Palgrave Handbook of Econometrics: Applied Econometrics

George Dotsis, Raphael N. Markellos and Terence C. Mills 955

implied volatility VXO over the period 1990–2007.^5 The relationship appears to

be negative, especially during times of market stress. Derivatives markets facil-

itate empirical estimation by providing an alternative source for backing-out

latent volatility (for example, Pan, 2002; Aït-Sahalia and Kimmel, 2006).

The rest of the chapter is structured as follows. In section 19.2 we discuss the
properties of some popular stochastic volatility models. Section 19.3 is devoted to
the econometric methods used for drawing inferences in stochastic volatility mod-
els. First, we review estimation methods when volatility is treated as unobserved,
such as efficient method of moments (EMM), Markov chain Monte Carlo (MCMC),
and methods based on the empirical characteristic function (ECF). Subsequently,
we discuss methods that incorporate information from the derivatives markets into
the estimation procedure. Lastly, we review some recent methods that allow infer-
ences to be made in stochastic volatility models using high frequency data. In
section 19.4 we conduct an empirical comparison of various stochastic volatility
models. Section 19.5 concludes and provides directions for future research.

19.2 Volatility specifications

We assume that the logarithms of an asset,yt=ln(St), and its latent volatility,
Vt, evolve over time according to the following general jump diffusion stochastic
volatility model:

dyt=

(

μ−

1

2

Vt

)

dt+

√

VtdWSt+yStdNtS

dVt=a(Vt,t)dt+σ(Vt,t)dWVt +ytVdNtV. (19.1)

Here,WStandWVt are standard correlated Brownian motions.NtSandNtVare Pois-
son processes uncorrelated with these Brownian motions, with constant intensities

λyandλv, andyStandytVare the jump sizes of the asset return and volatility, respec-

tively. WhenNtS=NVt, jumps in asset returns and volatility occur simultaneously,

and whenNtS=NtV, the jump times are independent. The termsa(·)andσ(·)are
the drift and diffusion functions of the volatility process, respectively. The parame-
ter vector of (19.1) is denoted as. For simplicity, we assume that the mean return
of the asset,μ, is constant, although some models allow the conditional mean
return to be a linear function of volatility, as in Merton (1980). In the late 1980s,
stochastic volatility models were developed by assuming that the processes driving
volatility and asset prices had continuous paths, that is, they followed diffusion
processes. By the late 1990s, new types of stochastic volatility models had been
introduced which were based on jump diffusion processes for the underlying asset
price and, more recently, there have appeared stochastic volatility models that are
founded on “double jump” processes, where both the underlying asset price and
volatility follow jump-diffusion processes. In the following sub-sections we discuss
various specifications of thea(·)andσ(·)terms that are nested within the general
specification (19.1).^6