I'm trying to grasp the rationale for calibrating stochastic volatility models (i.e. Heston model) to empirical IV data from market prices. Doesn't this assume that the options are fairly priced and that no risk premium exists? What if one's purpose is not to predict future market prices but rather the true expected value of the option? If I'm not a market maker, and I simply want to calculate the fair price for an option given a model, wouldn't relying on market data introduce bias?

Or are there parameters in the Heston model like vol of vol that are invariant with respect to things like shifted IV curves as a result of overpriced/underpriced options and risk premium.

Or is the process of calibrating an SV model to historical price data alone too difficult? In my mind the end goal is getting the best estimate of the p.d.f. of the underlying which can be used to calculate an option's value given expiration and strike, not simply to get a good fit to an IV surface.

Sorry if I'm being naïve, thanks!

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    $\begingroup$ What is the true value of an option? It will certainly not be dictated by what happened in the past. Furthermore, past prices are also simply market data. So relying on existing option quotes (or in many OTC markets on existing IVOL quotes) isn't really different in that aspect. Most importantly, how exactly would you get a smile from historical prices (of the underlying presumably, because if you look at options you have the exact same thing just old data). $\endgroup$
    – AKdemy
    Aug 5 at 9:01
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    $\begingroup$ The Heston model is a stochastic volatility model. What you are calibrating are the parameters that model the shape of the volatility. In order to get more accurate calibration results, you need the market implied volatility. The Heston model is used for what you described: calculating the fair value of the option price. It doesn't have any real world applications for predicting where the market goes. It is strictly an option pricing model, mostly used for exotic/path dependent options. $\endgroup$
    – Kevin K.
    Aug 5 at 13:30
  • $\begingroup$ @KevinK. I think I understand; the objective is to find the Heston parameters that would yield the market implied volatilities, which can then be used to calculate option prices with different payoff structures? I was thinking of finding Heston parameters using the underlying alone, then using the Heston model's probability function to solve for the option prices / IV's, and compare those to the market to determine if they are "fair" - in which case relying on market IV's would be "cheating". Thanks a ton. $\endgroup$ Aug 6 at 20:23
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    $\begingroup$ @LegendaryGeg exactly. The other parameters are still assumed to be constant, so the calibration process will only shape the volatility, which will in turn give you the option prices. $\endgroup$
    – Kevin K.
    Aug 7 at 22:13

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