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If I'd like to price options on variance/volatility in the Heston model. Is MC simulation and/or finite difference the only way to do it? Or is there an analytical expression for the probability density function of realized variance in the Heston model?

Thanks!

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The characteristic function of the Heston model is known in closed form. To obtain the option prices you have to perform numerical integration though.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.3204&rep=rep1&type=pdf

you can also check the already asked (and answered) question

Problem on Characteristic function in Heston model

which refers to this paper

Duffie, Pan, and Singleton (2000)

Now regarding your original question: here is a reference for options on variance which seems reasonable to me

Pricing Options on Realized Variance in Heston Model with Jumps in Returns and Volatility

they price a number of vol derivatives semi-analytically where only a numerical integration is necessary. This should help you

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  • $\begingroup$ Thanks for the reference to Sepp's paper. That was what I was looking for. Kind of surprising that there are multiple papers dealing with the characteristic function of the joint spot and instantaneous volatility process, but few focusing on the quadratic variation distribution. $\endgroup$ – ilovevolatility Nov 14 '18 at 0:39

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