Maximum Likelihood and Gaussian Estimation of Continuous Time Models in Finance
This paper overviews maximum likelihood and Gaussian methods of estimating continuous time models used in ﬁnance. Since the exact likelihood can be constructed only in special cases, much attention has been devoted to the development of methods designed to approximate the likelihood. These approaches range from crude Euler-type approximations and higher order stochastic Taylor series expansions to more complex polynomial-based expansions and inﬁll approximations to the likelihood based on a continuous time data record. The methods are discussed, their properties are outlined and their relative ﬁnite sample performance compared in a simulation experiment with the nonlinear CIR diﬀusion model, which is popular in empirical ﬁnance. Bias correction methods are also considered and particular attention is given to jackknife and indirect inference estimators. The latter retains the good asymptotic properties of ML estimation while removing ﬁnite sample bias. This method demonstrates superior performance in ﬁnite samples.