# Converting Annual Vol to Instantaneous Vol with Mean Reversion [closed]

Options Pricing and Mean Reversion

In the question above, in the accepted answer, the writer claims:

"For instance with a 100% mean reversion a 20% historical annual standard deviation would translate into approximately 30% instantaneous volatility."

I understand that the instantaneous volatility should be higher since annual changes will be less due to mean-reversion. But how did they get 20% annual std dev is approximately 30% instantaneous std dev?

Is there a formula I am missing?

• In the post that the other OP refers to in yet another link you find a formula: $$\hat{\sigma}=\sigma\sqrt{\frac{1-e^{-2\kappa T}}{2\kappa}}\,.$$ It matches those values for $T=1\,.$ Commented Apr 20, 2023 at 11:19
• No. $\sigma$ is the instantaneous volatility in the model that has mean reversion. As you wrote in OP that should be higher than the annual standard deviation $\hat{\sigma}$ that this model produces. $\sigma=30\%$ gives $\hat{\sigma}=20\%\,.$ Commented Apr 20, 2023 at 16:54