# Spot-Vol covariance and risk reversals

I have been looking into the covariance between log spot returns and log IV returns over a variety of tenors and lookback windows. However I am not sure how these numbers relate back to the outright risk reversal level and if they are directly comparable. Is there an accepted approach for turning the realised covariance into a risk reversal level, or deriving the implied covariance from the risk reversal?

• Well, there is a risk reversal with very specific strikes that is directly related to the implied covariance of asset returns and realised volatility. Would that do?
– user34971
Sep 8, 2021 at 12:43
• Sorry I am not quite sure what you are suggesting. Sep 8, 2021 at 13:05
• Let me have a try on this. Deriving implied covariance is model dependent, e.g. for local volatility dynamics $\frac{dS}{S}=\sigma(S,t)dW$, spot-vol correlation is either 1 or -1, so we can write $d\hat\sigma_{KT}=\frac{d\hat\sigma_{KT}}{dS} dS$. So spot-vol covariance under local volatility is $E[dSd\hat\sigma_{KT}]=\frac{d\hat\sigma_{KT}}{dS}E[dS^2]=\frac{d\hat\sigma_{KT}}{dS} \sigma(S,t)^2S^2dt$.
– ryc
Sep 8, 2021 at 17:03
• But local volatility do not have vol-of-vol nor spot-vol correlation parameters to let you tune its spot-vol covariance. If you estimated realized spot-vol covariance from market data and want to price risk-reversals based on your estimation, may need to resort to stochastic volatility models
– ryc
Sep 8, 2021 at 17:06
• @SurfaceTrader I wasn't suggesting anything, but asking if I give you two options/implied volatilities with specific strikes that together form a risk reversal and that the difference of these two implied vols can be shown to be the expected covariance between the spot return and the realized volatility, would that answer your question.
– user34971
Sep 8, 2021 at 17:36