Improving speed of convergence for the prices of European options in binomial trees with even numbers of steps

A new family of binomial trees with even numbers of steps as approximations to the Black–Scholes model is introduced. For this class of trees, the existence of asymptotic expansions for the prices of vanilla and digital European options is demonstrated. The rate of convergence is analyzed by discussion of the different cases of the spot stock price. As spacial cases, a tree with third order convergence is constructed in detail. The existence proof of any finite integer order of convergence is given.