Pricing Discretely Monitored Barrier Options under Markov Processes through Markov Chain Approximation

Abstract

The authors propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options with an underlying asset that evolves according to a one-dimensional Markov process, which includes diffusion and jump-diffusion processes. The prices and Greeks of a discretely monitored double barrier option are explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice—for example, time-dependent barriers and nonuniform monitoring time intervals. They provide numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature in a unified framework.

TOPICS: Derivatives, options

Key Findings

• ▪ Markov chain approximations to stochastic processes are used to develop a simple closed-form pricing formula for discretely monitored barrier options.

• ▪ This article explores price and Greek computations within the constant elasticity of a variance model and Merton’s and Kou’s jump-diffusions models.

• ▪ Numerical examples are provided to demonstrate the robustness and accuracy of the pricing technique, both to recreate specialized results in related barrier option literature and to extend to multiple novel settings.