How do you handle implied volatility performing a VaR Monte-Carlo simulation using a stochastic volatility process calibrated on the underlying

Question

Say you have a portfolio consisting of options each having a market implied volatility. If you now use some stochastic volatility model like GARCH to calibrate the real world volatility of the underlying, and then perform simulations of the correlated stock processes using the GARCH stochastic volatility. What volatility would you use when revaluing the options at the end of the simulation? Some possibilities I imagine are:

1.The same IV used at the start to get the market prices of the options.

2.The stochastic GARCH volatility at the end of the period.

3.The Implied volatility multiplied by $\frac{\sigma_{2}^{GARCH}}{\sigma_{1}^{GARCH}}$, where $\sigma_{1}^{GARCH}$ and $\sigma_{2}^{GARCH}$ is the volatility at the start and end period respectively.

Does any of these options make sense? Is there a "right" answer?

As a bonus question, what correlation would you use for the simulation? Can you use simply the standard EWM correlation as you would with constant volatility, or is there an equivalent to GARCH volatility for correlation?

Answer 1

The historical volatilities of the market factors is not the same as the implied volatilty used to price the options. The "implied volatility" is just one of the model inputs. It does not need to be similar to the historical volatility of the underlying.

The mark to market of an option is the premium that one would have to pay in the market for this option. Sometimes you can just observe this premium in the market. Other times, there is a widely recognized, but simplistic model that explains the option premium as the function of (underlying, risk-free interest rate, implied vol). People can back out the implied vol from option premium, and quote the implied vol, which conveys exactly the same information as the option premium if the underlying and interest rates are observable. A more sophisticated model might have some structure for the underlyings and interest rates and also perhaps additional model inputs. For example, if you choose not to assume that the underlying is normally distributed, then you could have some way of quantifying this assumptions, such as risk reversal. Then, hopefully, you have historical time series for all these market observables and model inputs, and can use their historical volatilties.