Journal of Computational Finance

Risk.net

Pricing in three-factor models using icosahedral lattices

Lynda A. McCarthy, Nick J. Webber

ABSTRACT

In models with three or more factors it may be difficult to value path-dependent Bermudan options by finite-difference or Monte Carlo methods. We describe a lattice method for three-factor models based on icosahedral branching that samples the joint distribution of the underlying state variables more uniformly than standard lattice schemes and does not oversample the tails. Two versions are presented: a true icosahedral branching suitable for non-Markovian processes, and a recombining oblate icosahedral branching suitable for Markov processes. It is found that the method is considerably faster than simple branching schemes. In the non-recombining lattice we show how time steps can be freely chosen while we are still able to adjust the location of nodes at the terminal time according to a low-discrepancy sequence to avoid clustering. Illustrating the method on three-factor affine and Heath–Jarrow–Morton interest rate models we find that when certain conditions are met the icosahedral lattice can value options with acceptable accuracy and speed.

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