11.5 Change of measure

Importance of Change of Measure

Change of measure lets you transform one probability measure into another while preserving the essential structure of a stochastic process. The payoff is practical: problems that are hard under one measure often become tractable under a different one.

The technique is central to financial mathematics, where switching between real-world and risk-neutral measures underpins derivative pricing and hedging. But it also appears throughout applied probability whenever you need to re-weight outcomes without losing the mathematical structure you're working with.

Radon-Nikodym Theorem

Equivalent Probability Measures

Two probability measures $P$ and $Q$ on a measurable space $(\Omega, \mathcal{F})$ are equivalent (written $P \sim Q$) if they agree on which events are impossible:

$P(A) = 0 \iff Q(A) = 0 \quad \text{for all } A \in \mathcal{F}$

In other words, equivalent measures share exactly the same null sets. They can disagree wildly on the probabilities of events, but they never disagree on whether an event is possible or impossible.

Absolute Continuity

A weaker condition than equivalence is absolute continuity. We say $Q$ is absolutely continuous with respect to $P$ (written $Q \ll P$) if:

$P(A) = 0 \implies Q(A) = 0 \quad \text{for all } A \in \mathcal{F}$

The implication only runs one direction here. Events that are impossible under $P$ must also be impossible under $Q$, but $Q$ is allowed to assign zero probability to events that $P$ considers possible. Equivalence is the special case where absolute continuity holds in both directions: $P \ll Q$ and $Q \ll P$.

Radon-Nikodym Derivative

The Radon-Nikodym theorem states that if $Q \ll P$, there exists a non-negative, $\mathcal{F}$-measurable function $\frac{dQ}{dP}$ such that:

$Q(A) = \int_A \frac{dQ}{dP} \, dP \quad \text{for all } A \in \mathcal{F}$

This function $\frac{dQ}{dP}$ is called the Radon-Nikodym derivative. Think of it as a density that re-weights outcomes: it tells you how much more (or less) likely each outcome is under $Q$ compared to $P$. Expectations transform accordingly:

$E_Q[X] = E_P\!\left[X \cdot \frac{dQ}{dP}\right]$

When $P \sim Q$, the Radon-Nikodym derivative is strictly positive $P$-a.s., and $E_P\!\left[\frac{dQ}{dP}\right] = 1$.

Girsanov Theorem

Brownian Motion Under Change of Measure

Girsanov's theorem is the workhorse for changing measure in continuous-time stochastic calculus. It tells you exactly how a Brownian motion transforms when you switch from $P$ to an equivalent measure $Q$.

Suppose $W_t$ is a standard Brownian motion under $P$, and $\theta_t$ is an adapted process satisfying appropriate integrability conditions (typically the Novikov condition: $E_P\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_s^2 \, ds\right)\right] < \infty$). Define the Radon-Nikodym derivative process:

$Z_t = \exp\!\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right)$

Then $Z_t$ is a $P$-martingale, and setting $\frac{dQ}{dP}\big|_{\mathcal{F}_T} = Z_T$ defines an equivalent measure $Q$.

Drift Transformation

Under the new measure $Q$, the process:

$\tilde{W}_t = W_t + \int_0^t \theta_s \, ds$

is a standard Brownian motion. Equivalently, the original Brownian motion decomposes as $W_t = \tilde{W}_t - \int_0^t \theta_s \, ds$, so it acquires a drift of $-\theta_t$ under $Q$.

The key steps for applying Girsanov:

1. Identify the drift you want to remove (or add)
2. Set $\theta_t$ equal to that drift process
3. Verify the Novikov condition to ensure $Z_t$ is a true martingale
4. Construct $\tilde{W}_t$ and rewrite your SDE under the new measure

Martingale Property Preservation

Girsanov's theorem is powerful precisely because it converts processes with drift into martingales. If $X_t$ satisfies $dX_t = \mu_t \, dt + \sigma_t \, dW_t$ under $P$, you can choose $\theta_t = \mu_t / \sigma_t$ (the market price of risk in financial applications) so that under $Q$:

$dX_t = \sigma_t \, d\tilde{W}_t$

The drift vanishes, and $X_t$ becomes a local martingale under $Q$. This is exactly what you need for risk-neutral pricing, where discounted asset prices must be martingales.

Applications in Finance

Risk-Neutral Pricing

Under the risk-neutral measure $Q$, the discounted price of any traded asset is a martingale. This means the price of a derivative with payoff $H$ at time $T$ is:

$V_0 = E_Q\!\left[e^{-rT} H\right]$

You don't need to estimate the real-world drift of the asset. The change of measure absorbs the drift into the Radon-Nikodym derivative, and all pricing reduces to computing expectations under $Q$.

Fundamental Theorems of Asset Pricing

These two theorems connect the economic concept of arbitrage to the mathematical concept of equivalent measures:

• First Fundamental Theorem (FFTAP): A market is arbitrage-free if and only if there exists at least one probability measure $Q \sim P$ under which discounted asset prices are martingales.
• Second Fundamental Theorem (SFTAP): An arbitrage-free market is complete (every contingent claim can be replicated) if and only if the risk-neutral measure $Q$ is unique.

So the existence of $Q$ rules out arbitrage, and the uniqueness of $Q$ guarantees that every derivative has a unique price.

Martingale Measures vs. Real-World Measures

• The real-world (physical) measure $P$ describes actual probabilities of market outcomes. You'd use it for risk management, forecasting, and statistical estimation.
• Martingale (risk-neutral) measures $Q$ are constructed so that discounted prices are martingales. You use them for pricing and hedging.

Change of measure is the bridge between these two perspectives. The Radon-Nikodym derivative $\frac{dQ}{dP}$ encodes the market price of risk, which is the compensation investors demand for bearing uncertainty.

Esscher Transform

Exponential Tilting

The Esscher transform is a specific change of measure defined by exponential tilting. Given a random variable $X$ and a parameter $\theta$, the new measure $Q$ has Radon-Nikodym derivative:

$\frac{dQ}{dP} = \frac{e^{\theta X}}{E_P[e^{\theta X}]}$

The denominator is just a normalizing constant ensuring $Q$ is a valid probability measure. By varying $\theta$, you shift the distribution of $X$: positive $\theta$ tilts probability mass toward larger values of $X$, and negative $\theta$ tilts it toward smaller values.

Moment Generating Functions

The normalizing constant in the Esscher transform is the moment generating function (MGF) of $X$ under $P$:

$M_X(\theta) = E_P[e^{\theta X}]$

So the Radon-Nikodym derivative is simply $\frac{dQ}{dP} = \frac{e^{\theta X}}{M_X(\theta)}$. This connection makes the Esscher transform especially convenient when the MGF has a known closed form (e.g., for normal, Poisson, or gamma distributions), since you can compute expectations under $Q$ directly from derivatives of $M_X$.

Semi-Martingale Processes

The Esscher transform extends naturally to stochastic processes. For a semi-martingale $X_t$ (which decomposes into a local martingale plus a finite-variation process), the Esscher transform preserves the semi-martingale structure. This makes it particularly useful for models driven by Lévy processes, where Girsanov-type results are more delicate. The Esscher transform provides a tractable way to identify a risk-neutral measure in markets with jumps.

Change of Numéraire

Money Market Account as Numéraire

A numéraire is a reference asset used to denominate prices. The default choice is the money market account $B_t = e^{\int_0^t r_s \, ds}$, which grows at the risk-free rate. Under the risk-neutral measure $Q$, the ratio $S_t / B_t$ is a martingale for any traded asset $S_t$.

But you're free to choose a different numéraire, and doing so changes the associated pricing measure. The general principle: if $N_t$ is any strictly positive traded asset, there exists a measure $Q^N$ (the $N$-forward measure) under which $S_t / N_t$ is a martingale for all traded assets $S_t$.

Forward Measures vs. Spot Measures

• Spot (risk-neutral) measure $Q$: numéraire is the money market account $B_t$. Discounted prices are martingales.
• $T$-forward measure $Q^T$: numéraire is the zero-coupon bond $P(t,T)$ maturing at $T$. Asset prices divided by $P(t,T)$ are martingales.

The Radon-Nikodym derivative connecting them is:

$\frac{dQ^T}{dQ}\bigg|_{\mathcal{F}_t} = \frac{P(t,T)}{B_t \cdot P(0,T)}$

Forward measures are especially useful for pricing instruments with payoffs at a fixed future date, because the bond numéraire absorbs the discounting.

Simplifying Option Pricing Formulas

Change of numéraire often turns a two-part pricing problem into something cleaner. A classic example is the European call option. Under the risk-neutral measure, the price is:

$C_0 = E_Q\!\left[e^{-rT}(S_T - K)^+\right]$

By splitting this into two terms and applying a change of numéraire (switching to the stock as numéraire for the first term), you arrive at the Black-Scholes formula:

$C_0 = S_0 \, \Phi(d_1) - K e^{-rT} \Phi(d_2)$

Each $\Phi(d_i)$ term is an exercise probability under a different measure. The change of numéraire makes the derivation more transparent and avoids messy direct computation.

Discrete-Time Change of Measure

Binomial Model Example

Change of measure works in discrete time too. In the binomial model, a stock moves up by factor $u$ or down by factor $d$ at each time step. Under the real-world measure $P$, the up-move has probability $p$. Under the risk-neutral measure $Q$, the up-move probability becomes:

$q = \frac{e^{r \Delta t} - d}{u - d}$

where $r$ is the risk-free rate and $\Delta t$ is the length of each step. This $q$ is chosen so that the discounted stock price is a martingale under $Q$.

Equivalent Martingale Measures

An equivalent martingale measure (EMM) in discrete time is a probability measure $Q \sim P$ under which the discounted price process $S_n / B_n$ is a martingale. The martingale condition at each node gives:

$E_Q\!\left[\frac{S_{n+1}}{B_{n+1}} \bigg| \mathcal{F}_n\right] = \frac{S_n}{B_n}$

In the binomial model, this equation has a unique solution for $q$ (as long as $d < e^{r\Delta t} < u$, which is the no-arbitrage condition). The uniqueness of $q$ reflects market completeness: every payoff can be replicated.

Cox-Ross-Rubinstein Formula Derivation

The CRR formula for a European call option follows directly from the risk-neutral measure:

1. At each of $n$ time steps, the stock moves up (probability $q$) or down (probability $1-q$)

2. After $n$ steps, the stock price is $S_0 \, u^k d^{n-k}$ if there were $k$ up-moves

3. The call payoff is $(S_0 \, u^k d^{n-k} - K)^+$

4. The option price is the discounted expected payoff under $Q$:

$C_0 = e^{-r n \Delta t} \sum_{k=0}^{n} \binom{n}{k} q^k (1-q)^{n-k} (S_0 \, u^k d^{n-k} - K)^+$

Only terms where $S_0 \, u^k d^{n-k} > K$ contribute. As $n \to \infty$ with appropriate scaling of $u$ and $d$, this converges to the Black-Scholes formula, connecting the discrete and continuous frameworks.