A Simple and Efficient Two-Factor Willow Tree Method for Convertible Bond Pricing with Stochastic Interest Rate and Default Risk

Abstract

Most numerical methods for option pricing (lattice methods, Monte Carlo simulation) focus on the market’s short-run dynamics, in which instantaneous returns cumulate to a probability distribution at maturity, as the Black–Scholes model does. If the risk-neutral density is available, it is possible to jump immediately to maturity, but this only works for options without path-dependence and for a small number of returns processes. The willow tree method offers a solution between these two cases. The option’s life is split into a small number of discrete chunks, and the price dimension at each of these time steps is represented by a finite and relatively small set of possible values. Each node allows a transition to every price node at the next step; thus, the “tree” resembles a willow, with many long branches connecting the sets of prices separated by discrete time intervals. For the problems it can handle, the willow tree approach has been shown to be accurate and much faster than procedures that require modeling an asymptotically infinite number of instantaneous transitions.

In this article, Lu and Xu show how to extend a willow tree to two stochastic factors, an underlying stock price and the interest rate, to apply the model to pricing convertible bonds. The procedure is amenable to many different returns processes. In a numerical illustration, the authors show that the two-factor willow tree is as accurate as a binomial or a Monte Carlo approach but requires much less computing time.