Pricing Discretely Monitored Barrier Options Under Markov Processes Using a Markov Chain Approximation

Abstract

We propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options whose underlying asset evolves according to a generic one-dimensional Markov process. This set of stochastic processes includes, but is not limited to, diffusion and jump diffusion processes commonly used in derivative pricing applications. The formula’s derivation combines the integral equation method, the Z−transform technique, and a continuous-time Markov chain approximation of the underlying Markov process. It does not require one to perform an intermediate numerical quadrature or related potentially runtime-intensive, error-prone, or otherwise complicated numerical procedure that may require a high degree of tuning to ensure appropriate accuracy (e.g. an inverse Laplace transform or inverse Z−transform). Rather, the price and Greeks of a discretely-monitored double barrier option may be 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. Examples including time-dependent barriers and non-uniform monitoring time intervals, may be seamlessly incorporated into this framework. In addition, by limiting the monitoring frequency to a large value, we also obtain an accurate closed-form formula for the price and Greeks of continuously-monitored double barrier options with time-dependent barriers under general Markov processes. Finally, we provide many 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.

Keywords: Discrete Barrier Options, Continuous-Time Markov Chains, Integral Equations, Z−Transform, Markov Process

JEL Classification: C58, G12, G13