How Equivalent Measure Changes Affect Diffusion Variance

Summary

The document asks whether changing probability measures changes the variance of a diffusion and why covariance between an asset and a numeraire does not alter that variance. The accepted response explains that an equivalent change of measure preserves events that occur with probability one. For Brownian-driven processes, quadratic variation is almost surely fixed, so the diffusion’s volatility is preserved under such a measure change.

The response distinguishes this pathwise quadratic variation from drift: covariance between the asset and the Radon–Nikodym density, or numeraire relationship, appears in the drift adjustment rather than the volatility. It cautions that the variance behavior may differ for processes other than Brownian motion. The explanation is conceptual and relies on the equivalence of the measures and the Brownian setting; it does not work through a specific asset model or generalize the result to every stochastic process.

Key ideas

• Equivalent probability measures preserve events that have probability one.
• Brownian quadratic variation is almost surely fixed, so a conventional equivalent measure change preserves diffusion volatility.
• Covariance with a numeraire or measure-change density affects the drift adjustment rather than volatility.
• The response cautions that variance behavior can differ for non-Brownian processes.

Tags

# Effect on variance of change of measure


# Effect on variance of change of measure












My current understanding: (a) changing the probability measure of a diffusion process does not change the variance. (b) for a general stochastic process the variance may change. Please confirm whether this is correct. Secondly, for (a) why isn't it important that there may be covariance between the asset being diffused and the value of the numeraire? (or the ratio of numeraires)?

## Answer by Arshdeep (score 1, accepted)

https://quant.stackexchange.com/a/76172

(i) is true if measures are equivalent i.e. if $Pr(A)=0$ or $1$ in the first measure then it has to be the same in the other measure.

Being equivalent is always true when you change measure through the conventional radon-nikodym derivative.

Here's another way to see it:

The total probability of asset paths whose quadratic variation equals $σ^2$ is 1 in the first measure so has to be 1 in the second. Therefore variance is preserved.

The above hinges on the fact that quadratic variation of brownian motion is almost surely (i.e. with probability 1) 1 per unit time

If the stochastic process is different from brownian motion this may change.

(ii) the covariance between asset and the radon-nikodym derivative (or asset ratios) shows up as drift and not as vol. Covariance reflects systemic tie-up between the asset and the numeraire, so it biases the asset in terms of drift according to whether the covariance is positive or negative.

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