Girsanov proved the price exists. MRT proves the hedge exists. The safety net of quantitative finance.
Two theorems, two guarantees
Chapter 18 gave us Girsanov's Theorem: a method to price any derivative by moving to the risk-neutral world. But pricing and hedging are two different questions. Knowing the fair price of an option is useful. Knowing that you can actually construct a portfolio that perfectly replicates it — regardless of what the market does — is essential. The Martingale Representation Theorem (MRT) provides this second guarantee.
Girsanov answers "how much does it cost?" The MRT answers "can I actually build it?" — and proves the answer is always yes, in a complete market.
1 · Brownian motion as the only driver
The MRT rests on a single, powerful idea: if a market's entire randomness comes from one Brownian Motion — and only one — then every possible fair bet in that market is simply a scaled version of that Brownian Motion. Nothing can happen in the market that is not, in some sense, a reflection of the Brownian wiggle.
THEOREM 19.1 — MARTINGALE REPRESENTATION THEOREM
Every Martingale is an Itô Integral
Let M(t) be any martingale whose information comes from a Brownian Motion W(t). Then there exists a process Γ(t) such that:
M(t) = M(0) + ∫₀ᵗ Γ(u) dW(u)
In plain English: If the Brownian Motion is the only source of randomness in the world, then every fair game (martingale) — no matter how exotic — can be written as a position in the Brownian Motion, scaled by some factor Γ(t).
Every possible derivative payoff — no matter how complex — flows through the MRT funnel and emerges as a scaled Itô integral. This is the mathematical guarantee of hedgeability.
The "No Jumps" implication: because Itô integrals are continuous processes, the MRT tells us that in a Black-Scholes world everything moves smoothly. If a stock suddenly jumps from ₹50 to ₹40 instantaneously — a "gap" — then the world's randomness is no longer driven by a single Brownian Motion alone, the MRT fails, and perfect hedging becomes impossible. This is why jump-diffusion models require more complex hedging strategies.
2 · Market completeness — can I hedge anything?
A market is called complete if every possible derivative payoff can be perfectly replicated using only the available instruments — typically the stock and the risk-free bank account. The MRT tells us exactly when completeness holds: two precise conditions must both be satisfied simultaneously.
Rule 1
Non-zero volatility: σ(t) ≠ 0
The stock must keep moving. If volatility drops to zero — even momentarily — the stock becomes indistinguishable from a bond and loses its hedging power entirely. A static stock cannot offset any option's risk.
Violation: σ = 0 → stock is a bond → no hedge possible for any option.
Rule 2
No hidden randomness
The derivative's payoff must depend only on the same Brownian Motion that drives the stock. If the payoff depends on a second, independent Brownian Motion — a hidden driver — you cannot hedge it using the stock alone.
Violation: payoff depends on W₂ independent of stock's W₁ → unhedgeable residual risk.
Complete market: one driver, one hedging instrument — perfect replication guaranteed. Incomplete market: hidden second driver leaves residual unhedgeable risk.
Real markets are never perfectly complete. Stochastic volatility models (Heston, SABR) introduce a second Brownian Motion that drives σ itself — making the market incomplete. Practitioners respond by adding variance swaps or VIX futures as additional hedging instruments, one per extra source of randomness. The number of hedging instruments needed always equals the number of independent Brownian drivers.
3 · The recipe for a perfect hedge
Given that a hedge exists (guaranteed by the MRT), how do we find it? The theorem provides a constructive recipe — a step-by-step procedure for identifying the exact number of shares Δ(t) to hold at every instant.
🔧 The MRT Hedge Construction Recipe
1
Discount and martingalise: Multiply the option value V(t) by the discount factor D(t) = e^(−rt). Under the risk-neutral measure ℙ̃, the product D(t)V(t) is guaranteed to be a Martingale.
D(t)V(t) is a ℙ̃-Martingale
2
Apply the MRT: Since D(t)V(t) is a Martingale adapted to the Brownian filtration, the MRT guarantees there exists some process Γ(t) — the derivative's "Brownian sensitivity" — such that:
D(t)V(t) = V(0) + ∫₀ᵗ Γ(u) dW̃(u)
3
Match the noise: The stock's discounted value D(t)S(t) also satisfies an Itô equation with the same dW̃ term. Set the coefficient of dW̃ in the portfolio equal to the coefficient in the derivative. The resulting stock position is:
Δ(t) = Γ(t) / (σ(t) · D(t) · S(t))
Setting the portfolio's noise equal to the option's noise and cancelling dW̃ gives the exact hedge ratio at every instant.
4 · The catch — existence ≠ construction
The MRT is an existence theorem, not a construction theorem. It tells you with certainty that a perfect hedge exists. It does not hand you the explicit formula for Γ(t). This distinction matters enormously in practice.
What MRT guarantees ✓
A replicating portfolio exists for every derivative
The hedge is perfect — zero residual risk in a complete market
The hedge is unique — there is only one correct Δ(t)
The replication is self-financing — no cash injections needed
What MRT does NOT give ✗
The explicit formula for Γ(t) or Δ(t)
A recipe for computing the hedge for a specific derivative
Any guidance when the market is incomplete
Anything useful when σ = 0 or when jumps occur
To actually compute Δ(t) in practice, you use one of three methods: the Greek Delta from the BSM formula (cₓ), the PDE methods from Chapters 13–14, or the advanced tool of Malliavin Calculus — which essentially provides a general formula for Γ(t) by differentiating random variables with respect to the Brownian path itself.
The MRT is analogous to an existence proof in pure mathematics: "We can show that a solution must exist, but finding it explicitly requires additional work." The theorem is powerful precisely because it separates the question of whether a hedge exists (answered universally by MRT) from the question of how to compute it (answered case-by-case by BSM Greeks or Malliavin Calculus).
5 · When the safety net breaks — incomplete markets
The MRT's guarantee is only as strong as the completeness conditions. In real markets, completeness fails in several well-documented ways. Understanding these failures is as important as understanding the theorem itself — because they define the limits of delta hedging.
Three completeness failures: zero volatility (stock useless as hedge), hidden randomness (stock cannot offset), price jumps (hedge unravels instantly at the gap).
Jump risk is the most practically significant failure mode. During market crashes, stocks gap through strike prices before a delta-hedger can rebalance. This is why practitioners supplement delta hedges with gamma hedges (using options to hedge options) and jump-risk premiums embedded in bid-ask spreads. The BSM model assumes continuous paths precisely because it needs the MRT — and the MRT requires no jumps.
Try it — complete vs incomplete market hedging
Simulate delta hedging in two scenarios. In the complete market (pure GBM), the replicating portfolio tracks the option perfectly. In the incomplete market (GBM + jump), the hedge breaks at the jump moment and a residual error remains. Change the jump size to see how much the hedge fails.
20%
15% (only in jump mode)
Final option value
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Portfolio value
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Hedge error
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MRT holds?
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Gold = BSM option price | Green = replicating portfolio | Red = stock path (scaled) | In jump mode, watch the portfolio diverge at the jump moment
Chapter 19 — the Martingale Representation Theorem in four ideas:
Theorem 19.1 (MRT): Every martingale adapted to a Brownian filtration can be written as an Itô integral M(t) = M(0) + ∫Γ dW. If Brownian Motion is the only driver, every fair bet is a scaled version of that motion.
Market completeness: A market is complete — every derivative can be perfectly hedged — if and only if (1) σ(t) ≠ 0 always, and (2) no hidden randomness exists. The number of hedging instruments needed equals the number of independent Brownian drivers.
The hedge formula: Δ(t) = Γ(t) / [σ(t)·D(t)·S(t)]. Found by discounting the option to create a martingale, applying MRT to identify Γ(t), then matching the dW̃ coefficients.
The catch: MRT is an existence theorem — it proves a hedge exists but does not compute it. Finding Γ(t) explicitly requires BSM Greeks, PDE methods, or Malliavin Calculus. When σ = 0, jumps occur, or hidden drivers exist, the theorem fails and perfect hedging is impossible.
The safety net: In a continuous-time GBM model with strictly positive volatility, the MRT is your mathematical guarantee that every risk is theoretically eliminable — provided you can trade continuously and the market is complete. The next step is extending this framework to multiple stocks and multiple Brownian drivers.