Pricing the American put option: A detailed convergence analysis for binomial models

Viewing binomial models as a discrete approximation of the respective continuous models, the interest focuses on the notions of convergence and especially "fast" convergence of prices. Though many authors were proposing new models, none of them could successfully explain better performance for their models, since they all lacked a measure of convergence speed. In the case of the European call option Leisen and Reimer[96] examined convergence speed by the order of convergence in a rigorous framework. However the analysis could not be transformed to the case of American put options. For the model of Cox, Ross and Rubinstein[79], Lamberton[95] addressed the same problem. For American put options he derived that the error is bounded by suitable sequences with order of convergence 1/2 from above and by 2/3 from below. However, the simulation results of Broadie and Detemple[96] suggest order one. One aim of this paper is to improve this result and extend it to different lattice approaches. We establish the result that the model of Cox, Ross and Rubinstein[79] converges with order one. From a general theorem follows for the models of Jarrow and Rudd[83] and Tian[93] that the error is bounded by order one from above and 1/2 from below. Thus none of these three models performs better in comparison to the other. In a further step an error representation is derived using the concept of order of convergence. This allows an error analysis of extrapolation. Moreover we study the Control Variate technique introduced by Hull and White[88]. Since the investigation reveals the need for smooth converging models in order to get smaller initial errors, such a model is constructed. The different approaches are then tested: simulations exhibit up to 100 times smaller initial errors.