An explicit finite difference approach to the pricing of barrier options

Abstract

A modified explicit finite difference approach to the pricing of barrier options is developed. To obtain accurate prices, the grid is constructed such that the barrier is located in a suitable position relative to horizontal layers of nodes on the grid. This means that the barrier passes through a horizontal layer of nodes for continuous-time barrier options and is located halfway between two horizontal layers of nodes for discrete-time barrier options. Both single and double barrier cases can be accommodated. The option price at each node on the grid may be obtained by implementing a standard trinomial tree procedure. As the initial asset price will generally not lie exactly on the grid, the current value of the option is obtained using a quadratic interpolation of the option prices at the three adjacent nodes. The approach is shown to be robust and to provide accurate option prices and hedge ratios (such as delta, gamma, and theta) regardless of whether or not the barrier is close to the initial asset price, and it works effectively for both continuous-time and discrete-time barrier options. This device of adjusting the grid so that the barrier and the asset price lie on the grid is well known in the numerical analysis area.