Stochastic Processes & Geometric Brownian Motion

An option is not priced from one forecast of where the stock finishes. The payoff depends on a price that can take many paths between today and expiration. Stochastic processes are the language for that uncertainty. Geometric Brownian motion is the classic model of one asset-price path, and it sits under Black-Scholes and under a Monte Carlo sketch.

A stochastic process is a value that changes through time and includes randomness. For a stock, write the price as S(t): the price depends on the time t.

A random walk is the picture. Someone takes a step every minute. The direction and size of the next step are not known, so the location later is a cloud of possibilities, not a single point. A stock is richer than that walk. The same idea holds: the future path is uncertain, and a model represents many possible paths.

A discrete random walk uses separate steps. In continuous-time finance, Brownian motion is the model for random changes that accumulate without a pause. A standard Brownian motion is written W(t).

Over a small interval dt, the random change dW has an average of zero, and its variance is tied to dt. Brownian motion is the mathematical source of continuous randomness. It is not a forecast of the next print.

Models split the move into two pieces. Drift is the systematic, average rate of change in the model. Volatility is the size of the random variation around that average.

Two assets can share a drift and still look nothing alike. The one with higher volatility produces a much wider fan of future paths.

The classic model for an asset price makes both the drift and the random shock proportional to the current price. A $10 move in a $100 stock is not the same event as a $10 move in a $1,000 stock, so the model works in percentages, not in a fixed dollar step.

Read the equation as two engines that move the price at each small step.

The next point on a path is those two pieces added together. Repeating the step builds one path. Repeating the whole path thousands of times builds a distribution of paths.

The payoff depends on the underlying at expiration. If that price is uncertain, there is a distribution of payoffs, not one payoff. Geometric Brownian motion is one way to generate the paths that feed that distribution.

A Monte Carlo sketch follows five steps. The model calculates. It does not predict.

1. Start from today's underlying price.
2. Set the assumptions: volatility, and a drift or a risk-free rate.
3. Draw many possible price paths.
4. Compute the option payoff on each path.
5. Average the discounted payoffs. That average is a theoretical value.

Monte Carlo draws paths for one long option. Changing volatility, time, the starting price, or the number of paths changes the fan you see.

Under the risk-neutral measure, the real-world expected return is replaced by the risk-free rate, with dividends handled by the model's assumptions.

That replacement does not mean investors expect the stock to earn the risk-free rate. It is a valuation framework: a derivative price can be written as a discounted expected payoff under a pricing measure. The risk-neutral pricing lesson is where that measure is built.

Geometric Brownian motion is a foundation. Markets do not follow it exactly. The classic version is a continuous path with one volatility held constant over the horizon.

• Volatility changes through time.
• News can jump the price. The path is not only small steps.
• Returns can have heavier tails than this model implies.
• Volatility clusters: a wild stretch is often followed by another.
• Listed option prices can imply a smile or a skew, not one constant volatility.

A smile needs a different volatility story than one constant sigma. That story is the local and stochastic volatility lesson.

Geometric Brownian motion is one assumption inside the classic Black-Scholes framework. From that process, no-arbitrage, and continuous-time math, some European options get a closed-form theoretical price. A useful chain is: stochastic process, then geometric Brownian motion, then a pricing model, then a theoretical price and the Greeks.

This page does not derive that closed form. It gives the path model the formula stands on. The numbers on Build, What If, and Monte Carlo stay with the calculator.