To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're only playing with the real world probabilities, and we can never determine the risk-neutral measure because Heston model is incomplete. So even if we can Monte-Carlo the stock price paths under the real world probabilities, what then? We still cannot decide on the risk-neutral measure.

I also have read somewhere about using variance/vol swaps to make the market complete again, but haven't seen a good explanation (or at least the rough scheme/intuition etc) on how to use var/vol swaps to determine the risk-neutral measure.

Could anybody help? Thanks!

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    $\begingroup$ The wiki page on the Heston model answers your question pretty well. $\endgroup$ – Raskolnikov Feb 5 '19 at 11:24
  • $\begingroup$ @Raskolnikov thanks. So it says the main idea is to use the var/vol swap to "calibrate" the risk-neutral measure. Would you care to provide any more concrete exposition on this method? $\endgroup$ – Vim Feb 5 '19 at 14:37
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    $\begingroup$ I have no experience with that particular method. The method I used was basically calibrating with vanilla options. Technically, you only need two financial instruments, the underlying asset and an option on the underlying or the underlying and a vol swap. But in practice, since markets are imperfect, people tend to use the complete vanilla option surface (cleaned up for outliers and such). Calibrating in that case amounts to a closest fit (through least squares for instance) to the vanilla option prices. $\endgroup$ – Raskolnikov Feb 5 '19 at 15:13
  • $\begingroup$ If however you only have the price of the asset, then several risk-neutral measures are compatible with it. Then, you can select one based on extra conditions you impose for theoretical reasons. $\endgroup$ – Raskolnikov Feb 5 '19 at 15:13
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    $\begingroup$ @Raskolnikov if my interpretation is correct, in practice, under the Heston model, we still assume the drift of the stock price is $(r-q)dt$ under the risk neutral measure; then what we calibrate are the params of the variance process under the risk neutral measure. But how can we just require the risk neutral price process still has the same drift as in the classical BS model? $\endgroup$ – Vim Feb 6 '19 at 6:22

First you assume that the Heston model under the risk neutral measure takes the following form:

$$ dv = \kappa (\theta - v) dt + \eta \sqrt{v} dW $$

Then you calibrate the model to the available options quoted in the market, i.e. find values for the Heston parameters $(\kappa, \theta, \eta)$ such that the options prices generated by the Heston model gives a good fit to the market.

Generating vanilla options prices with the Heston model can be done by MC, but for calibration purposes (and for vanilla options in general) there are semi-analytical methods which will greatly speed up your calibration procedure. Google for example "Heston" + "Fourier" + "Characteristic function".

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    $\begingroup$ Thanks. But my question is how to determine the risk neutral measure? $\endgroup$ – Vim Feb 4 '19 at 23:35
  • $\begingroup$ I've just explained above how to determine the risk neutral measure. $\endgroup$ – ilovevolatility Feb 5 '19 at 16:11
  • $\begingroup$ Sorry I read it as calibrations done under the real world probabilities. Could you be a bit more specific about where and how you get the risk neutral measure? $\endgroup$ – Vim Feb 5 '19 at 16:34

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