The martingale representation theorem

CM2 · Financial Engineering & Loss Reserving · 11 min read

The gap this theorem fills

Option pricing in CM2 is built from one argument: if a portfolio of shares and cash reproduces an option's payoff in every state, the option must cost what that portfolio costs. The binomial model constructs that portfolio explicitly, two equations in two unknowns, and the answer is the delta.

In continuous time nothing is constructed. The price appears as a discounted expectation under a different probability measure, and nobody has shown that a hedging strategy exists at all. The valuation formula might be a number with no portfolio behind it, in which case the no-arbitrage argument that justified it would be empty.

The martingale representation theorem is what closes that gap. It says the strategy exists, and it identifies it. Everything else on this page is there to make that sentence mean something.

Martingales

A process M_t is a martingale if its expected future value, given everything known now, is its value now:

E[M_t | F_s] = M_s for all s < t

In words: the best forecast of where it will be is where it is. It has no drift.

The conditioning on F_s — the information available at time s — is doing real work and should not be skipped over. A martingale is not a process that is constant, nor one whose unconditional mean is constant. It is one that, at every moment and from whatever point it has reached, is equally likely to go up as down in value terms. Standard Brownian motion is the example to keep in mind; so is the running total of a fair gambling game.

Two properties get used constantly. A martingale's unconditional expectation is its starting value, E[M_t] = M₀, which is why a martingale argument so often collapses a hard expectation into a known number. And a stochastic integral against Brownian motion, ∫φ_sdW_s, is always a martingale — because dW has mean zero whatever φ does, provided φ is known before the step it multiplies.

The fundamental theorem, in two parts

The first part is what makes the risk-neutral measure interesting:

Under Q, the DISCOUNTED price of any traded asset is a martingale.

That is the definition of Q restated. Saying "every asset earns the risk-free rate under Q" and saying "every asset's price divided by e^rt has no drift under Q" are the same statement. And it immediately gives the valuation formula, because a martingale's value now is the expectation of its value later:

e^−rtV_t = E_Q[e^−rTV_T | F_t], so V_t = e^−r(T−t)E_Q[payoff | F_t]

Every option price in the syllabus is that line. Black-Scholes is it with a lognormal payoff; the binomial price is it on a tree.

The second part is the representation theorem itself. If M_t is a martingale under Q, and W is Brownian motion under Q, then there is a process φ_t, known at each moment before it is used, such that

M_t = M₀ + ∫₀^t φ_s dW_s

In words: every martingale is a running total of bets on the same source of randomness, with the bet size varying.

The theorem is an existence result. It does not tell you what φ is; it tells you there is one. That sounds weak and is exactly what was needed.

Why that is the hedging strategy

Put the two parts together, which is the argument worth being able to reproduce.

The discounted option price is a martingale under Q, by part one. So by part two it can be written as its starting value plus an integral against dW. But the discounted share price is also a martingale under Q, and it too is driven by the same dW. So the integral against dW can be rewritten as an integral against the discounted share price:

dṼ_t = φ_t dS̃_t, where the tildes mean discounted

In words: every change in the discounted option value is φ times the change in the discounted share price.

And that is a portfolio. Holding φ_t shares at every moment, with the rest in cash, produces exactly the option's value changes — so it produces the option's payoff. φ is the delta.

Three conclusions follow, and they are what the theorem buys:

A hedge exists. The no-arbitrage price is a price somebody can actually charge and then deliver against.
The price is unique, because the replicating portfolio costs what it costs.
The market is complete — every payoff can be manufactured — and that is a consequence of there being exactly as many sources of randomness as there are traded assets to hedge with.

When it fails, which is the useful part

The theorem needs the market to be complete: one source of randomness, one asset to hedge with. Add a second, unhedgeable source and it breaks, and the consequences are the reason an actuary cares.

Stochastic volatility. If volatility has its own randomness and is not traded, there are two sources of noise and one hedging instrument. No perfect hedge exists, Q is no longer unique, and there is a range of arbitrage-free prices rather than one.
Jumps. A price that can leap cannot be hedged by continuous trading, because the hedge cannot be adjusted during the jump.
Mortality and lapse. This is the actuarial case. A guarantee that pays on death or surrender depends on randomness that no traded asset tracks, so it is intrinsically unhedgeable. The financial part can be hedged; the demographic part has to be diversified across policyholders and capitalised.

So the honest summary is that the theorem explains why option pricing is so clean in its own setting and why the insurance products built on options are not. Saying that is usually worth more than restating the theorem.

Where marks are lost

Saying the share price is a martingale. It is not, under either measure. Under Q the discounted price is. The undiscounted price grows at r under Q and at its own expected return in the real world.

Saying prices are martingales under the real-world measure. They are not, and if they were there would be no risk premium. The whole function of the change of measure is to remove a drift that really is there.

Claiming the theorem tells you the hedge. It proves one exists. Finding it is a separate calculation — differentiating the price, which is what the delta is.

Omitting the conditioning. E[M_t | F_s] = M_s. Writing E[M_t] = M_s is a different and weaker statement, and a question asking for the definition wants the conditioning.

Forgetting that φ must be known in advance. The integrand has to be adapted — decided before the step it multiplies. A strategy that uses tomorrow's price to choose today's holding manufactures arbitrage out of nothing, and the adaptedness condition is what rules it out.

Questions students ask

Why is a martingale the right object?

Because "no arbitrage" and "no drift after adjusting for the risk-free rate" are the same idea. If a discounted traded price had a predictable drift, you could trade on it and make money without risk. Martingale is the precise name for the absence of such a drift, which is why the whole theory is phrased in those terms rather than in terms of expectations and discount factors.

Is the risk-neutral measure a belief about the world?

No, and this is worth being firm about. Nobody thinks shares are expected to return the risk-free rate. Q is a calculating device: a reweighting of the possible futures under which discounted prices have no drift, so that a price can be computed as an expectation. The real-world measure contains the beliefs; Q contains the prices. The deflator page expresses exactly the same content while keeping the real probabilities in view.

Does the theorem work on a binomial tree?

Yes, and in the discrete case it is a statement you can verify by hand, which is why the worked example below is a single two-state step. The continuous version needs stochastic calculus to state; the content — that the increment of the discounted option value is a multiple of the increment of the discounted share value, and that the multiple is the delta — is visible in two subtractions.

What is Girsanov's theorem, which gets mentioned alongside it?

The other half of the machinery. Girsanov says how to change from the real-world measure to Q: it tells you what happens to a Brownian motion when the probabilities are reweighted, namely that it acquires a drift, and the amount of drift removed is the market price of risk. Girsanov produces Q; the representation theorem then produces the hedge under it. Together they are the formal version of the two things the binomial model does by hand.

A worked example

The theorem is an existence result in continuous time, so the way to see it is in discrete time, where everything can be checked. Take the two-period tree from the binomial option pricing page, reprinted so this page stands alone.

A share at ₹400 rises 25% or falls 20% each six months. The continuously compounded risk-free rate is 6% a year. Consider the one-year European put struck at ₹420. All figures are invented for this example.

q = (e^0.03 − 0.80)/(1.25 − 0.80) = (1.0304545 − 0.80)/0.45 = 0.5121212
put values: V₀ = 46.175039, V_up = 9.469196, V_down = 87.587124

Step 1: the discounted share price is a Q-martingale

Discount by e^−rt and check that the expected next value equals the current one.

E_Q[S_0.5] = 0.5121212 × 500 + 0.4878788 × 320 = 412.181814
e^−0.03 × 412.181814 = 0.9704455 × 412.181814 = 400.000000

Exactly S₀. In discounted terms, writing S̃ for the discounted share price:

S̃₀ = 400.000000
S̃_up = e^−0.03 × 500 = 485.222767
S̃_down = e^−0.03 × 320 = 310.542571

q S̃_up + (1−q) S̃_down − S̃₀ = 0.000000000

Zero to nine decimal places. No drift: that is the martingale property, and it holds because q was constructed to make it hold.

Step 2: so is the discounted option price

Ṽ₀ = 46.175039
Ṽ_up = e^−0.03 × 9.469196 = 9.189339
Ṽ_down = e^−0.03 × 87.587124 = 84.998533

q Ṽ_up + (1−q) Ṽ_down − Ṽ₀ = 0.000000000

Also zero. Nothing about the option was assumed here: it follows from the price having been computed as a discounted Q-expectation, which is the first part of the fundamental theorem in action.

Step 3: the representation, which is the theorem

The claim is that the change in the discounted option value is a fixed multiple of the change in the discounted share value, over this step, in both states — one number φ serving for both. Compute it from the two outcomes:

φ = (Ṽ_up − Ṽ_down) / (S̃_up − S̃_down)
= (9.189339 − 84.998533) / (485.222767 − 310.542571)
= −75.809194 / 174.680196 = −0.4339885

Now test it. The representation requires Ṽ − Ṽ₀ = φ(S̃ − S̃₀) in each state separately:

	φ × (change in discounted share)	Change in discounted option
Up	−0.4339885 × 85.222767 = −36.985700	9.189339 − 46.175039 = −36.985700
Down	−0.4339885 × (−89.457429) = 38.823494	84.998533 − 46.175039 = 38.823494

Both rows agree to six decimal places, with the same φ. That is the martingale representation theorem on a single step: the discounted option value is its starting value plus φ times the increment of the discounted share value, and one φ covers every state.

And φ is the delta

Compute the hedge ratio the ordinary way, from the undiscounted values as the binomial page does:

Δ = (V_up − V_down)/(S_up − S_down) = (9.469196 − 87.587124)/(500 − 320) = −0.4339885

The same number to seven decimal places. The discounting cancels, because both numerator and denominator were multiplied by the same e^−0.03. So the abstract integrand whose existence the theorem asserts is, concretely, the number of shares to hold — the thing a trading desk would actually put on. That identification is the whole content of the theorem, and here it can be checked in two subtractions and a division.

The contrast: under the real probabilities, none of this holds

Suppose the share really has a 60% chance of rising each period. Recompute the two drifts:

real-world E[S̃_0.5] − S̃₀ = 0.6 × 485.222767 + 0.4 × 310.542571 − 400 = +15.350688
real-world E[Ṽ_0.5] − Ṽ₀ = 0.6 × 9.189339 + 0.4 × 84.998533 − 46.175039 = −6.662022

Neither is zero, so neither is a martingale under P. The discounted share drifts up by ₹15.35, which is its risk premium. The discounted put drifts down by ₹6.66, because a put is a hedge and its holder accepts a negative expected return for the protection.

Notice the ratio of those two drifts: −6.662022 / 15.350688 = −0.4339885 — the delta again, and exactly so rather than approximately. The reason is one line: under P each drift is (p − q) multiplied by that quantity's own spread across the two states, and (p − q) = 0.0878788 cancels when you divide. So the real-world drifts are not zero, but they stand in exactly the proportion φ — which is why the same hedge that works under Q works in the real world too. The measure change removes the drift; it does not change the portfolio.

That is the sentence to take away. Q is a device for computing, and the hedge it identifies is a real portfolio of real shares that works whatever the probabilities actually are. The martingale representation theorem is the guarantee that such a portfolio is always there to be found.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

The martingale representation theorem, and why hedging is possible at all