Hi experts, I am looking for advise and help in some volatility mapping I am trying to do.

Ignoring the asset-class (FX, interest rate etc), I have a time-series of extremely short-dated (seconds) market-stream. These are converted into a series of rolling (instantaneous) volatility (e.g. stdev[A1:(A1, 30,0)].

I now have a set of X number of (instantaneous) volatilities, and I bucket these into a kind of 'histogram' of probabilities (y-axis) versus volatilities; where probability = (no. of counts of vols in that bucket)/(total no. of counts).

Now, is there a way to convert this pdf (prob versus vol) into the pdf of the underlying variable S under the risk-neutral black-scholes framework? More precisely, my understanding is that the standard pdf of a variable S can be estimated using the 2nd derivative of the call-price

pdf(K=strike) = d^2C/dK^2 where C = call-Price(S, K, rf, T-expiry, vol[K])

and we assume pdf(K = strike) = exp^1/2 * [S - mean/vol(K)]

how can I generate a function of vol[K] to reflect pdf[K=Strke] from the histogram/curve of pdf[vol]

Kind regards