Simulation of Wishart processes and generalised Heston models

Abstract

The topic of this work is in studying the Wishart processes in the extended Heston model framework, which is commonly used as a pricing tool in finance. A volatility process, which is driven by the Wishart process, is simulated by the exact splitting scheme of Ahdida and Alfonsi [2]. We enhance their approach by adding the component of a time integral of volatility to the existing generator of the Wishart process. We also avoid the expensive matrix exponentiation procedure, that Ahdida and Alfonsi [2] state about, by the change of measure approach from Malham et al. [50] and Gnoatto and Grasselli [28]. Using the composition techniques from Ninomiya and Victoir [55], we construct the three new schemes to sample the pairs of volatility and its time integral. One of the schemes has a local error of order two, whereas the other two schemes are based on the Strang splitting with the local errors of order three. The change of measure approach of Gnoatto and Grasselli [28] helps us to transfer between the restricted and full parameter cases of the Wishart processes. The sampled pairs of volatility and its time integral are then used during the pricing procedure. Comparison of the moment generating functions with Gnoatto and Grasselli [28], and comparison of the call option prices with Leung [45], are used to show a correctness of the implemented schemes.EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis and Computation (project reference EP/S023291/1) grant 2277802