George Dotsis, Raphael N. Markellos and Terence C. Mills 967Duffie and Singleton (1993), the simulated maximum likelihood (SML) of Santa-
Clara (1995) and Pedersen (1995a), the range-based QML estimation technique
of Alizadeh, Brandt and Diebold (2002) and the Markov chain approximation of
Chourdakis (2002).
We believe there are two avenues for future research. First, more work needs to
be done on the comparison of non-affine and affine volatility models. Two ques-
tions (at least) suggest themselves – does analytical tractability come at the cost of
empirical misspecification?, and what is the impact of non-affine specifications on
derivatives valuation? Second, despite this plethora of statistical stochastic volatil-
ity models there is still a lack of understanding about the economic underpinnings
of volatility. Carr and Wu (2008) show that the market price of volatility risk is large
and negative, and Bollerslev and Zhou (2007) find that the volatility risk premium
forecasts future excess returns. However, there is still no solid explanation of the
economic sources of volatility risk premia and there is also a lack of understand-
ing of the behavior of the pricing kernel as a function of both market returns and
return volatilities.
Notes
- There is a wide variety of interpretations of the term “volatility” within the financial and
econometrics literature: for example, variance, (annualized) standard deviation, (total)
risk, uncertainty, and so on. In the context of continuous-time models and throughout
this chapter, the term is used to describe the latent instantaneous variance. - Baillie (2006), for example, provides a recent survey of ARCH/GARCH modeling.
- See Sundaresan (2000) for a comprehensive review of the development and application
of continuous-time methods in finance and Aït-Sahalia (2007) for a survey of estima-
tion methods in continuous-time models. Merton (1990) is the benchmark book in
continuous-time finance. - Ghysels, Harvey and Renault (1996) provide an extensive review of stochastic volatility
models, but they are mainly concerned with models defined in discrete time. - VXO is the implied volatility of a synthetic at-the-money option on the S&P500 equity
index with a constant time to maturity of 30 calendar days to expiry. In 2003, the Chicago
Board Options Exchange (CBOE) introduced a new implied volatility index, coined the
VIX. This is calculated in a model-free manner as a weighted sum of out-of-money option
prices across all available strikes on the S&P500 index. Carr and Wu (2006) show that
the VIX represents the conditional risk-neutral expectation of the return volatility under
general market settings. In 2005 CBOE introduced futures on VIX and, in 2006, European
calls/puts written on forward VIX. - Psychoyios, Skiadopoulos and Alexakis (2003) provide a comprehensive review of
alternative volatility processes. - See Andersen, Benzoni and Lund (2002) for discussion on the simulation of diffu-
sion/jump stochastic volatility models. - The characteristic function methodology has also been used for parameter estimation
in discrete time independent and identically distributed (i.i.d) and autoregressive mov-
ing average (ARMA) processes by Feuerverger and McDunnough (1981a, 1981b) and
Feuerverger (1990). - The realized volatilities are based on intraday transactions on the S&P500 index. This
dataset has been used by Huang, Liu and Yu (2007) and is available from Professor
Yu’s web page (http://www.mysmu.edu/faculty/yujun/research.html). The construction