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,

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

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.