Journal of Computational Finance

Risk.net

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

Conall O’Sullivan and Stephen O’Sullivan

  • A novel efficient lattice method
    for the numerical pricing of derivative securities is introduced.
  • ATTs are shown to be more efficient than selected state-of-the-art implicit methods when pricing one-factor American options.
  • When faced with complex numerical pricing problems, ATTs offer a compelling alternative to conventional implicit techniques.

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 toN 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 OE0; 1 on the substep scale but aggregate to probabilities within the range OE0; 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.

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