Financial Mathematics · Lecture 9 of 12

Martingales and the Risk-Neutral Measure

Fair games, conditional expectation, and the astonishing result that we can price as if every investor were indifferent to risk, by changing the probability measure.

November 15, 2026By Diganta Bhattacharya5 min read

Information and conditional expectation

As time passes we learn more. The information available at time $t$ is a σ-algebra $\mathcal F_t$, and the growing family $(\mathcal F_t)$ is a filtration. $\mathbb E[X\mid\mathcal F_t]$ is the best forecast of $X$ given what we know at $t$. Two rules do almost all the work:

Tower property: $\mathbb E\big[\mathbb E[X\mid\mathcal F_t]\mid\mathcal F_s\big]=\mathbb E[X\mid\mathcal F_s]$ for $s\le t$.
Taking out what is known: if $Y$ is known at $t$, $\mathbb E[YX\mid\mathcal F_t]=Y\,\mathbb E[X\mid\mathcal F_t]$.

Martingales

Definition$M_t$ is a martingale if $\mathbb E|M_t| \lt \infty$ and $\mathbb E[M_t\mid\mathcal F_s]=M_s$ for all $s\le t$. Your best guess of its future value is its current value: a fair game.

Examples: $W_t$; $W_t^2-t$; $\exp(\theta W_t-\tfrac12\theta^2t)$ (by the MGF from Lecture 2). A practical test from Itô's lemma: an Itô process is a (local) martingale exactly when its $dt$ term is zero. Stochastic integrals $\int H\,dW$ are martingales because each increment has conditional mean zero.

Why martingales matterIf $M$ is a martingale then $M_0=\mathbb E[M_T]$: today's value is the expected terminal value. Pricing comes down to finding a quantity that is a martingale and whose terminal value is the payoff.

The problem with the real-world measure

Under the real-world measure $\mathbb P$, $d(e^{-rt}S)=e^{-rt}S\big((\mu-r)\,dt+\sigma\,dW\big)$. Investors demand $\mu \gt r$ for bearing risk, so the discounted stock drifts upward and isn't a martingale. If we priced an option as $e^{-rT}\mathbb E^{\mathbb P}[\text{payoff}]$, we'd need to know μ and also some risk premium for the option itself. That's hopeless in practice.

Changing measure: Girsanov

Define the market price of risk $\lambda=(\mu-r)/\sigma$ and a new measure $\mathbb Q$ by reweighting each path:

$$\frac{d\mathbb Q}{d\mathbb P}\Big|_{\mathcal F_T}=\exp\!\Big(-\lambda W_T-\tfrac12\lambda^2T\Big).$$

Girsanov's theorem: under $\mathbb Q$, $\widetilde W_t=W_t+\lambda t$ is a Brownian motion. This is the "exponential tilt shifts a normal's mean" fact from Lecture 2, applied to whole paths. Substituting:

$$dS=\mu S\,dt+\sigma S\,(d\widetilde W-\lambda\,dt)=rS\,dt+\sigma S\,d\widetilde W.$$

Under $\mathbb Q$ the stock grows at the risk-free rate and $e^{-rt}S_t$ is a martingale. $\mathbb Q$ is the risk-neutral (equivalent martingale) measure. It agrees with $\mathbb P$ on which events are possible and disagrees only on how likely they are.

The pricing formula

Risk-neutral valuationFor a payoff $H$ at $T$ that can be replicated by trading the stock and bond: $$V_t=e^{-r(T-t)}\,\mathbb E^{\mathbb Q}\big[H\mid\mathcal F_t\big].$$

Why it holds. A self-financing portfolio of stock and bond, once discounted, is a stochastic integral against the discounted stock, so it's a $\mathbb Q$-martingale. If it replicates $H$, its value today is the $\mathbb Q$-expectation of discounted $H$. By no-arbitrage the option must cost the same. This is the fundamental theorem of asset pricing: no arbitrage ⇔ an equivalent martingale measure exists. If the market is also complete, which means every payoff can be replicated, that measure is unique.

Interpretation$\mathbb Q$ is not a forecast. Nobody believes the stock is expected to earn $r$. $\mathbb Q$ is a bookkeeping device that bakes the cost of hedging into the probabilities. Q-probabilities are "prices of Arrow–Debreu securities, scaled up by $e^{rT}$".

Key takeaways

A martingale is a fair game: E[M_T | F_t] = M_t, so today's value is the expected future value.
Girsanov: reweighting paths by an exponential martingale shifts the drift of Brownian motion.
Under Q the stock earns r and discounted prices are martingales: V₀ = e^{−rT} E^Q[payoff].

Exercises

Show that W_t² − t is a martingale using Itô's lemma.
Show exp(θW_t − θ²t/2) is a martingale using the MGF.
Under Q, compute the forward price E^Q[S_T] and check it equals S₀e^{rT} from Lecture 3.