Change of Measure

Motivation: why this matters in quant finance

The pricing formula $V_0=e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{payoff}]$ hides a profound move: the expectation is not taken under the real-world probability measure $\mathbb{P}$ . It is taken under a pricing measure $\mathbb{Q}$ chosen so that discounted asset prices become martingales.

This is not a trick for changing notation. Historical data are observed under

\mathbb{P}

, where risky assets earn risk premia. Derivative prices are computed under

\mathbb{Q}

, where tradable risk premia disappear from drift. Geometric Brownian motion under

\mathbb{P}

has drift

\mu

; under

\mathbb{Q}

it has drift

r

. Girsanov's theorem explains how the same paths can support both descriptions.

Change of measure is the bridge between statistical modelling and no-arbitrage pricing. It is what lets quants estimate volatility from historical data, replace the drift by the risk-free rate, and then price an option without pretending the world is risk neutral.

The informal idea

A probability measure assigns weights to scenarios. Changing measure keeps the scenario space fixed and changes the weights. If a scenario is twice as likely under $\mathbb{Q}$ as under $\mathbb{P}$ , expectations under $\mathbb{Q}$ give that scenario twice the weight.

The object that records the reweighting is the Radon-Nikodym derivative:

Z=\frac{d\mathbb{Q}}{d\mathbb{P}}.

For any integrable payoff $X$ ,

\mathbb{E}^{\mathbb{Q}}[X]=\mathbb{E}^{\mathbb{P}}[ZX].

In continuous time, $Z$ becomes a density process $(Z_t)$ that reweights path probabilities progressively.

Formal definition

Let $(\Omega,\mathcal{F})$ be a measurable space with two probability measures $\mathbb{P}$ and $\mathbb{Q}$ .

$\mathbb{Q}$ is absolutely continuous with respect to $\mathbb{P}$ , written $\mathbb{Q}\ll\mathbb{P}$ , if $\mathbb{P}(A)=0$ implies $\mathbb{Q}(A)=0$ .
$\mathbb{P}$ and $\mathbb{Q}$ are equivalent, written $\mathbb{P}\sim\mathbb{Q}$ , if each is absolutely continuous with respect to the other.

\mathbb{Q}\ll\mathbb{P}

, the Radon-Nikodym theorem gives a non-negative random variable

Z=d\mathbb{Q}/d\mathbb{P}

such that

\mathbb{Q}(A)=\int_A Z\,d\mathbb{P}, \qquad \mathbb{E}^{\mathbb{Q}}[X]=\mathbb{E}^{\mathbb{P}}[ZX].

In a filtered setting, the density process is

Z_t=\mathbb{E}^{\mathbb{P}}\left[\frac{d\mathbb{Q}}{d\mathbb{P}}\mid \mathcal{F}_t\right].

Key properties

Null events stay null

Equivalent measures agree on which events are impossible in the almost-sure sense. This is crucial in finance: an arbitrage that occurs on a set with positive probability under one equivalent measure also has positive probability under the other.

Expectations are weighted expectations

The identity $\mathbb{E}^{\mathbb{Q}}[X]=\mathbb{E}^{\mathbb{P}}[ZX]$ turns measure changes into ordinary expectations with a likelihood-ratio weight. Monte Carlo likelihood-ratio methods are built on this formula.

Drift changes, volatility does not

For Brownian diffusions, Girsanov's theorem changes the drift term but leaves the quadratic variation intact. In Black-Scholes, this is why $\mu$ becomes $r$ while $\sigma$ remains the same volatility parameter.

Martingale measures price assets

A measure $\mathbb{Q}$ is an equivalent martingale measure if discounted tradable prices are martingales under $\mathbb{Q}$ . Under such a measure, no-arbitrage prices are discounted $\mathbb{Q}$ -expectations.

Worked examples

Example 1: finite-state reweighting

Suppose $\Omega=\{u,d\}$ , $\mathbb{P}(u)=0.6$ , $\mathbb{P}(d)=0.4$ , while $\mathbb{Q}(u)=0.5$ , $\mathbb{Q}(d)=0.5$ . Then

Z(u)=\frac{0.5}{0.6}, \qquad Z(d)=\frac{0.5}{0.4}.

For any payoff $X$ ,

\mathbb{E}^{\mathbb{Q}}[X]=0.6Z(u)X(u)+0.4Z(d)X(d).

Example 2: Black-Scholes drift shift

Under $\mathbb{P}$ ,

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t^{\mathbb{P}}.

Let $\theta=(\mu-r)/\sigma$ . Girsanov defines a new Brownian motion

W_t^{\mathbb{Q}}=W_t^{\mathbb{P}}+\theta t.

Substituting $dW_t^{\mathbb{P}}=dW_t^{\mathbb{Q}}-\theta dt$ gives

dS_t=rS_t\,dt+\sigma S_t\,dW_t^{\mathbb{Q}}.

The risk premium has moved from the drift into the density process.

Example 3: pricing a call

Once the stock has risk-neutral dynamics, a European call price is

C_0=e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)^+].

Evaluating this expectation under log-normal

S_T

gives the Black-Scholes formula.

Common confusions and pitfalls

"Risk-neutral means investors are risk-neutral." No. It is a pricing measure, not a psychological statement. Real investors can be risk averse; the measure change encodes risk premia into state prices.

"Changing measure changes the paths." The sample paths stay on the same measurable space. The weights assigned to those paths change.

"The physical drift is irrelevant to everything." It is irrelevant to replication-based derivative prices in the complete Black-Scholes model. It remains central for forecasting, portfolio choice, and risk management under

\mathbb{P}

"Any new measure is allowed." Pricing measures must be equivalent and must make discounted tradable prices martingales. Otherwise they either create arbitrage or assign impossible states positive weight.

Where this goes next

Radon-Nikodym Theorem: The measure-theoretic result behind density ratios.
Girsanov's Theorem: The continuous-time theorem that shifts Brownian drift.
Risk-Neutral Measure: The finance interpretation of equivalent martingale measures.
Risk-Neutral Valuation: How payoff expectations become arbitrage-free prices.
Black-Scholes Formula: The canonical drift-removal example.