Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model
International Journal of Theoretical and Applied Finance (IJTAF), 2010
Posted: 7 Jun 2010
Date Written: February 1, 2010
We deal with discretization schemes for the simulation of the Heston stochastic volatility model. These simulation methods yield a popular and flexible pricing alternative for pricing and managing a book of exotic derivatives which cannot be valued using closed-form expressions. For the Heston dynamics an exact simulation method was developed by Broadie and Kaya (2006), however we argue why its practical use is limited. Instead we focus on efficient approximations of the exact scheme, aimed to resolve the disadvantages of this method; one of the main bottlenecks in the exact scheme is the simulation of the Non-central Chi-squared distributed variance process, for which we suggest an efficient caching technique. At first sight the creation of a cache containing the inverses of this distribution might seem straightforward, however as the parameter space of the inverse Non-central Chi-squared distribution is three-dimensional, the design of such a direct cache is rather complicated, as pointed out by Broadie and Andersen. Nonetheless, for the case of the Heston model we are able to tackle this dimensionality problem and show that the three-dimensional inverse of the non-central chi-squared distribution can effectively be reduced to a one dimensional cache. The performed analysis hence leads to the development of three new efficient simulation methods (the NCI, NCI-QE and BK-DI scheme). Finally, we conclude with a comprehensive numerical study of these new schemes and the exact scheme of Broadie and Kaya, the almost exact scheme of Smith, the Kahl-Jäckel scheme, the FT scheme of Lord et al. and the QE-M scheme of Andersen. From these results, we find that the QE-M scheme is the most efficient, followed closely by the NCI-M, NCI-QE-M and BK-DI-M schemes, whilst we observe that all other considered schemes perform a factor 6 to 70 times less efficient than the latter four methods.
Keywords: Stochastic volatility, simulation, Heston, non-central chi-squared inversion, control variate
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