robust, realistic model for how a single stock price evolves, we can finally begin to tackle the primary goal of quantitative finance: pricing derivatives.

How can we use this model to find the fair price of a call option, whose payoff depends on the random future price $S_T$ ? This will require the full power of Itô's Lemma and the concept of a risk-neutral world.

Up Next: Next: Monte Carlo Simulation