Accelerated trinomial trees applied to American basket options and American options under the Bates model

Abstract

The accelerated trinomial tree (ATT) is a derivatives pricing lattice method that circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs), in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps, each of which contains an inner lattice/trinomial tree with N nonuniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, is constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk-neutral extended probabilities: extended in the sense that values can lie outside the range [0, 1] on the substep scale but aggregate to probabilities within the range [0,1] on the superstep scale. Hence, it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient or have comparable efficiency to competing implicit methods that use L time steps in pricing Black–Scholes American put options and two-dimensional American basket options. Crucially, this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs, which may be relatively complex in their implementation. We also extend ATTs to the pricing of American options under the Heston model and the Bates model in order to demonstrate the general applicability of the approach.