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!