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: $V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[H_T \mid \mathcal{F}_t].$

This formula is a theorem, not an assumption. The drift $\mu$ of the stock under $\mathbb{P}$ does not appear — it has been absorbed into the Radon-Nikodym derivative.

Change of Numeraire

Girsanov extends naturally to numeraire changes. Let $N_t > 0$ be any traded asset (e.g., a zero-coupon bond $P(t,T)$, a foreign currency account). Define the $N$-measure $\mathbb{Q}^N$ by: $\frac{d\mathbb{Q}^N}{d\mathbb{Q}}\bigg|_{\mathcal{F}_T} = \frac{N_T / N_0}{B_T},$ where $B_T = e^{rT}$ is the money-market account numeraire. Under $\mathbb{Q}^N$, the prices of all traded assets normalised by $N_t$ are martingales.

Applications:

• Forward measure ($N_t = P(t,T)$): eliminates discounting. Forward prices $F(t,T)$ are $\mathbb{Q}^T$-martingales. Useful for caplet pricing in one-factor interest rate models.
• Stock as numeraire: for exchange options and Margrabe's formula.
• Quanto products: foreign risk-neutral measure as numeraire, with Girsanov accounting for the FX drift adjustment.

Martingale Representation Theorem

In a Brownian filtration, any $\mathbb{Q}$-martingale $M_t$ can be represented as a stochastic integral: $M_t = M_0 + \int_0^t \phi_s \, d\widetilde{W}_s$ for some adapted process $\phi$. This is the martingale representation theorem (MRT). It implies:

1. Every contingent claim can be replicated by a self-financing trading strategy — market completeness.
2. The hedging strategy is given by $\phi_s = \partial V / \partial S \cdot \sigma S$ (the Delta).

In incomplete markets (e.g., stochastic volatility without a traded variance swap), the MRT fails: the filtration contains sources of randomness not driven by the traded assets, and claims on them cannot be replicated.

Limitations

Strict local martingales. If the Novikov condition fails, $Z_t$ is a strict local martingale: $\mathbb{E}[Z_T] < 1$. The corresponding $\mathbb{Q}$ is a sub-probability measure. Under the 3/2 stochastic volatility model and some CEV models with large elasticity, the stock price under the risk-neutral measure is a strict local martingale, causing put-call parity violations in the form of inflated call prices.

Non-uniqueness in incomplete markets. In incomplete markets with $d$ traded assets but $m > d$ sources of randomness, the market price of risk is a $(m-d)$-dimensional family. There are infinitely many equivalent martingale measures; the model does not pin down a unique price for non-replicable payoffs. In practice, one either assumes additional structure (e.g., zero price for volatility risk) or appeals to utility-based or model-calibration arguments.

Infinite horizon. On $[0,\infty)$, the Novikov condition must hold for every finite $T$, and additional care is needed to define the measure on the full path space. Kazamaki's condition (a weaker condition involving a supermartingale) is sometimes preferred.

Interview Angle

L1: What is the risk-neutral measure? Why do we price under $\mathbb{Q}$ rather than $\mathbb{P}$? Does the stock's expected return $\mu$ affect option prices?

The risk-neutral measure $\mathbb{Q}$ is a probability measure equivalent to $\mathbb{P}$ under which all discounted traded asset prices are martingales. Under $\mathbb{Q}$, the stock grows at the risk-free rate $r$, not at the physical drift $\mu$: $dS = rS \, dt + \sigma S \, d\widetilde{W}^{\mathbb{Q}}$.

We price under $\mathbb{Q}$ because the fundamental theorem of asset pricing (FTAP) states: a market is free of arbitrage if and only if an equivalent martingale measure exists. If we price any derivative as $V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[H_T \mid \mathcal{F}_t]$, the resulting price is the unique price consistent with no-arbitrage in a complete market — any other price would allow a riskless profit. Under $\mathbb{P}$, the discounted stock is not a martingale, so $\mathbb{E}^{\mathbb{P}}[\cdot]$ has no pricing interpretation without knowing the risk premium.

$\mu$ does not affect option prices. The Black-Scholes price depends on $\sigma, S_t, K, T-t, r$ — not on $\mu$. Two investors who disagree entirely about the stock's expected return but agree on $\sigma$ will construct the same delta-hedged portfolio, incur the same hedging costs, and arrive at the same no-arbitrage option price. Mechanically, Girsanov absorbs $\mu$ into the Radon-Nikodym density; the market price of risk $\theta = (\mu-r)/\sigma$ appears only in $Z_T$, not in the pricing formula. Note that $\mu$ does affect the probability that an option expires in-the-money under $\mathbb{P}$ — but this is irrelevant for pricing.

L2: State Girsanov's theorem precisely. What does the Novikov condition guarantee, and what goes wrong if it fails? Derive the risk-neutral dynamics of GBM from its physical measure using Girsanov.

Girsanov's theorem. Let $\theta = (\theta_t)$ be $\mathbb{F}$-adapted with $\mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\tfrac{1}{2}\int_0^T \theta_t^2 \, dt\right)\right] < \infty$ (Novikov). Define the Doléans-Dade exponential: $Z_t = \exp\!\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right).$ Set $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T} = Z_T$. Then: (i) $\mathbb{Q} \sim \mathbb{P}$; (ii) $\widetilde{W}_t = W_t + \int_0^t \theta_s \, ds$ is a standard $\mathbb{Q}$-Brownian motion.

Role of Novikov. $Z_t$ satisfies $dZ_t = -\theta_t Z_t \, dW_t$ (Itô's lemma) and is non-negative, so it is a non-negative local martingale. Every non-negative local martingale is a supermartingale: $\mathbb{E}[Z_T] \leq Z_0 = 1$. The Novikov condition upgrades this to a true martingale with $\mathbb{E}[Z_T] = 1$, which ensures $\mathbb{Q}(\Omega) = \mathbb{E}^{\mathbb{P}}[Z_T] = 1$ — a genuine probability measure. Without Novikov, $\mathbb{E}[Z_T] < 1$: the putative $\mathbb{Q}$ is a sub-probability measure integrating to less than 1. The resulting "pricing formula" would inflate derivative prices — e.g., a call price could violate $C \leq S_t$ — and the interpretation as a no-arbitrage price breaks down entirely.

Deriving risk-neutral GBM. Under $\mathbb{P}$: $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t^{\mathbb{P}}.$ Set $\theta_t = (\mu - r)/\sigma$ (constant). Novikov: $\mathbb{E}^{\mathbb{P}}[\exp(\theta^2 T/2)] = e^{(\mu-r)^2 T/(2\sigma^2)} < \infty$. Define $\mathbb{Q}$ with this $\theta$. Then $\widetilde{W}_t = W_t^{\mathbb{P}} + \theta t$ is a $\mathbb{Q}$-Brownian motion, so $dW_t^{\mathbb{P}} = d\widetilde{W}_t - \theta \, dt$. Substitute: $dS_t = \mu S_t \, dt + \sigma S_t(d\widetilde{W}_t - \theta \, dt) = (\mu - \sigma\theta) S_t \, dt + \sigma S_t \, d\widetilde{W}_t = r S_t \, dt + \sigma S_t \, d\widetilde{W}_t.$ Under $\mathbb{Q}$, the stock drifts at $r$ and $e^{-rt}S_t$ is a $\mathbb{Q}$-martingale. The physical drift $\mu$ has been entirely absorbed into the Radon-Nikodym derivative.

L3: What is a strict local martingale? Give a model where Novikov fails and describe the consequence for call pricing. How does the change-of-numeraire technique simplify pricing under the forward measure?

Strict local martingale. A local martingale $M$ is strict if it is not a true martingale, i.e., $\mathbb{E}[M_t] < M_0$ for some $t > 0$. Equivalently, the localising sequence of stopping times $\tau_n \nearrow \infty$ reveals mass that "escapes to infinity": the process can reach arbitrarily large values with positive probability, losing mass in the process.

CEV model with $\beta > 1$. Consider the risk-neutral dynamics $dS = rS \, dt + \sigma S^\beta \, d\widetilde{W}$ with $\beta > 1$. The stock price can reach $+\infty$ in finite time with positive probability (the diffusion coefficient grows faster than linear). Under this model, the discounted stock $e^{-rt}S_t$ is a strict local martingale under the putative risk-neutral measure: $\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] < S_0$. Consequences: the pricing formula $C = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+]$ gives call prices that violate put-call parity in the form $C - P \neq S_0 - Ke^{-rT}$. Specifically, $C$ is too low relative to the true no-arbitrage bound, and the apparent "call-put parity deficit" equals the mass lost at infinity. A market maker who prices using the strict-local-martingale measure is mispricing deep in-the-money calls and is exposed to arbitrage.

Forward measure and change of numeraire. Let $P(t,T)$ be the time-$t$ price of a zero-coupon bond maturing at $T$. The $T$-forward measure $\mathbb{Q}^T$ is defined by: $\frac{d\mathbb{Q}^T}{d\mathbb{Q}}\bigg|_{\mathcal{F}_T} = \frac{P(T,T)/P(0,T)}{B_T/B_0} = \frac{e^{-rT}}{P(0,T)} \cdot e^{rT} = \frac{1}{P(0,T) e^{rT}}.$ Under $\mathbb{Q}^T$, the forward price $F(t,T) = S_t / P(t,T)$ is a martingale. The call pricing formula transforms as: $C = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-rT}(S_T - K)^+\right] = P(0,T) \cdot \mathbb{E}^{\mathbb{Q}^T}\!\left[(F(T,T) - K)^+\right].$ This simplifies the computation because under $\mathbb{Q}^T$, $F(t,T)$ is a driftless martingale — no discounting factor appears inside the expectation. For interest rate products, the simplification is dramatic: a caplet with payoff $\delta(L(T, T+\delta) - K)^+$ paid at $T+\delta$ becomes, under the $(T+\delta)$-forward measure: $\text{Caplet} = \delta P(0, T+\delta) \cdot \mathbb{E}^{\mathbb{Q}^{T+\delta}}\!\left[(L(T, T+\delta) - K)^+\right].$ Under $\mathbb{Q}^{T+\delta}$, the LIBOR rate $L(T, T+\delta)$ is itself a martingale (by definition of the forward measure). If one models $L$ as log-normal under $\mathbb{Q}^{T+\delta}$, the expectation evaluates to the Black formula for caplets directly — no stochastic discount factor needed. This is the foundation of the LIBOR Market Model (BGM): choose each forward LIBOR rate to be log-normal under its own forward measure, then use Girsanov to convert between measures when simulating the full term structure.