I have the historical data for 1Year ATM Implied Volatility on SPX 500. I want to simulate the 1 year call option prices 1 year from now. What methods and approaches do I need to use? (Heston,GARCH, Black-Scholes etc...)
The best solution is to compute the implied volatility for a call that matures in two years then the implied volatility for one year call one year from now will be equal to:
You can find this formula in the wikipedia article about forward volatility:
Now in order to generate many volatilities, the only solution is to use a stochastic volatility model (I have a preference for Heston model) to generate 2 years IV and one year IV then use the formula above which is always valid. To do that, you need to estimate heston model parameters which uses as inputs european calls and puts prices.
The calibration procedure consists on minimising the distance between market options prices and prices given using the parameters of Heston model. You can find the calibration algorithm in the following article:
Once this is done, you generate as many volatilities as you want by simulating the heston equation.