Numerical Evaluation of Dynamic Behavior of Ornstein–Uhlenbeck Processes Modified by Various Boundaries and its Application to Pricing Barrier Options

In financial engineering, one often encounters barrier options in which an action promised in the contract is taken if the underlying asset value becomes too high or too low. In order to compute the corresponding prices, it is necessary to capture the dynamic behavior of the associated stochastic process modified by boundaries. To the best knowledge of the authors, there is no algorithmic approach available to compute such prices repeatedly in a systematic manner. The purpose of this paper is to develop computational algorithms to capture the dynamic behavior of Ornstein-Uhlenbeck processes modified by various boundaries based on the Ehrenfest approximation approach established in Sumita et al. (J Oper Res Soc Jpn 49:256–278, 2006). As an application, we evaluate the prices of up-and-out call options maturing at time τ _M with strike price K _S written on a discount bond maturing at time T, demonstrating the usefulness, speed and accuracy of the proposed computational algorithms.