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Assume the Heston Model with dynamics under the martingale measure $Q$ given by

\begin{align} dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\ dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{2,t}^Q. \end{align}

How, do I find the price of an European put option, i.e. $$e^{-rT}E^Q[(K-S_T)^+]$$.

My idea is to somehow transform this expression into something that looks like an European call-option (I guess that's the so-call put-call-symmetry). And this post seems to try to tackle the problem. My question is how the last step with the Girsanov Kernel is performed? Some more detailed steps on how to find the kernel and the relation beween the Wiener Processes under the different measures would be helpfull. $$$$

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    $\begingroup$ If your ultimate goal is to simply price a (put) option in the Heston framework, you can follow various paths: 1) Monte Carlo simulation (easy to implement, but slow); 2) numerical solution via a 2D-PDE (easy-ish to implement, still slow); 3) derivation of the characteristic function and either direct integration or using FFT methods ('lengthy' derivation). $\endgroup$
    – Kermittfrog
    Apr 5, 2022 at 6:49
  • $\begingroup$ My goal is to price the put-option using a combination of the call-formula and some kind of put-call-parity / put-call-symmetry. But I need to derive it and that is the reason why I need to understand the Girsanov-step. $\endgroup$
    – Landscape
    Apr 5, 2022 at 6:58
  • $\begingroup$ These references may help you with the "Girsanov-step" (a.k.a. Girsanov \textbf{throrem}). 1) math.stackexchange.com/questions/1011636/… 2) math.stackexchange.com/questions/1812152/… $\endgroup$
    – Yoda And Friends
    Apr 5, 2022 at 10:41
  • $\begingroup$ @Landscape . You don't need "some kind of put-call parity" because put-call-parity holds in every model. Therefore $\text{put}=e^{-rT}K-S_0+\text{call}$. Now use your favourite model to price the call. $\endgroup$
    – Kurt G.
    Apr 5, 2022 at 10:51

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