Bayesian Estimation of Continuous-Time Finance Models

A new Bayesian method is proposed for the analysis of discretely sampled diffusion processes. The method, which is termed high frequency augmentation (HFA), is a simple numerical method that is applicable to a wide variety of univariate or multivariate diffusion and jump-diffusion processes. It is furthermore useful when observations are irregularly observed, when one or more elements of the multivariate process are latent, or when microstructure effects add error to the observed data. The Markov chain-Monte Carlo-based procedure can be used to attain the posterior distributions of the parameters of the drift and diffusion functions as well as the posteriors of missing or latent data. Several examples are explored. First, posteriors of the parameters of a geometric Brownian motion are attained using HFA and compared with those obtained using standard analytical methods in a short Monte Carlo study. Second, a stochastic volatility model is estimated on a sample of S&P 500 returns, a problem for which posteriors are analytically intractable. Third, it is shown how the method can be used to estimate an interest rate process using data that suffer from severe rounding. Finally, extension of the method to jump-diffusions is described and applied to the analysis of the U.S dollar/German mark exchange rate. ∗I thank Michael Brandt, Valentina Corradi, Frank Diebold, Bjorn Eraker, Eric Jacquier, Craig MacKinlay, Robert Stambaugh, and participants at the 1998 North American summer meetings of the Econometric Society for their comments and help. All errors remain my responsibility.