In a world driven by a single
martingale
source — a Brownian motion W — there is only one way to be
a martingale. That is the content of the martingale representation theorem:
every martingale adapted to the Brownian filtration is, secretly, an
Itô integral
against W.
Precisely: if (\mathcal{F}_t) is the filtration generated by a
Brownian motion W, and (M_t) is any
martingale adapted to it (square-integrable), then there is a unique adapted process
(H_t) with
M_t = M_0 + \int_0^t H_s\, dW_s.
No drift term, no other noise source — just a single stochastic integral. The Brownian motion
is the only raw material, and H is the recipe for how much
of it to use at each instant.
From representation to a hedge, line by line
This abstract fact is the secret engine of replication — the reason every
option in a complete market can be manufactured by trading. Follow the logical chain.
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
H_T is, by the pricing rule,
\tilde{V}_t = \mathbb{E}_{\mathbb{Q}}\big[e^{-rT} H_T \mid \mathcal{F}_t\big],
and a conditional expectation of a fixed random variable is automatically a martingale (the
tower property: \mathbb{E}_{\mathbb{Q}}[\tilde{V}_t \mid \mathcal{F}_s] = \tilde{V}_s).
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
\tilde{V}_t = \tilde{V}_0 + \int_0^t H_s\, d\tilde{W}_s.
Step 3 — match the integrand against a trading strategy. A self-financing
portfolio holding \phi_t shares 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. So the
portfolio's increment is
d\tilde{V}_t = \phi_t\,\sigma \tilde{S}_t\, d\tilde{W}_t.
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 Itô representation is unique — forces them
equal, H_t = \phi_t\,\sigma\tilde{S}_t. Solving for the holding,
\boxed{\,\phi_t = \frac{H_t}{\sigma \tilde{S}_t}.\,}
The abstract integrand H_t is the hedge: it tells you
exactly how many shares \phi_t to hold at each instant to track the
option. The mathematics that guarantees a representation exists is the same mathematics that
guarantees a hedging strategy exists — they are one theorem. This holding
\phi_t is the option's delta.
Let (\mathcal{F}_t) be the filtration generated by a Brownian
motion W, and let (M_t) be a
square-integrable (\mathcal{F}_t)-martingale. Then there is a
unique adapted process (H_t) (with
\mathbb{E}\int_0^T H_s^2\,ds < \infty) such that
M_t = M_0 + \int_0^t H_s\, dW_s \qquad \text{for all } t \le T.
In a Brownian world, every martingale is an Itô integral against
W — and that integrand is the replicating strategy.
The representation theorem is exactly why the Black–Scholes market is
complete. The Step-2 integrand exists, so every claim can be
replicated; and it is unique, so there is one and only one fair price and
one and only one hedge. This dovetails with the second
fundamental theorem:
completeness ⇔ a unique equivalent martingale measure. One Brownian source, one martingale
measure, one hedge — the whole edifice stands on a single noise dimension.
The catch is the hypothesis: the filtration must be Brownian. The theorem
fails the moment there is a noise source the stock cannot span — a jump, a second
uncorrelated shock, stochastic volatility. Then some martingales are not integrals
against W, the integrand (the hedge) may not exist, the market
becomes incomplete, and the martingale measure is no longer unique. Markets
are complete precisely to the extent that there are as many tradeable assets as sources of
risk.
Looking forward, the integrand H_t — equivalently the holding
\phi_t = H_t/(\sigma\tilde{S}_t) — is the option's
delta, the first and most important of the
Greeks. Computing it explicitly for a call option recovers the
\Phi(d_1) of the Black–Scholes formula, where delta-hedging
becomes a concrete recipe a trader can run.