Girsanov's Theorem and Equivalent Martingale Measures

Setup

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t)_{t \in [0,T]}$. Let $(W_t)_{t \in [0,T]}$ be a standard Brownian motion under $\mathbb{P}$.

Convention. We work on a finite horizon $[0,T]$. All measure changes are defined on $\mathcal{F}_T$. The Novikov condition (stated below) ensures the change of measure is well-defined.

Equivalent Measures

Two probability measures $\mathbb{P}$ and $\mathbb{Q}$ on $(\Omega, \mathcal{F})$ are equivalent (written $\mathbb{P} \sim \mathbb{Q}$) if they agree on null sets: $\mathbb{P}(A) = 0 \iff \mathbb{Q}(A) = 0 \quad \forall A \in \mathcal{F}.$

Equivalent measures agree on what events are possible, but assign different probabilities. This is the minimal requirement for a change of measure to preserve the model structure: under any equivalent measure, the same paths exist; only their likelihoods change.

Absolute continuity $\mathbb{Q} \ll \mathbb{P}$ is the weaker condition requiring only the one-way implication $\mathbb{P}(A) = 0 \Rightarrow \mathbb{Q}(A) = 0$. For equivalence, we need both directions.

Radon-Nikodym Derivative

If $\mathbb{Q} \ll \mathbb{P}$ on $(\Omega, \mathcal{F})$, the Radon-Nikodym theorem guarantees a unique (a.s.) non-negative random variable $Z$ such that: $\mathbb{Q}(A) = \mathbb{E}^{\mathbb{P}}[Z \cdot \mathbf{1}_A], \qquad \forall A \in \mathcal{F}.$

We write $Z = d\mathbb{Q}/d\mathbb{P}$ (the likelihood ratio or density). For equivalence, $Z > 0$ $\mathbb{P}$-a.s.

The density process $(Z_t)_{t \in [0,T]}$ is defined as: $Z_t = \mathbb{E}^{\mathbb{P}}\!\left[\frac{d\mathbb{Q}}{d\mathbb{P}} \,\Bigg|\, \mathcal{F}_t\right].$

By the tower property, $Z_t$ is a non-negative $\mathbb{P}$-martingale with $Z_0 = 1$ and $Z_T = d\mathbb{Q}/d\mathbb{P}$. The change-of-measure formula for conditional expectations is: $\mathbb{E}^{\mathbb{Q}}[X \mid \mathcal{F}_t] = \frac{\mathbb{E}^{\mathbb{P}}[Z_T X \mid \mathcal{F}_t]}{Z_t}, \qquad X \in L^1(\mathbb{Q}).$

Girsanov's Theorem

Theorem. Let $\theta = (\theta_t)_{t \in [0,T]}$ be an $\mathbb{F}$-adapted process satisfying the Novikov condition: $\mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_t^2 \, dt\right)\right] < \infty.$

Define the Doléans-Dade exponential (Girsanov kernel): $Z_t = \mathcal{E}\!\left(-\int_0^\cdot \theta_s \, dW_s\right)_t = \exp\!\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right).$

By Itô's lemma, $Z_t$ satisfies $dZ_t = -\theta_t Z_t \, dW_t$, so $Z_t$ is a non-negative local martingale. The Novikov condition upgrades it to a true martingale with $\mathbb{E}[Z_T] = 1$.

Define $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T} = Z_T$. Then:

1. $\mathbb{Q}$ is a probability measure equivalent to $\mathbb{P}$.
2. The process $\widetilde{W}_t = W_t + \int_0^t \theta_s \, ds$ is a standard Brownian motion under $\mathbb{Q}$.

Interpretation. Under $\mathbb{P}$, the process $\widetilde{W}_t$ has drift $\theta_t \, dt$. The change of measure to $\mathbb{Q}$ absorbs this drift into the likelihood ratio, leaving $\widetilde{W}_t$ drift-free (a martingale under $\mathbb{Q}$) — and by Lévy's characterisation, a $\mathbb{Q}$-Brownian motion.

Novikov Condition: Role and Scope

The Novikov condition ensures $Z_t$ is a true martingale (not merely a local martingale). Without it, one may have $\mathbb{E}^{\mathbb{P}}[Z_T] < 1$, meaning $\mathbb{Q}$ as defined is a sub-probability measure. In financial terms, this corresponds to the stock price being a strict local martingale under the putative risk-neutral measure, which causes call prices to exceed the forward — a pathological but theoretically possible outcome.

Black-Scholes. With constant market price of risk $\theta = (\mu - r)/\sigma$, the Novikov condition is satisfied trivially: $\exp(\theta^2 T / 2) < \infty$.

Heston model. The Feller condition $2\kappa\bar{v} > \xi^2$ (ensuring the variance process stays positive) is related to but does not directly imply the Novikov condition for the market price of volatility risk. Verification requires separate analysis and depends on the chosen form of the volatility risk premium.

Application: Risk-Neutral Pricing

Consider a stock under the physical measure $\mathbb{P}$: $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t^{\mathbb{P}}.$

Let the continuously compounded risk-free rate be $r$. Set the market price of risk: $\theta_t = \frac{\mu - r}{\sigma}.$

By Girsanov, define $d\mathbb{Q}/d\mathbb{P} = Z_T$ with this $\theta$. Then $\widetilde{W}_t = W_t^{\mathbb{P}} + \theta t$ is a $\mathbb{Q}$-Brownian motion, and: $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t^{\mathbb{P}} = \mu S_t \, dt + \sigma S_t \, (d\widetilde{W}_t - \theta \, dt) = r S_t \, dt + \sigma S_t \, d\widetilde{W}_t.$

Under $\mathbb{Q}$, the stock drifts at the risk-free rate. The discounted price $e^{-rt}S_t$ is a $\mathbb{Q}$-martingale (the defining property of an equivalent martingale measure).

No-arbitrage pricing formula. By the fundamental theorem of asset pricing (FTAP), the absence of arbitrage implies the existence of an equivalent martingale measure. The time-$t$ price of any attainable contingent claim with $\mathcal{F}_T$-measurable payoff $H_T$ is: $V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[H_T \mid \mathcal{F}_t].$