Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

120 Optimizing Optimization

independent of past realizations while empirical evidence shows that conditional
homoskedasticity is often violated in financial data. In particular, we observe
volatility clusters on returns series. Such behavior is captured by autoregressive
conditional heteroskedastic models (ARCH) 9 and their generalization (GARCH
models), 10 where the innovations are conditionally stable Paretian distributed.
Several empirical experiments by Rachev and Mittnik (2000) have reported the
typical behavior problem of time-series modeling.
Assume that daily stock returns follow ARMA( p , q )-GARCH( s , u ) processes;
i.e., assume:

ra ar b

z

jt j ji

i

p

jt i ji

i

q

jt i jt

jt jt j

,   









,,, ,,,

,,,

0

11

∑∑εε

εσtt

jt j ji

i

s

jt i ji

i

u

σσε,,cc,, d,,jt i

2

0

1

2

1

 ^2







∑∑

where r j, (^) t is the daily return of the stock j ( j  1, ... , 30) at day t ( t  1, ... ,
1,837). Since several studies have shown that ARMA-GARCH filtered residuals
are themselves heavy tailed, then it makes sense to assume that the sequence of
innovations z j (^) , (^) t is an infinite-variance process consisting of i.i.d. random vari-
ables in the domain of normal attraction of a stable distribution with index
of stability α (^) j belonging to (0,2). That is, there exist normalizing constants
hR()jT ∈ 
and
kRj()T ∈ such that:
1
h 1
zk S
j
T jt
t
T
j
T d
() , j jjj
() ()

∑  ⎯→⎯ α σβμ, , ,
where the constants hj()T have the form
hLTTj()T  j()j
α
and L j ( T ) are slowly
varying functions as T →. S (^) α j ( σ (^) j , β (^) j , μ (^) j ) is a stable Paretian distribution with
index of stability, α (^) j  (0,2], skewness parameter, β (^) j  [  1,1], scale parameter,
σ (^) j  ℜ (^)  , and location parameter, μ (^) j  ℜ.^11 In particular, we can easily test differ-
ent distributional hypotheses for the innovations of ARMA(1,1)-GARCH(1,1):
raar b z
c
jt j j jt j jt jt jt jt jt
jt j
,,,, ,, ,, ,,
,
  ; 

01111 
2
εεεσ
σ ,,,,cdjjtσε^2 jjt,,011 112
(5.1)^
9 See Engle (1982).
10 See Bollerslev (1986).
11 Refer to Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000) for a general discus-
sion of the properties and use of stable distributions.