Here is a strange claim: every fair game driven by Brownian information is secretly a trading strategy. Suppose someone hands you a martingale (M_t) — any fair game at all, however exotic, as long as the only information feeding it is the path of a single Brownian motion W. Perhaps it is a conditional expectation of some payoff, perhaps a cooked-up functional of the whole path so far. The martingale representation theorem says: whatever it is, it can be manufactured by continuously trading against W. There is an adapted process (H_t) — a "position size", chosen at each instant using only information available at that instant — such that

No drift term, no second noise source, no residual randomness left over — just a single Itô integral. In a world whose only raw material is one Brownian motion, there is only one way to be a martingale: accumulate H-weighted Brownian increments. The process H is the recipe for how much of the noise to hold at each instant.

If W were (after Girsanov) the noise driving a stock price, then "trading against W" literally means trading the stock — and the abstract integrand H becomes a hedge. That single translation is why every option in the Black–Scholes world can be replicated, and it is the whole reason this theorem sits at the heart of quantitative finance. This page does three things: states the theorem honestly, verifies it by hand on martingales you already know, and then walks the finance identification line by line.

The theorem, stated properly

Read the hypotheses as carefully as the conclusion. The filtration must be the one the Brownian motion itself generates — every scrap of information in the world must be readable off the path of W. That is a genuinely strong assumption: it rules out a world containing a coin flip independent of W, a jump process, or a second Brownian motion. The conclusion is correspondingly strong — the theorem doesn't say the martingale is approximately a stochastic integral, or a stochastic integral plus corrections. It says it is one, exactly, with nothing left over.

Notice also what the theorem does not give you: a formula. It is a pure existence and uniqueness statement. It promises that H is out there and that there is only one of it — but it hands you no recipe for writing H down. (Finding it is a separate job, done with Itô's lemma or a PDE — we return to this below, because in finance the division of labour matters enormously.)

Check it by hand: the theorem in action

Existence theorems feel abstract until you catch one red-handed. Here are two martingales you already know — let's find their integrands explicitly and watch the representation come true.

Example 1 — the compensated square: M_t = W_t^2 - t. You met this as the canonical "quadratic martingale". The theorem insists it must be a stochastic integral. Which one? Apply Itô's lemma to f(x) = x^2:

so subtracting the t kills the drift exactly:

There it is — the representation, concretely: M_0 = 0 (check: W_0^2 - 0 = 0) and integrand \boxed{H_s = 2W_s}. It is adapted (it only needs the current value of W), and square-integrable: \mathbb{E}\int_0^T (2W_s)^2 ds = \int_0^T 4s\, ds = 2T^2 < \infty. Every hypothesis and every promise of the theorem, verified by hand. Notice the trading reading: to manufacture the fair game W_t^2 - t, hold a position of size 2W_s at each instant — double your exposure when the path is high, go short when it is negative.

Example 2 — a conditional expectation: M_t = \mathbb{E}[W_T \mid \mathcal{F}_t]. By the tower property this is a martingale for free; by independent increments it evaluates to

And W_t = 0 + \int_0^t 1\, dW_s: representation found, with the simplest possible integrand \boxed{H_s = 1}. Hold one unit of the noise, always. This example matters more than it looks — in finance, "the conditional expectation of a terminal payoff" is exactly what a price process is, and the theorem is about to tell us that every such price can be manufactured by holding H units of the tradeable asset.

A martingale as a stochastic integral

The figure draws one Brownian path W (faint) together with the martingale M_t = \int_0^t H_s\, dW_s it generates through a simple adapted integrand H — here piecewise-constant, its level re-drawn every fifth of the way along. Watch how the two curves relate: wherever H is large, M amplifies W's wiggles; wherever H is negative, M moves against W, mirroring it. That is what "a position of size H_s in the noise" looks like.

Two things never change, however you refresh. First, M carries no drift — it merely accumulates H-weighted Brownian increments and wanders around its start M_0 = 0, a fair game at every instant. Second, M inherits its entire randomness from W: given the driving path and the recipe H, there is nothing left to chance. The representation theorem is the astonishing converse of this picture — every Brownian-adapted martingale, however it was defined, is secretly one of these curves for some H. Refresh for a fresh \omega.

From representation to a hedge, line by line

Now the payoff. This abstract fact is the secret engine of replication — the reason every option in a complete market can be manufactured by trading, and therefore the reason it has one arbitrage-free price. Follow the logical chain slowly; every step is short, and the last one is the entire derivatives industry.

Step 1 — the discounted price is a \mathbb{Q}-martingale. Work under the risk-neutral measure \mathbb{Q}. The discounted value of a claim paying X_T at maturity is, by the pricing rule,

and a conditional expectation of a fixed random variable is automatically a martingale — the tower property gives \mathbb{E}_{\mathbb{Q}}[\tilde{V}_t \mid \mathcal{F}_s] = \tilde{V}_s. (This is Example 2 again, with W_T upgraded to an option payoff.)

Step 2 — represent it as an integral against the \mathbb{Q}-Brownian motion. By Girsanov the discounted price is driven by a \mathbb{Q}-Brownian motion \tilde{W}; the filtration is Brownian, so the representation theorem applies to the \mathbb{Q}-martingale \tilde{V}_t. There exists a unique adapted H_t with

Pause on how remarkable this already is: the option's value process — defined purely as an expectation, with no trading in sight — has just been rewritten as something that accumulates by riding the noise.

Step 3 — match the integrand against a trading strategy. A self-financing portfolio holding \phi_t shares (and the rest in the bank account) has discounted value moving as d\tilde{V}_t = \phi_t\, d\tilde{S}_t, and the discounted stock satisfies d\tilde{S}_t = \sigma \tilde{S}_t\, d\tilde{W}_t (its drift died with the change of measure — that was Girsanov's job). So the portfolio's increment is

Step 4 — read off the hedge. Comparing the integrand from Step 2 with the portfolio's integrand from Step 3 — both multiply the same d\tilde{W}_t, and the representation is unique — forces them equal: H_t = \phi_t\,\sigma\tilde{S}_t. Solving for the holding,

The abstract integrand H_t is the hedge: divide it by the stock's own noise loading \sigma\tilde{S}_t and you get exactly how many shares \phi_t to hold at each instant so that your portfolio tracks the option perfectly — not on average, but along every single path. The mathematics that guarantees a representation exists is the same mathematics that guarantees a hedging strategy exists — they are one theorem. And this holding \phi_t has a famous name: it is the option's delta, the first of the Greeks.

Existence beats the formula

Here is the point students miss on first reading: the theorem never tells you what H_t is — and that is fine, because existence is the expensive part. Before you invest any effort computing a hedge, you need to know a perfect hedge is even possible. That is a structural question about the market — can trading one stock span all the randomness? — and no amount of calculus answers it. The representation theorem answers it once and for all: in a one-Brownian world, yes, always, for every square-integrable claim. Vanilla call, digital, lookback on the whole path — hedgeable, before you've written a single formula.

Computing the hedge is a separate, easier job, and it is what the Black–Scholes formula machinery does. When the claim is Markovian the price is a smooth function V(t, S_t), and Itô's lemma computes the representation explicitly: the d\tilde{W} coefficient of \tilde V_t comes out as

For a European call this evaluates to the famous \Delta = \Phi(d_1). So the division of labour is: martingale representation guarantees the hedge exists and is unique; Itô and the PDE tell you what it is. A derivatives desk lives on both halves — the first is why the business is possible, the second is what the traders' software actually evaluates, thousands of times a second.