Novikov's Condition

Girsanov's theorem, the tool that converts real-world drift into risk-neutral drift, hinges on a specific exponential process being a genuine martingale — not just something that looks like a fair game locally, but one whose expected value truly stays constant all the way out. If that exponential is only a local martingale (see Local Martingales and Semimartingales), the change of measure silently fails to be a valid probability measure at all, and every price built on top of it is wrong. Novikov's condition is the standard, checkable test for whether you're safe.

The analogy: a fuel gauge that might be lying

Imagine a process that's supposed to represent "total probability" — it should always sum to exactly 1, the way a fuel gauge should always show how much fuel is actually left. A local martingale version of this process can leak probability mass without you noticing, the way a fuel gauge with a slow, undetectable leak reads lower over time even though nothing looks visibly wrong at any single instant. Novikov's condition is a diagnostic you can run in advance — check one integral, and it certifies the gauge has no leak, guaranteeing the process really is a genuine martingale that stays exactly at 1 in expectation forever.

Writing it down

Girsanov's theorem defines a candidate change-of-measure process, the stochastic exponential,

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

where $\theta_t$ is the "market price of risk" process driving the shift from real-world to risk-neutral drift. $Z_t$ is always a local martingale, automatically, but it is only a genuine martingale — with $E[Z_t]=1$ for all $t$, the property that lets it define a valid probability measure via $dQ/dP = Z_T$ — under an extra condition. Novikov's condition says: if

$$E\left[\exp\left(\frac{1}{2}\int_0^T \theta_s^2\, ds\right)\right] < \infty,$$

then $Z_t$ is guaranteed to be a genuine martingale on $[0,T]$. In words: as long as the expected value of this specific exponential — built from the squared market-price-of-risk, integrated over the whole horizon — stays finite, you're safe; the change of measure is valid and Girsanov's theorem applies cleanly. If $\theta_t$ grows too explosively (as a function of the underlying process), that expectation can diverge, and the change of measure quietly fails.

The exponential curve above is the shape of the trouble: Novikov's condition is checking whether an exponential of a squared, integrated quantity stays finite in expectation. Drag the rate up and watch how fast an exponential can blow up — that's exactly the failure mode Novikov's condition is built to rule out before it happens.

Worked example 1: constant $\theta$, easily satisfied

Suppose $\theta_t = \theta$ is a constant (the simplest, most common case — a fixed market price of risk). Then $\int_0^T \theta_s^2\,ds = \theta^2 T$, a deterministic (non-random) number, so $E[\exp(\tfrac12 \theta^2 T)] = \exp(\tfrac12\theta^2 T)$, which is finite for any finite $\theta$ and $T$. Concretely, with $\theta = 0.3$ and $T=1$: $\exp(0.5 \times 0.09 \times 1) = \exp(0.045) \approx 1.046$ — a small, finite number, so Novikov's condition holds trivially. This is why the standard Black-Scholes-style change of measure, with a constant market price of risk, is never in doubt: the condition is automatically satisfied whenever $\theta_t$ is bounded.

Worked example 2: state-dependent $\theta$, where it can fail

Now let $\theta_t$ depend on the process itself, say $\theta_t = c \cdot X_t$ for some diffusion $X_t$ with unbounded range — a market price of risk that grows with the level of some factor. Novikov's condition requires $E[\exp(\tfrac12 c^2\int_0^T X_s^2\,ds)] < \infty$. If $X_t$ is, for instance, a diffusion whose tails are heavy enough (growing too fast relative to how the constant $c$ scales it), this expectation can diverge for $c$ above some threshold — a concrete, checkable boundary between "the change of measure is a genuine martingale" and "it silently degenerates into a strict local martingale that isn't a valid probability measure." Practically, a modeler checks this by bounding $\theta_t$ (verifying it can't grow too fast in the tails of the model) rather than attempting the full integral by hand — but the qualitative lesson is the same either way: an unbounded, state-dependent market price of risk is exactly the setting where this check stops being a formality.

What this means in practice

• Risk-neutral pricing is only valid if Novikov's condition (or a weaker sufficient alternative) holds — models with bounded market-price-of-risk (most textbook models) are automatically fine, but models with highly state-dependent volatility-of-volatility or leverage effects need this checked explicitly.
• A failure here is not cosmetic — if $Z_t$ is only a strict local martingale, $E[Z_T] < 1$, meaning the "risk-neutral measure" you constructed isn't actually a valid probability measure (it doesn't integrate to 1), and any price computed as an expectation under it is systematically biased.
• Bubble models deliberately exploit this failure: some asset-price bubble models are built with a discounted price process that is a strict local martingale on purpose, which is mathematically what "the price can jump-drop to a fundamental value with positive probability" looks like.