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?

Thanks in advance.

  • 1
    $\begingroup$ 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 $T=1\,.$ $\endgroup$
    – Kurt G.
    Commented Apr 20, 2023 at 11:19
  • $\begingroup$ @KurtG. Is the formula backwards? Plugging in T = 1 and K = 1 (I assume 100% mean reversion means the rate coefficient is 1), get about 0.66 for the square root term. But, doesn't sigma represent our normal (unadjusted) volatility of 20% so that the adjusted volatility is 20% * 0.66 which is obviously not 30%. So should the formula instead have sigma and sigma_hat switched? $\endgroup$
    – jmac
    Commented Apr 20, 2023 at 16:08
  • $\begingroup$ 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 $\hat{\sigma}=20\%\,.$ $\endgroup$
    – Kurt G.
    Commented Apr 20, 2023 at 16:54


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