Palgrave Handbook of Econometrics: Applied Econometrics

964 Continuous-Time Stochastic Volatility Models

breakthrough paper, Aït-Sahalia (2008) derives transition density approximations
for multivariate diffusions. In Aït-Sahalia and Kimmel (2007) this method is applied
for maximum likelihood estimation of a variety of stochastic volatility models,
both affine and non-affine. Aït-Sahalia and Kimmel (2007) provide approxima-
tions for the joint density of asset returns and a vector of option prices or the joint
density of asset prices and an implied volatility index, which is used as a proxy for
latent volatility.

19.3.4 Integrated volatility

Recent advances in high frequency financial econometrics offer alternative meth-
ods for making inferences about latent stochastic volatility dynamics. Results from
the theory of quadratic variation (e.g., Andersenet al., 2001; Barndorff-Nielsen and
Shephard, 2002) show that, when the sampling frequency becomes high, realized
integrated volatility converges to the unobserved true integrated volatility. Figure
19.3 shows the evolution of the daily integrated volatility for the S&P500 over the
period 1993 to 2004.^9 Integrated volatility over a period [t,T] is computed from
summing high frequency log returns and is given by:

lim

N→∞

∑^2 N

i= 1

(

yt+i

2 n(T−t)

−yt+i− 1

2 N(T−t)

) 2

a.s.

→ IVt,T=

1

T−t

∫T

t

Vsds. (19.24)

Expression (19.24) contains rich information with respect to the stochastic process
followed by the unobserved latent volatility. However, the expression is valid only
under the assumption that the sample paths of the asset are continuous. In the
presence of jumps in asset returns, integrated volatility also includes a jump com-
ponent (see Barndorff-Nielsen and Shephard, 2006). For affine models, it is feasible

2,50E-03

2,00E-03

1,50E-03

1,00E-03

5,00E-04

0,00E+00

05/01/93 05/01/95 05/01/97 05/01/99 05/01/01 05/01/03

Figure 19.3 Daily integrated volatilities of the S&P500 over the period January 5, 1993, to
December 31, 2004