George Dotsis, Raphael N. Markellos and Terence C. Mills 957The conditional means of the two models are identical but, from (19.5), it can be
seen that the conditional variance of SV2 depends on the level of volatility, thus
making the process heteroskedastic.
19.2.2 Affine jump diffusions
Another popular class comprises the affine jump diffusion models, which incorpo-
rate a jump component in asset returns and/or volatility. Here we examine the case
where jump times in asset returns and volatility occur simultaneously and jump
sizes are correlated.
SV3 dV=k
(
θ−Vt)
dt+σdWVt +ytVdNtV.In SV3,NSt =NVt, the volatility jump size is drawn from an exponential distri-
bution,f(yV)=ηe−ηy
V(^1) {y≥ 0 }, and the returns jump size follows the conditional
distributionyS|yV:N(μS+ρyV,σS^2 ). The SV3 model has been used, for example,
by Duffie, Pan and Singleton (2000), Eraker, Johannes and Polson (2003), Eraker
(2004) and Broadie, Chernov and Johannes (2007). It allows for a rapidly moving
and persistent factor to drive asset returns during times of market stress. Under
this volatility parameterization the distribution of stock prices in (19.1) can again
be derived from the characteristic function. The conditional mean and variance of
SV3 are:
Et
(
Vt
)
=Vte−k(T−t)+θ
(
1 −e−k(T−t)
)
- λv
kη
(
1 −e−k(T−t)
)
(19.6)
Vart
(
VT
)
=Vt
(
σ^2
k
)(
e−k(T−t)−e−^2 k(T−t)
)
(
σ^2 θ
2 k
)(
1 −e−k(T−t)
) 2
λvσ^2
2 k^2
(
1 −e−kτ
) (^21)
η
λv
k
(
1 −e−^2 kτ
) 1
η^2
. (19.7)
19.2.3 Non-affine diffusions
Non-affine models are not particularly popular in the option pricing literature as
they do not provide closed form formulae for option pricing. Neither the distri-
bution nor the characteristic function of (19.1) can be obtained in closed form.
However, these specifications have been widely used for econometric estimation
purposes. Here we assume again that the dynamics in (19.1) do not incorporate a
jump component.
SV4 dVt=k
(
θ−Vt)
dt+σVtγdWtVSV5
d(
lnVt)
=k(
θ−(
lnVt))
dt+σdWtVdVt=kVt(
θ+
σ^2
2 k
−lnVt)
dt+σVtdWVt.