Pricing Discrete European Barrier Options Using Lattice Random Walks

Abstract

This paper designs a numerical procedure to price discrete European barrier options in Black-Scholes model. The pricing problem is divided into a series of initial value problems, one for each monitoring time. Each initial value problem is solved by replacing the driving Brownian motion by a lattice random walk. Some results from the theory of Besov spaces show that the convergence rate of lattice methods for initial value problems depends on two factors, namely the smoothness of the initial value (or the value function) and the moments for the increments of the lattice random walk. This fact is used to obtain an efficient method to price discrete European barrier options. Numerical examples and comparisons with other methods are carried out to show that the proposed method yields fast and accurate results.

Discrete-Time Quadratic Hedging of Barrier Options in Exponential Lévy Model

“…For barriers in close proximity of the stock price the Markov chain representation of stock prices developed by [15] is more appropriate. Other papers dealing with discrete monitoring in the log-normal framework include [23], [25], [28], [29] and [38]. [1] describe a systematic way of handling discretization errors by means of quadrature.…”