Autoregressive Heteroscedasticity Model and Its Variants 231
return distributions are known to be conditionally leptokurtotic, only a
few studies have investigated the option pricing problem with GARCH
dynamics and non-Gaussian innovations using alternative assumptions
about the residuals.^14
Multivariate Extensions of ARCH/GARCH Modeling
Thus far we have applied ARCH and GARCH models to univariate time
series. Let’s now consider multivariate time series, for example the returns
of an ensemble of stocks or of stock indexes.
Let’s assume that returns are normally distributed. Although a univari-
ate normal distribution is completely characterized by two parameters—a
mean and a variance—a multivariate normal is characterized by a vector of
means and by a matrix of covariances.
Empirical studies of time series of stock returns find that the con-
ditional covariance matrix is time-varying. Not only is the variance of
each individual return time-varying, the strength of correlations between
stock returns also time-varying. For example, Figure 11.9 illustrates the
time-varying nature of correlations of the S&P 500 universe. We com-
puted correlations between daily returns of stocks that belong to the S&P
500 universe over a 100-day moving window from May 25, 1989, to
December 30, 2011. Between these dates there are 5,699 trading days.
As correlations are symmetrical in the sense that correlation between
returns of stocks A and B is the same as correlations between returns of
stocks B and A, for each trading day there are 500 × 499/2 = 12,4750
correlations. We average all these correlations to obtain an average daily
correlation C.
Figure 11.9 illustrates that correlations are not time constant. Aver-
age correlation ranges from a minimum of 0.1 to a maximum of 0.7 and
exhibits an upward trend. In addition there are periodic fluctuations around
the trend. These facts suggest modeling the covariance matrix with ARCH/
GARCH-like models.
(^14) See Christian Menn and Svetlozar T. Rachev, “Smoothly Truncated Stable Distri-
butions, GARCH-Models, and Option Pricing,” Mathematical Methods of Opera-
tions Research 63, no. 3 (2009): 411–438; Peter Christoffersen, Redouane Elkamhi,
Bruno Feunou, and Kris Jacobs, “Option Valuation with Conditional Heterosce-
dasticity and Nonnormality,” Review of Financial Studies 23 (2010): 2139–2183;
and Young-Shin Kim, Svetlozar T. Rachev, Michele Bianchi, and Frank J. Fabozzi,
“Tempered Stable and Tempered Infinitely Divisible GARCH Models,” Journal of
Banking and Finance 34, no. 9 (2010): 2096–2109.