QuantPrep | AI-Powered Learning for Quantitative Finance

We now have this powerful, pure model of randomness. But how do we use it to model a real-world asset, like a stock price ( $S_t$ )?

It's tempting to just say, "A stock is random, so $S_t = W_t$ ." This is wrong, and it fails for two major reasons. This lesson is one of the most important in all of finance. We are going to build, step-by-step, the correct model for a stock price by starting with logical, real-world assumptions.

The Problem: Why Simpler Models Fail

Our goal is to create an SDE (Stochastic Differential Equation) for our stock price, $S_t$ . We know this will have the general form:

dS_t = a dt + b dW_t

Where `a` is the Drift (the predictable, "trend" part) and `b` is the Diffusion (the random, "jiggle" part). Our job is to find the correct formulas for `a` and `b`.

Failed Model #1:

S_t = W_t

This is the simplest guess. It fails immediately:

It has zero drift: $W_t$ follows $N(0, t)$ . Its mean is 0. This model predicts the stock will, on average, always return to 0. Real stocks have a positive expected return (e.g., 8% per year).
It goes negative: $W_t$ is a bell curve centered at 0. It can easily go to -10 or -100. Real stock prices have limited liability and cannot go below $0.

Failed Model #2: Arithmetic Brownian Motion (

dS_t = \mu dt + \sigma dW_t

)

This is the next logical guess. We'll just add a "drift" coefficient $\mu$ (a constant) and a "diffusion" coefficient $\sigma$ (a constant).

a = $\mu$ (e.g., an average drift of +$8 per year)

b = $\sigma$ (e.g., a random jiggle of ±$20 per year)

This model fixes our first problem (it now has a positive drift $\mu$ ). But it's still deeply, fundamentally flawed because:

It can still go negative. A $10 stock with a -$20 random shock becomes -$10.
The volatility is constant in dollar terms. A $10 stock jiggles by ~±$20, while a $1000 stock also jiggles by ~±$20. This is not how markets work.

The "Aha!" Moment: Building the Correct Model

This leads us to our core insight, which is the foundation of the Black-Scholes model:

A stock's drift (return) and diffusion (volatility) are not fixed dollar amounts. They are proportional to the stock's current price, $S_t$ .

Building the Coefficients

1. Building the Drift Term `a`

We expect a percentage return, $\mu$ . A stock with $\mu = 8\%$ should drift by $8/year if $S_t = 100$ , or $4/year if $S_t = 50$ . Therefore:

$a = \mu S_t$

2. Building the Diffusion Term `b`

We expect a percentage volatility, $\sigma$ . A stock with $\sigma = 20\%$ should have a "jiggle" of ~±$20 if $S_t = 100$ , or ~±$10 if $S_t = 50$ . Therefore:

$b = \sigma S_t$

The Final Model: Geometric Brownian Motion (GBM)

We now have our final, logical ingredients. Let's plug them into our general SDE, $dS_t = a dt + b dW_t$ :

dS_t = (\mu S_t) dt + (\sigma S_t) dW_t

This is the Geometric Brownian Motion (GBM) equation. It's called "Geometric" because the drift and diffusion are multiplied by the current price $S_t$ . This model elegantly solves our problems:

The random jiggle ( $\sigma S_t dW_t$ ) is a percentage of the price, which is realistic.
The model creates a "natural barrier" at $0 because as $S_t$ approaches 0, the diffusion term also shrinks to 0, preventing the price from becoming negative.

Lesson 1.4: A Model for Stocks (Geometric Brownian Motion)

The Problem: Why Simpler Models Fail

The "Aha!" Moment: Building the Correct Model

1. Building the Drift Term `a`

2. Building the Diffusion Term `b`

The Final Model: Geometric Brownian Motion (GBM)