The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

This thesis aims to address the Kusuoka approximation (K-approximation) within the Gatheral model using Yamada’s method, also known as the Gaussian K-approximation. Our approach begins by transforming the original Gatheral model into a model with independent Wiener processes through Cholesky decomposition. Subsequently, the system is reformulated into its Stratonovich form, facilitating the definition of vector fields and their exponentials. We will assess whether the system satisfies the Uniformly Finitely Generated (UFG) condition. Additionally, based on our calculations, a simulation code will be developed to compare our results with those obtained by Yamada.

In this thesis, our research focus on a weak second order stochastic Runge–Kutta method applied to a system of stochastic differential equations known as the Gatheral Model. We approximate numerical solutions to this system and investigate the rate of convergence of our method. Both call and put options are priced using Monte-Carlo simulation to investigate the order of convergence. The numerical results show that our method is consistent with the theoretical order of convergence of the Monte-Carlo simulation. However, in terms of the Runge-Kutta method, we cannot accept the consistency of our method with the theoretical order of convergence without further research.