Numerical exploration of dynamic behavior of Ornstein–Uhlenbeck processes via Ehrenfest process approximation

Recently Ornstein–Uhlenbeck (O–U) processes have been drawing much attention in financial engineering for modeling stochastic behavior of spot interest rates. While transition probabilities of the O–U processes are readily accessible, it is numerically cumbersome to quantify their dynamic behavior much needed in certain applications, e.g., computing the prices of barrier options and the like in financial engineering. The purpose of this paper is to develop numerical procedures for evaluating distributions of first passage times and the historical maxima of the O–U processes via the Ehrenfest process approximation. Using the fact that a sequence of Ehrenfest processes with appropriate scaling and shifting converges in law to an O–U process, it is shown that first passage times and the historical maximum of the Ehrenfest processes converge in law to those of the O–U process. Through analysis of the spectral structure of the Ehrenfest process, efficient numerical algorithms are developed, thereby providing effective approximation tools for capturing the dynamic behavior of the O–U process. The proposed numerical algorithms are systematic in that the needed computations can be done repeatedly for different values of the underlying parameters with little alterations. Some numerical results are also exhibited, demonstrating speed and accuracy of the algorithms.