Analytical Pricing American Call Option with Dividend

American options provide early exercise opportunities to pose the additional difficulty to obtain the closed-form solution. In this study, a recursive formula is developed for determining the optimal exercise price of American call options with dividend based on the backward dynamic programming recursions and the Black-Scholes model combined with the Martingale pricing and the Girsanov Theorem. This recursive formula can be applied to two kinds of American call option pricing methods. One is analytical (closed-form) solution method, and the other is numerically analytical solution method. Numerically analytical solution method is an approach using the recursive formula through backward iterations only so that it is unable to obtain a general closed-form solution of America call option. The other one applies the backward recursive formula and the Taylor Series expansion to determine a general closed-form solution of the American call option