The Martingale Representation Theorem

An option pricing model isn't just "here's a fair price" — it's also, implicitly, a claim that the option can be perfectly recreated by trading the stock. That second claim is the entire justification for delta hedging. It sounds like it should require a separate, case-by-case argument for every payoff shape — but there's a single theorem guaranteeing it works for any Brownian-driven payoff, all at once.

The analogy: any fair outcome can be built from bets on one coin

Suppose someone hands you a complicated final payout that depends, in some intricate way, on the entire path of a sequence of fair coin flips — but whose expected value, given everything so far, never changes (a fair game start to finish). The martingale representation theorem's message: no matter how complicated that payout's dependence on the coin flips is, you can always replicate it exactly by continuously placing simple bets on the coin flips themselves, sized appropriately at each moment, using only information available at that moment. Nothing exotic is ever required beyond trading the coin.

Writing it down

Let $W_t$ be Brownian motion and $\mathcal{F}_t$ its filtration (everything observable up to time $t$). If $M_t$ is any martingale with respect to this filtration (built purely from $W_t$'s own randomness, with a finite-variance endpoint), then there exists a unique, appropriately measurable process $\phi_t$ such that

$$M_t = M_0 + \int_0^t \phi_s\, dW_s.$$

In words: every such martingale can be written as its starting value plus an Itô integral against Brownian motion — some quantity $\phi_t$, computable from information available up to time $t$, that says how much "exposure to $W_t$" the martingale has at each instant. Nothing needs to be added beyond trading $W_t$ itself; there is no residual, unhedgeable piece.

The financial translation is immediate. Let $V_t = E[\text{payoff} \mid \mathcal{F}_t]$ (discounted, under the risk-neutral measure) be the fair value process of a derivative. It's a martingale by construction. The theorem says $V_t = V_0 + \int_0^t \phi_s\, dW_s$ for some $\phi_t$ — and $\phi_t$, translated back into units of the underlying asset, is the hedge ratio (delta). The theorem is the formal reason a replicating, self-financing hedging strategy exists for any Brownian-driven payoff at all, not just the ones with a closed-form formula.

Every path sampled above is one realization of the noise $W_t$ that a hedge has to be built against. The theorem says: whatever fair-value process $V_t$ you're trying to track, there's always some $\phi_t$, computable from the path so far, that turns continuous trading in $W_t$ into an exact replication of $V_t$ — the theorem is what guarantees this is possible before you've even picked a specific payoff.

Worked example 1: representing $W_T^2 - T$ explicitly

Let $M_t = E[W_T^2 - T \mid \mathcal{F}_t]$. Since $\{W_t^2-t\}$ is itself already a martingale (a standard fact — verified the same way as in Quadratic Variation), $M_t = W_t^2 - t$ directly. Apply Itô's lemma to $g(W_t)=W_t^2$: $d(W_t^2) = 2W_t\,dW_t + dt$, so $dM_t = d(W_t^2 - t) = 2W_t\,dW_t + dt - dt = 2W_t\,dW_t$. Reading off the representation $dM_t = \phi_t\,dW_t$ gives $\phi_t = 2W_t$ exactly — an explicit, closed-form hedge ratio, found in two lines by just applying Itô's lemma and matching terms.

Worked example 2: a numeric hedge ratio at a point in time

Continuing the same example, suppose at $t=3$ the path has reached $W_3 = 1.5$. The hedge ratio there is $\phi_3 = 2 \times 1.5 = 3$. Concretely: a desk holding a claim worth $M_t = W_t^2-t$ should, at this instant, hold a position of size 3 units of "exposure to $dW_t$" — if $W_t$ moves up by a small amount $\Delta$, the claim's value moves by approximately $3\Delta$, and the replicating portfolio needs exactly that much sensitivity to stay matched. An instant later, once $W_t$ has moved to a new value, $\phi_t$ is recomputed as $2W_t$ at the new level — this constant rebalancing, driven by an always-available $\phi_t$, is exactly what continuous delta hedging is doing in practice.

What this means in practice

• It's the theorem behind "markets are complete" in a Brownian-driven model: any payoff, however path-dependent, can in principle be perfectly replicated by trading the underlying continuously, which is the deep reason Black-Scholes-style models can price any claim on the stock, not just vanilla options.
• Delta hedging is a direct application: the $\phi_t$ in the representation, once converted from "exposure to $W_t$" into "shares of stock," is precisely the option's delta at each instant.
• Numerical hedge ratios for exotic or path-dependent payoffs (where no closed form exists) are computed by estimating this same $\phi_t$ via simulation — Monte Carlo Greeks are a numerical stand-in for exactly this representation.