Random Walks and Brownian Motion: How Finance Models Uncertainty

The Drunk and the Stock Price

Here is a thought experiment: a person leaves a pub and stumbles home, taking one step forward or backward at random each second. Where will they be after a thousand steps?

You cannot predict the exact position, but you can say a lot about the distribution of possible positions. The expected position is where they started (zero). The spread grows with the square root of the number of steps: ( \sigma \sqrt{n} ).

This is a random walk, and it is the simplest model of how stock prices move. Each tick is a small random step. Over time, the price drifts and diffuses, with the uncertainty growing as ( \sqrt{t} ).

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From Discrete to Continuous

A random walk takes discrete steps at discrete times. Brownian motion (or a Wiener process) is the continuous-time limit: infinitely many infinitesimally small steps.

Properties of Brownian motion ( W_t ):

1. ( W_0 = 0 ) — starts at zero
2. Independent increments: ( W_t - W_s ) is independent of the past for ( s < t )
3. Normally distributed increments: ( W_t - W_s \sim N(0, t-s) )
4. Continuous paths: no jumps (though real markets do jump)

The variance grows linearly with time, so the standard deviation grows with ( \sqrt{t} ). This is why the volatility of an asset over a year is roughly ( \sigma_{\text{daily}} \times \sqrt{252} ) (there are about 252 trading days in a year).

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Geometric Brownian Motion — The Stock Price Model

Raw Brownian motion can go negative, which is no good for stock prices. Geometric Brownian motion (GBM) fixes this:

[ dS = \mu S , dt + \sigma S , dW ]

In English: the stock price changes by a drift term (( \mu S , dt )) plus a random shock (( \sigma S , dW )), both proportional to the current price.

The solution is:

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

The key features:

• Prices are always positive (thanks to the exponential)
• Returns are lognormally distributed
• Log returns are normally distributed and additive

This is the model underlying Black-Scholes. It is wrong in detail — real returns have fatter tails and volatility is not constant — but it is a remarkably good first approximation.

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Why the Square Root of Time Matters

The ( \sqrt{t} ) scaling of Brownian motion has profound practical consequences:

• Daily volatility of 1% → annual volatility of ~16% (since ( 1% \times \sqrt{252} \approx 16% ))
• Option values scale with ( \sqrt{T} ) — an option with 4 times the maturity has roughly twice the time value
• VaR over 10 days ( \approx ) VaR over 1 day ( \times \sqrt{10} ) — the Basel committee uses exactly this

This is not some abstruse mathematical fact. It is used every day by risk managers, traders, and regulators worldwide.

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Stochastic Calculus — A Taste

Ordinary calculus does not work for Brownian motion because the paths are nowhere differentiable (infinitely jagged). Itô calculus extends calculus to handle this.

The key result is Itô's lemma: if ( S ) follows GBM and ( f(S) ) is a smooth function, then:

[ df = f'(S) , dS + \frac{1}{2} f''(S) \sigma^2 S^2 , dt ]

That extra ( \frac{1}{2} f'' \sigma^2 S^2 , dt ) term does not exist in ordinary calculus. It is the correction for randomness, and it is where the ( \sigma^2/2 ) in the GBM solution comes from.

Itô's lemma is the starting point for deriving the Black-Scholes PDE, the Greeks, and most continuous-time financial models. You do not need to master it immediately, but knowing it exists — and what it does — is important.

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Simulating Random Walks

One of the best ways to build intuition is simulation. In Python:

import numpy as np

# Simulate GBM paths
S0, mu, sigma, T, n_steps = 100, 0.08, 0.2, 1.0, 252
dt = T / n_steps
n_paths = 1000

Z = np.random.randn(n_paths, n_steps)
S = np.zeros((n_paths, n_steps + 1))
S[:, 0] = S0

for t in range(n_steps):
    S[:, t+1] = S[:, t] * np.exp((mu - 0.5*sigma**2)*dt + sigma*np.sqrt(dt)*Z[:, t])

Run a thousand paths and you will see the fan of possibilities — most paths drift upward (positive ( \mu )), but some crash. The width of the fan grows with ( \sqrt{t} ). This visual is worth more than pages of equations.

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Beyond GBM

GBM is a starting point, not the final word. Real markets exhibit:

• Fat tails: extreme events are more common than the normal distribution predicts
• Volatility clustering: big moves follow big moves (GARCH, stochastic volatility models)
• Jumps: sudden large moves that Brownian motion cannot produce (jump-diffusion models)
• Mean reversion: some quantities (interest rates, volatility) tend to revert to a long-term level

These extensions build on the GBM foundation. Understanding GBM first makes the extensions comprehensible.

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The Connection to Derivatives Pricing

Here is the punchline: if stock prices follow GBM, then using Itô's lemma and the no-arbitrage principle, you can derive the Black-Scholes equation:

[ \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} - rC = 0 ]

This partial differential equation, combined with the boundary condition at expiry, gives you the famous Black-Scholes formula. The entire derivation flows from Brownian motion → Itô's lemma → no-arbitrage → pricing equation.

That is why random walks and Brownian motion are not abstract curiosities. They are the mathematical foundation of how the £600 trillion derivatives market is priced.

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Digging Deeper

Quantt takes you from random walk simulations through to stochastic calculus and derivatives pricing, all with interactive Python exercises and clear explanations. The mathematics is connected to finance at every step, so you always know why you are learning something.

For a wonderful visual introduction, 3Blue1Brown's probability videos are excellent. And the Wikipedia page on Brownian motion has a surprisingly good mathematical treatment.