Binomial trees as the number of time steps is increased (or equivalently as the time step tends to 0), converge to the exact value for an option. So why do people use FDM for pricing options (for example an American Put), if Binomial Trees give already accurate results and converges quickly?
Actually recombining binomial trees are only a particular case of an explicit FDM scheme. But they have obvious limitations, the foremost being that they cannot accomodate local volatilities. Also 1/2 explicit 1/2 implicit FDM schemes (Crank-Nicolson) have faster convergence with respect to the size of the time step. And FDM schemes can accomodate all sorts of boundary conditions including Dirichlet which is necessary to accurately price barrier options.
Our company chose to use FDM for calculating American Options. According to colleagues I talked with, binomial trees are efficient and accurate When there are a small number of option values. But it has a couple of weaknesses:
(1) Binomial tree models are generally inefficient when cash dividends should be taken into consideration;
(2) Compared with FDM, binomial trees are less efficient for multiple options valuations;
(3) Additionally, binomial trees are inefficient in valuing American options compared with European options.
Please correct me if I'm wrong.