Quant Finance Foundations, Part 7. Previously: Pricing an Option With a Binomial Tree.
Brownian motion is the mathematical language for continuous randomness, and it is what turns a discrete model into a usable one. The binomial tree gave an exact answer in a world with two outcomes per period. Real prices, however, move continuously.
Stochastic calculus has a reputation for being inaccessible, largely because most treatments open with measure theory. Fortunately, the core intuitions do not require it. This post builds them in order, and by the end Itô’s lemma — usually the wall people hit — should look almost obvious.
Start with a coin
Flip a fair coin. Heads, step +1. Tails, step −1. Repeat. That is a random walk, and it has two properties that carry through everything that follows.
The expected position is always zero. Up and down are equally likely, so on average you go nowhere.
The spread grows with the square root of time. This is the property that matters. After n steps, the standard deviation of your position is √n. After 100 steps, roughly 10 units from the origin. After 400 steps, roughly 20 — four times the steps, only twice the spread.
That square root is the single most useful piece of intuition in this post. It is why annual volatility of 20% implies monthly volatility of 20% / √12 ≈ 5.8%, not 20% / 12 ≈ 1.7%. Practitioners scale volatility by the square root of time constantly, and the reason is right here in the coin flips.
Shrink the steps to get Brownian motion
Now make the steps smaller and more frequent — infinitely small, infinitely often, keeping the total variance per unit time fixed. The limit is Brownian motion, written W(t). Accordingly, it behaves as the coin walk would suggest:
- Starts at zero.
- Increments over disjoint intervals are independent — no memory.
- The change over an interval of length t is normally distributed with mean 0 and variance t.
- Paths are continuous — no jumps.
And one strange property: the paths are continuous but nowhere differentiable. Zoom in on any point and the path never smooths into a line the way an ordinary curve does. It stays equally jagged at every scale.
This is not a technicality to be waved past. It is precisely why ordinary calculus fails here and a new one is needed. You cannot speak of the derivative of a Brownian path, because it does not exist anywhere.
Making it look like a stock
Brownian motion is not yet a usable stock model. Two problems: it can go negative, and it does not drift upward. Fix both by modelling the return rather than the price:
dS = μS dt + σS dW
Read this as: the small change in the stock price over a small interval has two parts. A predictable part, μS dt — drift, the expected return. And a random part, σS dW — the shock, scaled by volatility σ.
Both terms are proportional to S, which encodes something sensible: a stock at 100 and a stock at 1,000 have comparable percentage moves, not comparable absolute moves. This is geometric Brownian motion, the standard model underneath Black-Scholes. Because changes are proportional, the price can approach zero but never cross it.
The one thing that breaks ordinary calculus
Here is the crux. In ordinary calculus, when you take a small increment dx and square it, (dx)² is negligible — second order, discarded without a thought.
For Brownian motion this fails, and the coin walk already told us why. Over a time step of length dt, the random move has standard deviation √(dt). So the move itself is of size √(dt), and its square is of size dt — the same order as the drift term.
Formally: (dW)² = dt.
The squared random term does not vanish. It survives, and it is exactly the same size as the terms you are keeping. Every unusual feature of stochastic calculus follows from this one fact.
Itô’s lemma
| Ordinary calculus | Brownian motion | |
|---|---|---|
| Size of a step over dt | proportional to dt | proportional to √(dt) |
| Squared step | (dt)² — negligible | (dW)² = dt — survives |
| Chain rule | first order only | needs a second-order term |
Itô’s lemma is therefore the chain rule, corrected for the term that refuses to disappear.
Ordinary chain rule: if f depends on x, then df = f′(x) dx. For a function of a stochastic process, you must carry the second-order term:
df = f′(S) dS + ½ f″(S) (dS)²
Since (dS)² = σ²S² dt, substituting gives the full form:
df = [μS f′(S) + ½ σ²S² f″(S)] dt + σS f′(S) dW
That highlighted middle piece — ½σ²S²f″(S) — is the Itô correction. It has no counterpart in ordinary calculus, and it is where volatility enters option pricing.
Why the correction is economically real
The correction is not a formal artefact. Consider a stock that either doubles or halves with equal probability. Expected price after one period: (2 + 0.5)/2 = 1.25, a 25% expected gain. But the expected log return is (log 2 + log 0.5)/2 = 0. Two defensible answers to “what is the expected return,” differing by an amount that depends on volatility.
That gap is the Itô correction in concrete form. Whenever a payoff is a curved function of the underlying — and an option’s payoff is exactly that — the curvature interacts with volatility to change the expected value. The lemma is the bookkeeping that keeps this straight.
The practical consequence: a convex position benefits from volatility, a concave position suffers from it. On a desk this is called gamma, and it is the subject of the Greeks post later in this series.
Where this leads
Apply Itô’s lemma to an option’s value as a function of the stock, then construct a portfolio of the option and Δ shares chosen so the random dW terms cancel exactly. The portfolio becomes instantaneously riskless, so by no-arbitrage it must earn the risk-free rate. Writing that condition out gives the Black-Scholes partial differential equation.
That is the same replication argument from the binomial tree, translated to continuous time. The tree cancelled risk by solving two equations; the continuous version cancels it by choosing Δ to eliminate a differential. Same idea, different machinery.
Try it yourself
Watch square-root-of-time scaling emerge from something you built, in this order:
- Simulate 1,000 coin-flip walks of 100 steps each, in a spreadsheet or a few lines of Python.
- Record the final position of every walk.
- Plot the distribution — it will look normal, with standard deviation near 10.
- Rerun the whole thing with 400 steps instead of 100.
- Confirm the standard deviation roughly doubles rather than quadrupling.
Having watched √t emerge from something you built yourself, volatility scaling stops being a rule to memorise and becomes a fact about how randomness accumulates.
Frequently asked questions
Why does volatility scale with the square root of time?
Because independent random steps accumulate in variance rather than in size. After n steps the spread is √n, which is why annual volatility divided by √12 gives the monthly figure.
Why is Brownian motion nowhere differentiable?
Its paths stay equally jagged at every scale, so zooming in never smooths them into a line. As a result, ordinary calculus cannot be applied and Itô calculus is needed.
What does the Itô correction actually represent?
It captures the interaction between a payoff’s curvature and volatility. On a trading desk the same effect is called gamma.
Why model returns rather than prices?
Because proportional changes keep the price positive and reflect how real assets behave. Hence geometric Brownian motion underpins Black-Scholes.
Next in this series: Black-Scholes line by line — what each input does to the price, and which assumptions break in practice.