Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method

In this paper we propose a Bayesian method to estimate the hyperbolic diffusion model. The approach is based on the Markov chain Monte Carlo (MCMC) method with the likelihood of the discretized process as the approximate posterior likelihood. We demonstrate that the MCMC method Provides a useful tool in analysing hyperbolic diffusions. In particular, quantities of posterior distributions obtained from the MCMC outputs can be used for statistical inference. The MCMC method based on the Milstein scheme is unsatisfactory. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the discrete-time financial econometrics literature, such as the Taylor effect, a slowly declining autocorrelation function of the squared returns, and thick tails.

[1]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[2]  Michael M. Sørensen,et al.  A hyperbolic diffusion model for stock prices , 1996, Finance Stochastics.

[3]  Sylvia Richardson,et al.  Markov chain concepts related to sampling algorithms , 1995 .

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

[5]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

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

[7]  G. Mil’shtein.  A Method of Second-Order Accuracy Integration of Stochastic Differential Equations , 1979 .

[8]  Jun Yu,et al.  Bugs for a Bayesian Analysis of Stochastic Volatility Models , 2000 .

[9]  John Geweke,et al.  Using Simulation Methods for Bayesian Econometric Models , 1999 .

[10]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[11]  David J. Spiegelhalter,et al.  Introducing Markov chain Monte Carlo , 1995 .

[12]  Siddhartha Chib,et al.  MARKOV CHAIN MONTE CARLO METHODS: COMPUTATION AND INFERENCE , 2001 .

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

[14]  Ola Elerian,et al.  A note on the existence of a closed form conditional transition density for the Milstein scheme , 1998 .

[15]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[16]  Tina Hviid Rydberg.  Generalized Hyperbolic Diffusion Processes with Applications in Finance , 1999 .

[17]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[18]  Philip Heidelberger,et al.  Simulation Run Length Control in the Presence of an Initial Transient , 1983, Oper. Res..

[19]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[20]  John Geweke,et al.  Federal Reserve Bank of Minneapolis Research Department Staff Report 249 Using Simulation Methods for Bayesian Econometric Models: Inference, Development, and Communication , 2022 .

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

[22]  Nicholas G. Polson,et al.  The Impact of Jumps in Volatility and Returns , 2000 .

[23]  T. Rydén,et al.  Stylized Facts of Daily Return Series and the Hidden Markov Model , 1998 .

[24]  M. Sørensen,et al.  Martingale estimation functions for discretely observed diffusion processes , 1995 .

[25]  A. Sokal.  Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms , 1997 .

[26]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..