A Flexible Lattice Model for Pricing Contingent Claims under Multiple Risk Factors

Abstract

The original binomial model is an easy-to-apply approximation procedure for valuing options under Black-Scholes assumptions. There is a single stochastic factor and the volatility, interest rate, and other parameters are known. However, weakening those assumptions typically produces a non-recombining lattice that blows up asymptotically when the number of time steps is increased for closer replication of the underlying continuous-time process. Various extensions of the basic lattice structure have been developed over the years. In this article, Russo and Staino provide a very general lattice model in the form of a “forward-shooting grid” that can handle three correlated risk factors: volatility, interest rate, and stock price. An innovation in the model is that volatility and the riskless interest rate are the primary state variables, while the asset price (whose returns process is a function of both volatility and the riskless rate) is treated as an auxiliary variable. The lattice determines the possible evolution of the volatility and interest rate, and the stock price is carried along as a set of possible values falling into discrete buckets at each node. The trivariate branching structure is represented in a lattice that allows eight branches from each node. This accommodates many of the standard continuous-time models, including non-zero correlation among the stochastic factors. A simulation exercise shows striking improvement in performance relative to earlier models in the literature.