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Novikov’s Condition for a Girsanov Change of Measure in Heston

Article Quant Q&A · Author: alexcrespao
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Summary

The document asks whether Novikov’s condition holds for a proposed Girsanov kernel in the Heston model. The stock’s market price of risk leads to an exponential-integrability requirement involving the time integral of the reciprocal variance. The variance follows a square-root mean-reverting process, and the question assumes the Feller condition and independent Brownian drivers. A separate kernel component is set to zero as one possible choice.

No proof or answer is supplied, so the central mathematical issue remains open in this document. The key challenge is whether the reciprocal-variance integral has a sufficiently well-behaved exponential moment; the stated Feller condition alone should not be presented here as establishing that result. The material is useful as a focused question about equivalent martingale measures and measure-change conditions in stochastic volatility models, but it provides no derivation, parameter restrictions, or conclusion about when the required expectation is finite.

Key ideas

  • The proposed stock-risk kernel is inversely proportional to the square root of variance.
  • Novikov’s condition becomes an exponential-moment test involving integrated reciprocal variance.
  • The question assumes a square-root mean-reverting variance process and the Feller condition.
  • The document does not prove finiteness or give parameter conditions for the expectation.

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Full text
# Novikov condition in Heston model


# Novikov condition in Heston model












Consider the Heston model

\begin{equation} \begin{cases} dS_t = S_t(\mu dt + \sqrt{v_t} dW^1_t) \\ dv_t = k(\theta - v_t)dt + \sigma \sqrt{v_t} dW_t^2 \\ dB_t = rB_t dt \end{cases} \end{equation}

where $W^1$ and $W^2$ are indipendent brownian motions, Feller's condition is satisfied and $t \in [0,T]$. I'd like to find an equivalent martingale measure $Q$ using Girsanov's theorem. A suitable Girsanov's kernel is

\begin{equation} \begin{cases} \phi_t = -\frac{\mu-r}{\sqrt{v_t}} \text{ (necessary)}\\ \psi_t = 0 \text{ (for example)} \end{cases} . \end{equation}

To apply Girsanov's theorem one has to check the Novikov's condition $$\mathbb{E}\Big[e^{\frac{1}{2}\int_0^T \phi_t^2 dt}\Big] = \mathbb{E}\Big[e^{\frac{(\mu-r)^2}{2}\int_0^T \frac{1}{v_t} dt}\Big] < +\infty.$$

How is this proven?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.