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For stochastic volatility models like Heston, it seems like the standard approach is to calibrate the models from option prices. This seems a bit like a chicken and an egg problem -- wouldn't we prefer a model, based only on historical data, that we can use to price options? I don't see that as frequently.

For the Heston model, I see the method of maximum liklihood used to calibrate against historical data. However, this requires that the conditional probability distribution be derived (which for the Heston model is widely available). For other more complicated models, this seems to get more complicated.

Are there other approaches, other than maximum likelihood or fit to observed options prices, used to calibrate stochastic volatility models?

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  • $\begingroup$ I find Heston's model extremely difficult to estimate in practice based on historical time series, because volatility is latent. Even if we know the conditional distribution (which we do) this is a function of the volatility which is unobserved. $\endgroup$ – Kiwiakos Nov 20 '15 at 10:42
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I suggest taking a look at optimal hedge Monte Carlo and by extension the garam model from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1428555.

The basic issue is that a risk premium exists in option markets which is like Vol also unobserved. The better your guess of the future realized vol distribution, the better your guess of the risk premium currently implied by option markets. Then the basic question is does your forecast of option pnl relative to the risk compensate you enough to trade?

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On the historical vs implied calibration:

First, we model prices have to be arbitrate free. If you estimate the parameters based on history they do not necessarily have to agree with the current market prices. But then that's arbitrage. Or is it your historic estimation wrong? Instead of pondering whichever it is you can just fit your model to the market and voila - the market prices are consistent with the model.

Secondly, there is a story of two worlds - risk free and real world. When you're calibrating to the current prices your model is a tool that gives you a way to price other options using the risk neutral methodology. On the other hand, when you're estimating it from historical perspective you have to make sure that history repeats and that real world is still the same. But that model is NOT a tool, its an possibility.

Taking these two into account I think the answer is - it depends. If you are a large firm (ie. bank or some other market maker ) that makes commission bucks over large volumes of derivatives it might be extremely pricey to believe in your historical model and provide quotes that are not consistent with the market. On the other hand, a fund or another trading firm that makes money from longing the rising assets and shorting the plummeting ones - then you might have the bankroll to believe in the right model.

On the historical vs implied calibration:

Have no experience on historical calibration, so can't say much. Just that I would try calibrating the Heston model to the option prices for each day that you have and then somehow average (maybe moving-average) the parameters?

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