I'm looking to simulate the stochastic price and volatility process (Heston model) using some form of Euler method for Monte Carlo approximation of option prices. The results that I get are acceptable for deep in the money options and at the money options but not very satisfying at all for deep out of the money options. I want to reduce the variance for faster convergence and the importance sampling method seems suitable but the problem doesn't seem to be trivial at all.

Does anyone have an idea or reference on where to start?

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    $\begingroup$ Most of what you have asked for here is available via QuantLib - is that a possibility? It has interfaces into many languages, can simulate Heston processes and price vanilla options either using the semi-analytical expression or via Monte Carlo. $\endgroup$
    – StackG
    Nov 19 '20 at 7:01
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    $\begingroup$ Unless your question is purely academically driven (i.e. to learn more about importance sampling for Heston), I strongly recommend @StackG 's proposal. $\endgroup$ Nov 19 '20 at 8:35
  • $\begingroup$ You could try to find some potential hints at this in the literature. A quick search came up with this summary paper with examples : degruyter.com/view/journals/math/15/1/article-p679.xml $\endgroup$ Nov 19 '20 at 8:42
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    $\begingroup$ @StackG I have already coded the stochastic processes and calibration myself so QuantLib is not needed. Afaik QuantLib doesn't have any variance reduction methods implemented. I could be wrong though. $\endgroup$
    – spud
    Nov 19 '20 at 11:31
  • $\begingroup$ @Kermittfrog I have read that paper already but they have addressed the problem of slow convergence for MC simulations very briefly with a simple antithetic variables approach. I don't have slow convergence problems all across the board though. In fact, pricing of deep ITM and ATM options are very good but as expected deep OTM options perform poorly. My discretization scheme is by Zhu (2010) and similar to the QE scheme presented in the paper. $\endgroup$
    – spud
    Nov 19 '20 at 11:36

You can potentially try multi-level MC to help with the Euler discretization step size and thus reduce the overall work done when increasing the number of paths. Check out https://archive.siam.org/meetings/uq16/giles.pdf


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