90 AN ESSAY ON STOCHA ST IC VOLATILITY AND T HE YIELD CURVEthe news revealed at timet. That is whyσtis unobserved, contrarily toνt
which is observed at timet.
Perhaps the view initiated by Nelson (1990) that ARCH models are
approximations of diffusion models created confusion in the financial lit-
erature about the relation between stochastic and conditional volatility. Let
us see how this approximation works. Assume the following GARCH (1,1)
process:
ν^2 n+ 1 =w+βν^2 n+αε^2 n (5.10)withε=μν,μ∼N(0,1). We can write (5.10) as:
ν^2 n+ 1 −ν^2 n=w−(1−αE(μ^2 )−β)νn^2 +αν^2 n(μ^2 −E(μ^2 )) (5.11)This equation converges weakly in distribution to:
dν(τ)^2 =(ω−φν(τ)^2 )dτ+ψν(τ)^2 dz(τ) (5.12)withzta Wiener process. The convergence between the parameters of the
discrete equation (5.10) to the parameters of the continuous equation (5.12)
is the following:
limh−^1 wh=ω
limh−^1 (1−αh−βh)=φ
limh−1
2√
2 αh=ψ(5.13)withha very small time increment expressed in fraction of year. The
√
2 term
in the equation ofψis explained by the fact thatξ=μ^2 −E(u^2 ) is a chi-square
variable with one degree of freedom and a variance of 2. The sequenceξn
is consequently a sequence of chi-square variables and is the discrete time
approximation of the Brownian increments dz.^6
Fornari and Mele (2005) have generalized Nelson’s (1990) model to the
class of CEV-ARCH models. They show how volatility diffusions may be
approximated by these models by assuming a distribution for the innovation
which encompasses the normal distribution: the generalized error distribu-
tion (ged). Let us assume the following CEV-ARCH model for the interest
raterin its continuous form:
dr(τ)=(ι−θrτ)dτ+ν(τ)√
r(τ)dz 1dν(τ)δ=(ω−φν(τ)δ)dτ+ψν(τ)δη(
ρdz 1 +√
1 −ρ^2 dz 2)
(5.14)withdz 1 anddz 2 , two Wiener processes with a correlation ofρ, the correlation
being performed by a Cholesky decomposition. Equation (5.14) is the con-
tinuous representation of the CEV-ARCH model. There are three additional
parameters to estimate in comparison with standardARCH models:δ,ηand