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In the option pricing process, Black-Scholes (1973) solved a partial differential equation and introduced a model to determine the price of an option. While dealing with many problems in financial engineering, the application of Partial Differential Equations (PDEs) is fundamental to explain the changes that occur in the evolved systems. In this paper, we consider the European call option pricing problem that involves a two-dimensional Black-Scholes PDE. We transform the final time condition presented in [7] and compare the numerical prices using Crank-Nicolson scheme with analytic approximation prices obtained for a European basket option. Conclusions related to different parameters effects are given based on obtained results.

Among other limitations, the celebrated Black--Scholes option pricingmodel assumes constant volatility and constant interest rates, which is not supportedby empirical studies on for example implied volatility surfaces. Studiesby many researchers such as Heston in 1993, Christoffersen in 2009, Fouque in2012, Chiarella--Ziveyi in 2013, and the authors' previous work removed the constantvolatility assumption from the Black--Scholes model by introducing one ortwo stochastic volatility factors with constant interest rate. In the present paperwe follow this line but generalize the model by considering also stochasticinterest rate. More specifically, the underlying asset process is governed by amean-reverting interest rate process in addition to two mean-reverting stochasticvolatility processes of fast and slow mean-reverting rates respectively. The focusis to derive an approximating formula for pricing the European option using adouble asymptotic expansion method.