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

This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closed-form 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 numerical variance stabilizing transformation, computing derivatives of the simulated loglikelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications with annualized parameter values, where the diffusion model has an unknown transition density and has no analytical variance stabilizing transformation. The closed-form expansion, 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 tail of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closed-form approach when the observation frequency is lower. Both methods performs better when the volatility level is lower, but the simulation method is more robust to the volatility nature of the diffusion model. When applied to two well known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different loglikelihood in the case of higher volatility. ∗Osnat Stramer is Associate Professor (E-mail: [email protected]) and Jun Yan is Assistant Professor