The Almost-Exact Scheme (AES), as proposed by Oosterlee and Grzelak, has been applied to the Heston stochastic volatility model to show improved error convergence for small time-steps, as opposed to the classical Euler-Maruyama (EM) scheme, in European option pricing. This idea has been extended to the double Heston stochastic volatility model, to show similar improved results for Bermudan options. In this thesis, we extend this idea even further and develop an Almost-Exact Scheme to the Gatheral double mean reverting (DMR) model, to show improved error convergence for American put options. We illustrate that, because of the complexity of the dynamics of our model, a direct application of the AES is not possible, and therefore derive a diffusion trick, so we can instead use a partial implementation of the AES. Our partial implementation has two variants. In the first variant, we implement the AES on the long-run mean process combined with the Milstein scheme on the variance process. In the second variant, the Milstein scheme is replaced by a second order refinement. We name these two schemes AEMS and AEMS-SOR respectively. We conduct extensive simulation studies to evaluate the proposed schemes. The results indicate improved error convergence of the proposed scheme for small time-steps when time-to-maturity is equal to half a year, but does not seem to differ much from the EM scheme for a shorter time-to-maturity.

Black and Scholes have proposed a model for pricing European options where the underlying asset follows a so-called geometric Brownian motion which assumes constant volatility. The proposed Black-Scholes model has an exact solution. However, it has been shown that such an assumption of constant volatility is not realistic, and numerous extensions have been developed. In addition, models usually do not have a closed-form solution which makes pricing a challenging task. The thesis focuses on pricing Bermudan options under two stochastic volatility Heston-type models using an Almost-Exact scheme for simulation. Namely, we focus on deriving the Almost-Exact scheme for Heston and Double Heston model and numerically study the behaviour of the scheme. We show that the AES works well when the number of simulated steps is equal to the number of exercise dates which makes it efficient.