Martingale Representation

Every fair game on a Brownian filtration is a bet on that Brownian motion — and the bet size is the hedge.

The idea

Martingale representation theorem. Let $W$ be a Brownian motion and let $M$ be a martingale adapted to its filtration — meaning everything known at time $t$ comes from having watched $W$ up to time $t$. Then there is a process $H$ with $M_{t} = M_{0} + \int_{0}^{t} H_{s}\, dW_{s}.$

The right-hand side is an Itô integral, in which $H_{s}$ is the stake placed just before the increment $dW_{s}$ arrives. The theorem is therefore a statement about sources of randomness: when nothing is random except $W$, the only way to build a fair game is to bet on $W$ itself, and every adapted martingale is such a bet for some choice of stake $H$.

This is what permits a bank to sell an option and promise to deliver its payoff. Price the option as a fair game, and the theorem supplies an $H$ that reproduces the payoff by trading — a hedging strategy, not merely a fact about processes.

One limitation must be carried from the start: the theorem asserts that $H$ exists; it does not say what $H$ is.

Ways to work on it

• Walkthrough. Replicate a payoff on a one-step tree, then see the general theorem it becomes.
• Practice. Hedging strategies on one-step trees, the Black–Scholes delta as a share count, and what the theorem does and does not promise.
• Hardest. Attempt to hedge a payoff in a market with a third state.

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