In this thesis, we consider the pricing problem of an American put option. We introduce a new market model for the evolution of the underlying asset price. Our model adds a new parameter to the well known Heston model. Hence we name our model the extended Heston model. To solve the American put pricing problem we adapt the idea developed by Fouque et al. (2000) to derive the asymptotic formula. We then connect the idea developed by Medvedev and Scaillet (2010) to provide an asymptotic solution for the leading order term P0. We do numerical analysis to gain insight into the accuracy and validity of our asymptotic approximation formula.