Advances in Risk Management

54 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

1982), the utilization of these models has become widespread, ranging from
asset management (Manganelli, 2004), to derivatives pricing (Duan and
Zhang, 2001) and risk management (Manganelli, Ceci and Vecchiato, 2002).
A stochastic process is calledGARCH(p,q) if its time-varying conditional
variance is heteroscedastic with both autoregression and moving average
(Bollerslev and Engle, 1993):

yt=εt, εt∼N(0,σ^2 ) (3.16)

h^2 =α 0 +

∑q

i= 1

αiε^2 t−i+

∑p

j= 1

βjh^2 t−j (3.17)

In equation (3.17) autoregression in the GARCH(p,q) process squared resid-
uals has an order ofq, and the moving average component has an order
ofp.
One of the features that has traditionally made GARCH models popular
is the fact that parameters of the model can be straightforwardly estimated,
since construction of the ML function is made direct from the fact that the
model is formulated “in terms of the distribution of the one step ahead
prediction error” (Shephard, 2005). The conditional log-likelihood ofyt+ 1 is
(Campbell, Lo and McKinley, 1997; Hull, 1999; Noh, 1997):

LT(y 1 ,...,yT)=Tt= 1 lt(yt+ 1 ;q) (3.18)

where

lt(yt+ 1 ;q)=log

[

N

(

yt+ 1

ht

)]

−

log (h^2 t)

2

is the one-step conditional log-likelihood function,θis the vector of the
parameters of the model and N(·) is a standard normal density function.
Parameters can then be estimated by maximization of equation (3.18).
Throughout our discussion we consider the following GARCH(p,q)
process for a portfolio ofnassets (Manganelli, 2004):

yt=

√

htεt εt∼N(0, 1) (3.19)

ht=ztθ (3.20)

whereytis the return of a portfolio composed byn+1 assets calculated as
yt=

∑n+ 1
i= 1 aiyt,i, whereaiandyt,iare the weight and the return respectively
of asseti;zt=(1,yt^2 − 1 ,...,y^2 t−q,ht− 1 ,...,ht−q) andθ=(a 0 ,a 1 ,...,aq,b 1 ,...,bp)
We now derive the expression of the differential importance of weights
on portfolio volatility.