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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...)

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  • $\begingroup$ The only data you have to make your prediction/estimate is a time-series of 1y ATM vols? $\endgroup$ – Daneel Olivaw Feb 10 at 20:11
  • $\begingroup$ I want to use a model which should simulate implied volatilites so that I can have statistics (percentiles, mean , median etc..) on 1 year call prices 1 year from now.. $\endgroup$ – Sarat Muppana Feb 10 at 20:15
  • $\begingroup$ Yes, but what data is available to you? $\endgroup$ – Daneel Olivaw Feb 10 at 20:16
  • $\begingroup$ I have historical 1y ATM vols and vols for other tenors also. Is that sufficient and what else you think I might need? $\endgroup$ – Sarat Muppana Feb 10 at 20:18
  • $\begingroup$ This is far from being a trivial problem and there are multiple approaches. At the bare minimum, you would also need historical data about the S&P price. Then you could calibrate a Heston model to both the S&P and its vol. Ideally, rather than historical time series (ie. real world measure), you would need current option data for the S&P along a) different strikes and b) different maturities, to get a model calibrated to the risk-neutral measure. Once the Heston calibrated, you could simulate your S&P and its vol up to 1y and calculate the future option price. $\endgroup$ – Daneel Olivaw Feb 10 at 20:26
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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:

$$\sqrt{2*vol^2_{2y}-vol^2_{1y}}$$

You can find this formula in the wikipedia article about forward volatility:

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:

Heston calibration

Once this is done, you generate as many volatilities as you want by simulating the heston equation.

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  • $\begingroup$ Hi Thanks for the answer. my goal is to simulate (may be 50000) 1yr Implied volatlilies 1 year from now so that I can have a range of call prices. your method just gives me 1 answer.. $\endgroup$ – Sarat Muppana Feb 10 at 20:29
  • $\begingroup$ what are your inputs? $\endgroup$ – Valometrics.com Feb 10 at 20:31
  • $\begingroup$ I can gather the inputs. Please let me know various approaches and inputs. Thanks $\endgroup$ – Sarat Muppana Feb 10 at 20:34
  • $\begingroup$ I edited the answer. $\endgroup$ – Valometrics.com Feb 10 at 21:15

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