RAYMOND THÉORET, PIERRE ROSTAN AND ABDELJALIL EL-MOUSSADEK 91ρ;ηis the coefficient of elasticity. The GARCH(1,1) model is a special case
of equation (5.14) for whichδ=2,η=1 andρ=0.
Let us consider the following discrete time approximation of equa-
tion (5.14):
νδn+ 1 =w+ανδηn|μn|δη+βνδn+αE(|μ|δη)(νδn−νηδn) (5.15)This process converges weakly in distribution to:
dσ(τ)δ=(ω−φσ(τ)δ)dτ+ψσ(τ)δηdz 2 (5.16)Fornari and Mele (2005) assume thatμobeys to the ged distribution. Let us
write:
nδ,υ=2δ
υ−^1 ∇δυ(
δ+ 1
υ)(υ−^1 )with∇δυ= (υ
− (^1) )
2 δυ(3υ−^1 )
and, the gamma function. When the distribution is
assumed normal, the coefficient of the gamma functionυis equal to 2 and
consequently:∇ 22 =
( 1
2
)
2
( 3
2
)=1. At the limit, we have the following relations
between the coefficients of equations (5.15) and (5.16):
φ= 1 −nδ,υ
[
( 1 −γ)δ+( 1 +γ)δ
]
α−β
ψ=
√(
mδ,υ−n^2 δ,υ
)(
( 1 −γ)^2 δ+( 1 +γ)^2 δ
)
− 2 n^2 δ,υ( 1 −γ)δ( 1 +γ)δ
withmδ,υ=
22 υδ−^1 ∇^2 υδ(^2 δυ+^1 )
(υ−^1 )
According to equations (5.15) and (5.16), the CEV-ARCH model con-
verges weakly in distribution to a continuous diffusion model. But this link
is only an approximation. We must assume a distribution to prove this con-
vergence and the choice of this distribution has a great impact on the relation
between the coefficients of an ARCH model and the corresponding diffu-
sion model. Anyway, for simulating stochastic volatility via the Monte Carlo
method, we must have estimates of the parameters of the diffusion process
governing stochastic volatility; and, as we saw, ARCH models are a way to
compute them.
It is relevant to repeat that stochastic volatility is forwards-looking while
conditional volatility as estimated by anARCH model is backwards-looking.
Stochastic volatility is based on the uncertainty of the news revealed to
markets at timetas evidenced by equation (5.9): it anticipates this informa-
tion. On its side, conditional volatility as computed by an ARCH model is
based on observed information incorporated in market prices. Conditional