Palgrave Handbook of Econometrics: Applied Econometrics

George Dotsis, Raphael N. Markellos and Terence C. Mills 963

and estimates various diffusion/jump diffusion stochastic volatility models. His
method does not require any simulations and, in contrast to previous approaches,
also allows the filtration of latent volatility.

19.3.3 Derivatives markets

The development of derivatives markets provides an alternative source for filtering
latent volatility and estimating the parameters. Some studies use only infor-
mation from the derivatives markets, whereas others use both option prices and
historical returns.
For affine diffusion/jump diffusion stochastic volatility models, European option
prices can be obtained in closed form, up to numerical integration, using the char-
acteristic function methodology. For example, the price of a European call option
that depends on the parameter vectoris given by:

C(·;∗)=SP 1 −Xe−rtP 2. (19.22)

Here,P 1 andP 2 are cumulative probability functions that can be calculated by
Fourier inversion of the characteristic function (see Duffieet al., 2000) and∗is
the set of risk neutral parameters. For a non-affine process, option prices can be
calculated either by Monte Carlo simulation (for example, Christoffersen, Jacobs
and Mimouni, 2006) or by solving the associated PDE. Many studies (for example,
Bakshi, Cao and Chen, 1997; Broadieet al., 2007) calibrate option pricing models
with stochastic volatility to observed options prices via least squares:

SSE(t)=min

θ

∑N

i= 1

(

Oi−Oˆi(∗)

) 2

, (19.23)

whereNis the number of options at datet,Oiis the observed option price with
strike priceK, andOˆi()is the model’s implied price given the parameters∗. Here
we should note that the set of parameters∗is not the same as the set obtained
from historical returns because option prices are calculated under the risk-neutral
probability measure. Hence, option prices contain volatility and jump risk premia
that are incorporated into certain parameters. For example, under the assumption
that the volatility risk premium is linear in volatility (Heston, 1993), the parameters
of the SV2 model from calibration would bek∗=k+λandθ∗=kθ/(k+λ), whereλ
is the market price of volatility risk. Calibration of the stochastic volatility model to
option prices alone does not allow the identification of the risk premium parameter.
Another drawback of this approach is that the model has to be recalibrated every
day, which will usually produce different parameter estimates.
Another strand of the literature uses options prices and historical returns simul-
taneously. The advantage of this approach is that the time seriesVtcan be taken
from option prices and it is also feasible to identify risk premia. This approach also
imposes consistency between option prices and the time series properties of the
underlying returns (see Bates, 1996a). Some well known studies that use option
prices and historical returns are Chernov and Ghysels (2000), who use EMM, Pan
(2002), using a GMM procedure, and Eraker (2004), employing MCMC. In a recent