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I am just wondering if there is any problem with the so-called "exact" Heston simulations? So far what I have seen are the good things about it, what are the disadvantages? Because if it is so perfect, why is everyone not using the "exact" simulation since it would reduce the discretization error?

Thanks!

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    $\begingroup$ From this abstract: One paper by Broadie and Kaya (2006) stands out in that it proposes what the authors call “an exact simulation method” by which they mean that there it has no bias and therefore no additional time steps are required. In this paper we address the major drawback of their method: that it is only effectively applicable to derivative payoffs which depend only on observations of the underlying at very few points in time. $\endgroup$ Mar 21, 2013 at 12:57

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The exact simulation scheme by Broadie & Kaya (2006) is a ground-breaking research piece, but the algorithm is slow and complicated to implement. IMHO, that's why it is not so popular among practitioners.

The algorithm is "exact" in the sense that you can 'jump' any arbitrary time step as opposed to the time-discretization (e.g., Euler/Milstein or Andersen (2008)'s QE) scheme where you have to jump a small time step (therefore, many jumps) in order to control bias. In the exact scheme, however, you have to sample the integrated variance from given the terminal variance. This step is quite slow because the distribution of the integrated variance is given by its Fourier transform (i.e., characteristic function), so you need to numerically invert to obtain the CDF. Caching the CDF is not a viable option here because the CDF is unique given the terminal variance. So the quick0-and-dirty time-discretization scheme is still a practical solution in terms of performance and implementation effort. Lord et al (2010) and Van Haastrecht & Pelsser (2010) have some comparison results including the exact scheme, so please take a look.

But there is another important algorithm improving the drawback of the exact scheme. Glasserman & Kim (2011) express the integrated variance as the infinite sums of gamma random variables, removing the bottleneck in the exact scheme. Of course, you can't do the infinite sums in numerical implementation. But, as long as you use a reasonable number (e.g., <10) of terms, the truncation error is quite small.

References:

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  • $\begingroup$ Thanks for the detailed answer! $\endgroup$
    – AZhu
    Jun 12 at 16:37

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