Weak convergence of tree methods, to price options on defaultable assets

Abstract

We discuss a practical method to price and hedge European contingent claims on assets with price processes which follow a jump-diffusion. The method consists of a sequence of trinomial models for the asset price and option price processes which are shown to converge weakly to the corresponding continuous time jump-diffusion processes. The main difference with many existing methods is that our approach ensures that the intermediate discrete time approximations generate models which are themselves complete, just as in the Black-Scholes binomial approximations. This is only possible by dropping the assumption that the approximations of increments of the Wiener and Poisson processes on our trinomial tree are independent, but we show that the dependence between these processes disappears in the weak limit. The approximations thus define an easy and flexible method for pricing and hedging in jump-diffusion models using explicit trees for hedging and pricing.