The European option can be exercised only at the expiration date while an American option can be exercised on or at any time before the expiration date.
In this paper, we will study the numerical solutions of a class of complex partial diﬀerential equations (PDE) systems with free boundary conditions. This kind of problems arise naturally in pricing (ﬁnite-maturity) American options, which is applies to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump-diﬀusion (HEJD) and the ﬁnite moment log stable (FMLS) models. Developing eﬃcient numerical schemes will have signiﬁcant applications in ﬁnance computation. These equations have already been solve by the Hybrid Laplace transformﬁnite diﬀerence methods and the Laplace transform method(LTM). In this paper we will introduce a method to solve these equations by Tau method. Also, we will show that using this method will end up to a faster convergence. Numerical examples demonstrate the accuracy and velocity of the method in CEV models.