A Comparative Study of Likelihood Approximations for Univariate Diffusions

Maximum likelihood estimation of the parameters of stochastic differential equations commonly used in finance requires numerical approximation of their transitional probability density functions. This article undertakes a comparative study of the accuracy of Hermite polynomial expansion approximations for univariate diffusions and checks how the accuracy of the existing methods responds to increasing the order of the approximation. It is found that one class of expansion dealing with irreducible diffusions is particularly problematic due to the need to evaluate a number of troublesome integrals. A Gaussian quadrature is introduced which resolves the problem and improves the reliability of the expansion. A simulation study demonstrates all the methods in action and provides insight into the practical aspects of using these expansions. An empirical application using data on the VIX indicates that the proposed method based on the Gaussian quadrature performs very well when applied to financial data.