Trade-Offs in Exact Heston Simulation
Summary
The document explains why Broadie–Kaya exact simulation is not automatically the practical choice for the Heston stochastic volatility model. Its main advantage is that it can simulate across arbitrary time steps without the time-discretization bias associated with schemes such as Euler, Milstein, or Andersen’s QE method. This can reduce the need for many small simulation steps.
The main drawback described is computational cost and implementation complexity. Sampling integrated variance conditional on terminal variance requires numerical inversion of a characteristic function, and the resulting distribution cannot readily be reused across different terminal variances. The answer therefore presents time-discretized methods as practical when speed and implementation effort matter. It also points to the Glasserman–Kim gamma expansion as an alternative: a finite truncation approximates the infinite sum and introduces truncation error. The document cites comparison studies but gives no performance measurements or detailed accuracy results, so method choice depends on the simulation’s precision and runtime needs.
Key ideas
- Broadie–Kaya simulates Heston paths across arbitrary time steps, avoiding ordinary time-discretization bias.
- Its integrated variance sampling step relies on numerical inversion and can be slow.
- The conditional distribution varies with terminal variance, limiting the usefulness of caching.
- Time-discretized schemes trade some bias control for simpler and faster implementation.
- A truncated gamma expansion offers an alternative approximation with truncation error.
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# Problems with exact Heston simulations # Problems with exact Heston simulations 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! ## Answer by jaehyukchoi49 (score 3, accepted) https://quant.stackexchange.com/a/71126 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: - Andersen L (2008) Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance 11:1–42. https://doi.org/10.21314/JCF.2008.189 - Broadie M, Kaya Ö (2006) Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes. Operations Research 54:217–231. https://doi.org/10.1287/opre.1050.0247 - Glasserman P, Kim K-K (2011) Gamma expansion of the Heston stochastic volatility model. Finance Stoch 15:267–296. https://doi.org/10.1007/s00780-009-0115-y - Lord R, Koekkoek R, Dijk DV (2010) A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance 10:177–194. https://doi.org/10.1080/14697680802392496. - Van Haastrecht A, Pelsser A (2010) Efficient, almost exact simulation of the Heston stochastic volatility model. Int J Theor Appl Finan 13:1–43. https://doi.org/10.1142/S0219024910005668.
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