# Implied volatility surface modelling in filtered historical simulation

What is the best way to model implied volatility surface in filtered historical simulation (other than keeping it constant)? Is it appropriate to apply GARCH-like model to every point on the surface? Any reduction techniques? Any suggestions will be appreciated.

Thanks.

• A possible way: simulate paths with FHS until expiry date $T$, then return an array of underlying prices from such paths. You can fit a parametric density to this array, like a 2-Lognormal mixture, by means of the EM algorithm. Once you have the Lognormal mixture 5 parameters, you can get the option prices for that maturity using something like this. From option prices, invert B&S and get the implied volatility, as usual. Repeat for different values of $T$ to get a surface. – Lisa Ann May 16 at 13:52
• @LisaAnn, how to map implied volatilities onto volatility smile ("moneyness") in this method? Will it be the same result if I simply integrate GARCH volatilities along each path up to maturity to get an estimation of the implied volatility? Thanks. – marain May 17 at 19:36