Kim, Shephard, and Chib (1998) provided a Bayesian analysis of stochastic volatility models based on a fast and reliable Markov chain Monte Carlo (MCMC) algorithm. Their method ruled out the leverage effect, which is known to be important in applications. Despite this, their basic method has been extensively used in the financial economics literature and more recently in macroeconometrics. In this paper we show how the basic approach can be extended in a novel way to stochastic volatility models with leverage without altering the essence of the original approach. Several illustrative examples are provided.

This chapter develops Markov Chain Monte Carlo (MCMC) methods for Bayesian inference in continuous-time asset pricing models. The Bayesian solution to the inference problem is the distribution of parameters and latent variables conditional on observed data, and MCMC methods provide a tool for exploring these high-dimensional, complex distributions. We first provide a description of the foundations and mechanics of MCMC algorithms. This includes a discussion of the Clifford-Hammersley theorem, the Gibbs sampler, the Metropolis-Hastings algorithm, and theoretical convergence properties of MCMC algorithms. We next provide a tutorial on building MCMC algorithms for a range of continuous-time asset pricing models. We include detailed examples for equity price models, option pricing models, term structure models, and regime-switching models. Finally, we discuss the issue of sequential Bayesian inference, both for parameters and state variables.

In this paper we develop a general simulation-based approach to filtering and sequential parameter learning. We begin with an algorithm for filtering in a general dynamic state space model and then extend this to incorporate sequential parameter learning. The key idea is to express the filtering distribution as a mixture of lag-smoothing distributions and to implement this sequentially. Our approach has a number of advantages over current methodologies. First, it allows for sequential parmeter learning where sequential importance sampling approaches have difficulties. Second

This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es- timating continuous-time asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tutorial on building MCMC algo- rithms and show how to estimate equity price models with factors such as stochastic expected returns, stochastic volatility and jumps, multi-factor term structure models with stochastic volatility, time-varying central tenclancy or jumps and regime switching models.

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The object of this paper is to produce distributional forecasts of physical volatility and its associated risk premia using a non-Gaussian, non-linear state space approach. Option and spot market information on the unobserved variance process is captured by using dual ‘model-free’ variance measures to define a bivariate observation equation in the state space model. The premium for di¤usive variance risk is defined as linear in the latent variance (in the usual fashion) whilst the premium for jump variance risk is specified as a conditionally deterministic dynamic process, driven by a function of past measurements. The inferential approach adopted is Bayesian, implemented via a Markov chain Monte Carlo algorithm that caters for the multiple sources of nonlinearity in the model and the bivariate measure. The method is applied to empirical spot and option price data for the S&P500 index over the 1999 to 2008 period, with conclusions drawn about investors’required compensation for variance risk during the recent financial turmoil. The accuracy of the probabilistic forecasts of the observable variance measures is demonstrated, and compared with that of forecasts yielded by

Corporation (in alphabetical order). This financial support enables us to issue CARF Working

The object of this paper is to model and forecast both objective volatility and its associated risk premium using a non-Gaussian state space approach. Option and spot market information on the unobserved volatility process is captured via nonparametric, ‘model-free ’ measures of option-implied and spot price-based volatility, with the two measures used to define a bivariate observation equation in the state space model. The risk premium parameter is specified as a conditionally deterministic dynamic process, driven by past ‘observations ’ on the volatility risk premium. The inferential approach adopted is Bayesian, implemented via a Markov chain Monte Carlo (MCMC) algorithm that caters for the non-linearities in the model and for the multimove sampling of the latent volatilities. The simulation output is used to estimate predictive distributions for objective volatility, the instantaneous risk premium and the conditional risk premium associated with a one month option maturity. Linking the volatility risk premium parameter to the risk aversion parameter in a representative agent model, we also produce forecasts of the relative risk aversion of a representative investor. The methodology is applied both to artifically simulated data and to empirical spot and option price data for the S&P500 index over the 2004 to 2006 period.

Simulation smoothing involves drawing state variables (or innovations) in discrete time state-space models from their conditional distribution given parameters and observations. Gaussian simulation smoothing is of particular interest, not only for the direct analysis of Gaussian linear models, but also for the indirect analysis of more general models. Several methods for Gaussian simulation smoothing exist, most of which are based on the Kalman filter. Since states in Gaussian linear state-space models are Gaussian Markov random fields, it is also possible to apply the Cholesky Factor Algorithm to draw states. This algorithm takes advantage of the band diagonal structure of the Hessian matrix of the log density to make efficient draws. We show how to exploit the special structure of state-space models to draw latent states even more efficiently.

This chapter reviews the major contributions over the last two decades to the literature on the Bayesian analysis of stochastic volatility (SV) models (univariate and multivariate). Bayesian inference is performed by tailoring Markov chain Monte Carlo (MCMC) or sequential Monte Carlo (SMC) schemes that take into account the specific modeling characteristics. The popular univariate stochastic volatility model with first order autoregressive dynamics (SV) is introduced in Section 1, which provides a detailed explanation of efficient MCMC and SMC algorithms. We briefly describe several extensions to the basic SV model that allows for fat-tailed, skewed, correlated errors as well as jumps (Markovian or not, smooth or not) in both observation and volatility equations, and the leverage effect via correlated errors. Multivariate SV models are presented in Section 2 with particular emphasis on Wishart random processes, cholesky stochastic volatility models and factor stochastic volatility models. Section 3 contains several illustrations of both univariate and multivariate SV models based on both MCMC and SMC algorithms. Section 4 concludes the chapter. 1 Univariate SV models Univariate stochastic volatility (SV) asset price dynamics results in the movements of an equity index St and its stochastic volatility vt via a continuous time diffusion by a Brownian motion