George Dotsis, Raphael N. Markellos and Terence C. Mills 96119.3.1.2 Markov chain Monte Carlo
The Bayesian Markov chain Monte Carlo method was initially applied by Jacquier,
Polson and Rossi (1994) for the estimation of discrete-time stochastic volatility
models. However, the method has also become very popular in the estimation of
continuous-time models. Johannes and Polson (2006) provide an excellent survey
of MCMC applications in a variety of continuous-time asset-pricing models. In the
context of stochastic volatility, MCMC has been applied by, for example, Jones
(2003), Eraker, Johannes and Polson (2003) and Eraker (2004).
The output of the Bayesian MCMC is the posterior density,p(V,θ|y), of the
parameters and latent variables conditional on the data. The method quantifies
parameter uncertainty and model risk, filters out latent volatility, jump times and
jump sizes, and avoids optimization routines. However, under the MCMC it is
difficult to make comparisons across non-nested models such as SV2 and SV5.
Under the Bayesian approach the posterior densityp(V,θ|y)is proportional to:
p(y|V,)p(V|)p(), (19.17)wherep(y|V,)is the full likelihood,p(V|)is the density of the latent variable,
andp()is the prior density of the parameters. Sampling from the posterior density
is difficult because of the latent variable and, especially, because of the high dimen-
sion of the density. The MCMC method samples from the posterior by forming a
Markov chain overandVthat converges in distribution top(V,θ|y). In practice,
the posterior is further decomposed into the two conditional densities,p(V|y,)
andp(|y,V)(see Johannes and Polson, 2006). The two most popular MCMC
algorithms for sampling from the two conditionals are the Gibbs sampler and
Metropolis–Hastings. If the conditionals can be sampled directly then the MCMC
is performed with the Gibbs sampler; otherwise Metropolis–Hastings is applied.
19.3.2 Characteristic function methods
In stochastic volatility models the conditional density is rarely available in closed
form. However, for models that belong to the affine class it is feasible to derive the
conditional characteristic function. The advantage of the characteristic function
methodology is that usually it does not require discretization of the continuous-
time process. The characteristic function is a powerful tool that encodes the same
information as the conditional density. Though the characteristic function solves
the same Kolmogorov forward and backward equations as does the conditional
density, the boundary condition for the characteristic function is smoother and
this allows the derivation of the characteristic function in a tractable form.^8
The joint characteristic function conditional on current stock price and volatility
is defined as:
φ(
iω 1 ,iω 2 ,yt,Vt,τ;θ)
=E[
eiω^1 yt+τ+iω^2 Vt+τ|xt]. (19.18)
To keep notation simple, we concentrate on specification (19.1) without jumps. The
characteristic function of the process must satisfy the following partial differential