On Simulated Likelihood of Discretely Observed Diffusion Processes and Comparison to Closed-Form Approximation

Abstract

This article focuses on two methods to approximate the log-likelihood of discretely observed univariate diffusions: (1) the simulation approach using a modified Brownian bridge as the importance sampler, and (2) the closed-form approximation approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating the numerical variance stabilizing transformation, computing derivatives of the simulated log-likelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications under a benchmark model which has an unknown transition density and has no analytical variance stabilizing transformation. The closed-form approximation, particularly the second-order closed-form, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tails of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closed-form approximation when the observation frequency is lower. Both methods perform better when the volatility level is lower, but the simulation method is more robust to the volatility level. When applied to two well-known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different log-likelihoods in the case of higher volatility.