MCMC Analysis of Diffusion Models With Application to Finance

This article proposes a new method for estimation of parameters in diffusion processes from discrete observations. The method is based on Markov-chain Monte Carlo methodology and applies to a wide class of models including systems with unobservable state variables and nonlinearities. The method is applied to the estimation of parameters in one-factor (CEV) interest-rate models and a two-factor model with a latent stochastic volatility component (SV). The CEV model is found to do a poor job in capturing the time-varying volatility interest-rate data. The SV model provides vastly superior fit to that of the CEV model. The article also presents a simulation study that demonstrates that the method provides accurate parameter estimates of the SV model at moderate sample sizes.

[1]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[2]  Arnold Zellner,et al.  Bayesian Analysis of Regression Error Terms , 1975 .

[3]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

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

[5]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[6]  L. Collatz,et al.  Differential equations: An introduction with applications , 1986 .

[7]  Peter Guttorp,et al.  Estimation in models for security prices , 1987 .

[8]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

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

[10]  Valentine Genon-Catalot,et al.  Maximnm contrast estimation for diffusion processes from discrete observations , 1990 .

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

[12]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[13]  N. Yoshida.  Estimation for diffusion processes from discrete observation , 1992 .

[14]  Nicholas G. Polson,et al.  A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling , 1992 .

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

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

[17]  W. Gilks,et al.  Adaptive rejection sampling from log-concave density functions , 1993 .

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

[19]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .

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

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

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

[23]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

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

[25]  Yacine Aït-Sahalia.  Nonparametric Pricing of Interest Rate Derivative Securities , 1995 .

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

[27]  W. Gilks,et al.  Adaptive Rejection Metropolis Sampling Within Gibbs Sampling , 1995 .

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

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

[30]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[31]  T. Bollerslev,et al.  MODELING AND PRICING LONG- MEMORY IN STOCK MARKET VOLATILITY , 1996 .

[32]  Yacine Aït-Sahalia.  Nonparametric Pricing of Interest Rate Derivative Securities , 1996 .

[33]  Bo Martin Bibby,et al.  On Estimation for Discretely Observed Diffusions: A Review , 1996 .

[34]  Charles J. Geyer,et al.  Estimation and Optimization of Functions , 1996 .

[35]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[36]  G. J. Jiang,et al.  A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model , 1997, Econometric Theory.

[37]  T. Andersen,et al.  Estimating continuous-time stochastic volatility models of the short-term interest rate , 1997 .

[38]  Mathieu Kessler.  Estimation of an Ergodic Diffusion from Discrete Observations , 1997 .

[39]  Michael Sørensen,et al.  Estimating Functions for Discretely Observed Diffusions: A Review , 1997 .

[40]  Timothy G. Conley,et al.  Short-term interest rates as subordinated diffusions , 1997 .

[41]  Richard Stanton.  A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk , 1997 .

[42]  T. Andersen,et al.  Stochastic Volatility and Mean Drift in the Short Rate Diffusion : Sources of Steepness , Level and Curvature in the Yield Curve , 1998 .

[43]  Yacine Aït-Sahalia.  Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach , 1998 .

[44]  G. Roberts,et al.  Adaptive Markov Chain Monte Carlo through Regeneration , 1998 .

[45]  Christopher T. Downing.  Nonparametric Estimation of Multifactor Continuous Time Interest-Rate Models , 1999 .

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

[47]  Chris Downing.  Nonparametric estimation of multifactor continuous time interest rate models , 1999 .

[48]  George Tauchen,et al.  Using Daily Range Data to Calibrate Volatility Diffusions and Extract the Forward Integrated Variance , 1999 .