Calibrating Stochastic Volatility Models to Implied Volatility Surfaces

Summary

The document raises a modeling question: why calibrate a stochastic volatility model, such as Heston, to option-implied volatility data rather than to historical underlying-price data? It distinguishes fitting observed market option prices from estimating an option’s expected or fair value. The concern is that market prices may reflect risk premia or mispricing, so using them could import those effects into the model when the aim is to estimate the underlying’s probability distribution.

It also asks whether some model parameters, such as volatility of volatility, are less affected by shifts in the implied volatility curve, and whether historical-only calibration is simply too difficult. No answer or evidence is included, so the document presents the calibration tradeoff as an open question rather than recommending a method. It does not establish that market prices are fair, that parameters are invariant to risk premia, or that historical calibration yields a better distribution estimate.

Key ideas

• The document contrasts calibrating stochastic volatility models to implied volatility surfaces with fitting historical underlying prices.
• Market option prices may reflect risk premia or mispricing, potentially affecting a model calibrated to them.
• The stated objective matters: fitting market prices differs from estimating an option’s expected value.
• The document leaves open whether particular model parameters are stable under changes in the implied volatility curve.
• It provides no answer or evidence comparing the calibration approaches.

Tags

# Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?


# Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?












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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