Asset prices don't move in smooth, predictable lines. Their paths are jagged and uncertain, a concept often called a “random walk.” To model this mathematically, we need a tool that captures continuous, unpredictable movement. This tool is Brownian motion, the cornerstone of stochastic calculus in finance.

Formally, we use a standard Brownian motion, or Wiener process, denoted by $W_t$. It's a continuous-time stochastic process with three defining properties:

The second property is the most critical. It means that the movement of a stock price from today to tomorrow has nothing to do with its movement from yesterday to today. The variance also scales linearly with time. The longer the time interval, the larger the potential swing in price.

In financial markets, information arrives sequentially. What we know today includes everything we knew yesterday, plus any new information. This ever-growing collection of knowledge is formalized by the concept of a filtration.

Think of a filtration, denoted by $\mathcal{F}_t$, as the set of all available information up to time $t$. It’s a mathematical representation of an expanding universe of knowledge. As time $t$ increases, the information set $\mathcal{F}_t$ grows, meaning $\mathcal{F}_s$ is a subset of $\mathcal{F}_t$ for any $s < t$.

With the concepts of a Wiener process and a filtration in hand, we can introduce one of the most important ideas in quantitative finance: the martingale. A martingale is a stochastic process that models a “fair game.”

Imagine a betting game. If the game is fair, your expected wealth tomorrow, given everything you know today, should be exactly your wealth today. There's no predictable edge. A martingale formalizes this. A process $M_t$ is a martingale with respect to a filtration $\mathcal{F}_t$ if its expected future value, conditioned on the information available at an earlier time $s$, is simply its value at time $s$.

This concept is the bedrock of risk-neutral pricing, the main framework for valuing derivatives. By adjusting the probabilities of future outcomes, we can transform the expected return of an asset into the risk-free rate, making its discounted price process a martingale. This allows us to price complex options by calculating a simple expected value.

One of the strangest properties of Brownian motion is that its path is nowhere differentiable. At no point can you draw a unique tangent line. This mathematical roughness perfectly captures the volatile, unpredictable nature of real-world asset prices. But how do we measure this roughness?

We use a concept called quadratic variation. For a typical, smooth function from calculus, if you divide an interval into smaller and smaller pieces and sum the squared changes in the function's value, the sum goes to zero. The path is just not long enough for the sum to build up.

This non-zero quadratic variation, $[W, W]_t = t$, is a fundamental departure from classical calculus. It is the reason we need a new set of rules for differentiation and integration, leading to tools like Itô's Lemma, which are essential for deriving famous results like the Black-Scholes equation.

Finally, we can combine these ideas into a practical model for stock prices. We can't use a simple Wiener process because stock prices can't be negative, and their returns should be proportional to their current price. Instead, we use Geometric Brownian Motion (GBM).

The $S_t$ term in both parts ensures that the magnitude of the price change is proportional to the current price, a key feature of financial assets. The $dW_t$ term injects the random, jagged behavior we observe in markets. This model, while not perfect, provides a powerful and tractable foundation for pricing a vast range of financial derivatives.