Bayesian Estimation of Continuous-Time Finance Models 1 Introduction

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.

[1]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[2]  Philippe Jorion.  On Jump Processes in the Foreign Exchange and Stock Markets , 1988 .

[3]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[4]  Peter E. Rossi,et al.  Bayesian Analysis of Stochastic Volatility Models , 1994 .

[5]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[6]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[7]  A. Gallant,et al.  ESTIMATION OF CONTINUOUS-TIME MODELS FOR STOCK RETURNS AND INTEREST RATES , 1997, Macroeconomic Dynamics.

[8]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[9]  Andrew L. Rukhin,et al.  Tools for statistical inference , 1991 .

[10]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[11]  Yacine Ait-Sahalia.  Testing Continuous-Time Models of the Spot Interest Rate , 1995 .

[12]  A. Pedersen.  A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations , 1995 .

[13]  Edward W. Frees,et al.  Estimating the Volatility of Discrete Stock Prices , 1988 .

[14]  G. Casella,et al.  Explaining the Gibbs Sampler , 1992 .

[15]  Peter E. Rossi,et al.  Stochastic Volatility: Univariate and Multivariate Extensions , 1999 .

[16]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[17]  W. Torous,et al.  On Jumps in Common Stock Prices and Their Impact on Call Option Pricing , 1985 .

[18]  John Geweke,et al.  Simulation-based Bayesian inference for economic time series , 1996 .

[19]  Jón Dańıelsson.  Stochastic volatility in asset prices estimation with simulated maximum likelihood , 1994 .

[20]  Bjørn Eraker.  MCMC Analysis of Diffusion Models With Application to Finance , 2001 .

[21]  L. Hansen,et al.  Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes , 1993 .

[22]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[23]  Campbell R. Harvey,et al.  An Empirical Comparison of Alternative Models of the Short-Term Interest Rate , 1992 .

[24]  D. Duffie,et al.  Simulated Moments Estimation of Markov Models of Asset Prices , 1990 .

[25]  P. Honoré.  Pitfalls in Estimating Jump-Diffusion Models , 1998 .

[26]  Gary Gottlieb,et al.  Implications of the Discreteness of Observed Stock Prices , 1985 .

[27]  C. Ball,et al.  Estimation Bias Induced by Discrete Security Prices , 1988 .

[28]  Louis O. Scott.  Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

[29]  Siddhartha Chib,et al.  Markov Chain Monte Carlo Simulation Methods in Econometrics , 1996, Econometric Theory.

[30]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[31]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[32]  A. Zellner.  An Introduction to Bayesian Inference in Econometrics , 1971 .

[33]  W. Torous,et al.  A Simplified Jump Process for Common Stock Returns , 1983, Journal of Financial and Quantitative Analysis.

[34]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[35]  Joel Hasbrouck,et al.  Security Bid/Ask Dynamics with Discreteness and Clustering: Simple Strategies for Modeling and Estimation , 1998 .

[36]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .