Advances in Risk Management

RAYMOND THÉORET, PIERRE ROSTAN AND ABDELJALIL EL-MOUSSADEK 89

markets collapses.^5 We will use this distribution in the empirical section of
this paper.
To take into account these results, let us add and subtract E log(U^2 t)in
equation (5.5). We have:

log(x^2 t)=E log(Ut^2 )+ht+[log(Ut^2 )−E log(Ut^2 )] (5.6)

We can rewrite (5.6) to estimate it as:

log(x^2 t)= 1. 27 +ht+ςt (5.7)

withςt=[log(U^2 t)]−E log(U^2 t)].
In equation (5.7),ςis an innovation with a logarithmicχ^2 distribution.
Its expectation is:

E(ςt)=E[log(U^2 t)−Elog(U^2 t)]=Elog(Ut^2 )−Elog(U^2 t)= 0

and its variance:

V(ςt)=E(ς^2 t)=E[log(Ut^2 )−Elog(U^2 t)]^2 = 0. 5 π^2 ≈ 4. 93

In equation (5.7),htis the stochastic volatility expressed in logarithmic

form, i.e.σt=

√

e

ht

(^2). It is unobserved and must be filtered. To do so, we use
the extended Kalman filter in this paper. We are now in a position to compare
the concept of stochastic volatility with the concept of conditional volatility
of the ARCH models. In a GARCH (1,1) model, the conditional volatilityνt
may be expressed as:
yt=c+εtνt
ν^2 t=β 0 +β 1 ν^2 t− 1 +β 2 ε^2 t− 1 (5.8)
withyta price variable andc, a constant. If we examine equation (5.8), we
note that the conditional volatility is observed at timetbecause it is condi-
tional on observations made at time (t−1). But according to equation (5.3),
stochastic volatility is unobserved at timetbecause its equation contains
an innovation, contrarily to conditional volatility whose equation has no
innovation. This is the major difference between stochastic volatility and
conditional volatility.
Taylor (1994) reported that there is a mixing variableMtwhich relates
stochastic volatility to conditional volatility. This variable is distributed like
an inverse gamma. We can write:
σt=(Mtν^2 t)
(^12)
(5.9)
withσt, the stochastic volatility andνt, the conditional volatility. It is con-
sequently a random variable which links stochastic volatility to conditional
volatility. This random variableMtrepresents the uncertainty associated to