The Application Of MCMC Method In Term Structure Models

Dynamic asset pricing theory use arbitrage and equilibrium arguments to derive thefunctional relationship between asset prices and the fundamentals of the economy: state variables,structural parameters and market prices of risk. Continuous-time models are the centerpiece ofthis approach due to their analytical tractability. In many cases, these models lead to closed formsolutions or easy to solve differential equations for objects of interest such as prices or optimalportfolio weights.Empirical analysis of dynamic asset pricing models tackles the inverse problem: extractinginformation about latent state variables, structural parameters and market prices of risk fromobserved prices. The Bayesian solution to the inference problem is the distribution of theparameters,, and state variables, X, conditional on observed prices, Y. This posterior distribution,p，, combines the information in the model and the observed prices and is the key toinference on parameters and state variables.This chapter describes Markov Chain Monte Carlo (MCMC) methods for exploring theposterior distribution generated by continuous-time asset pricing models. MCMC samples fromthese high-dimensional, complex distribution by generating a Markov Chain over，X,whose equilibrium is p θ，X Y. The Monte Carlo method uses thesesamples for numerical integration for parameter estimation, state estimation and modelcomparison.Characterizing p θ，X Y in continuous-time asset pricing models is difficult for a varietyof reasons. First, prices are observed discretely while the theoretical models specify that pricesand state variables evolve continuously in time. Second, in many cases, the state variables arelatent from the researcher’s perspective. Third, p θ，X Y is typically of very high dimensionand thus standard sampling methods commonly fail. Fourth, especially for the interest rate termstructure models, parameters are non-normal and non-standard. In this chapter we will show thatMCMC methods tackle all of these issues.In the second and third part of the chapter: we will give a brief overview of the Bayesian inference and MCMC methods. In the fourth part of the chapter, we will focus on the specificapplication of the MCMC methods in the Regime switching model.