Pricing American Options by Willow Tree Method Under Jump-Diffusion Process

Abstract

Numerical solution methods for option pricing fall into two broad classes, either the lattice framework or Monte Carlo simulation. Monte Carlo simulation is problematic for American options, while the Binomial and Trinomial approaches have difficulty incorporating stochastic jumps. The probability density at expiration that is generated by a lattice typically has different skewness and kurtosis than is embedded in option market prices. The Binomial can be fitted imposing a terminal density with the right moments, using the Johnson system of distributions for example, but the problem still remains that the number of distinct nodes to calculate increases quadratically with the number of time steps. This article proposes the “Willow Tree” method, which combines a small number of large time steps that connect to many possible next period nodes, with the constraint that the terminal density, constructed from the Johnson system, has the same mean, volatility, skewness, and kurtosis as the risk neutral density extracted from prices in the options market. The result is a much faster and more accurate valuation than may be achieved through existing approaches.