956 Continuous-Time Stochastic Volatility Models
19.2.1 Affine diffusions
The must popular stochastic volatility (SV) models are the so-called affine mod-
els. Broadly speaking, affine models are characterized by linearity of the drift and
variance functions in (19.1) and provide computational tractability that leads to
closed or semi-closed solutions in a variety of applications (see Duffie, Pan and
Singleton, 2000). We examine the following affine specifications of the general
stochastic volatility model in (19.1) whenNtS=NtV=0.
SV1 dVt=k(θ−Vt)dt+σdWtV
SV2 dVt=k(θ−Vt)dt+σ
√
VtdWVt.In the SV1 model, also called the Ornstein–Uhlenbeck process, volatility follows
a Gaussian mean-reverting process. The parameterkcaptures the speed of mean
reversion,θis the long-run mean of volatility andσis the volatility of volatility.
SV1 was first used by Vasicek (1977) for term structure modeling and has also
been used for option pricing by Hull and White (1987), Scott (1987), Stein and
Stein (1991), and Brenner, Ou and Zhang (2006), among others. This is the only
stochastic volatility model for which the distribution of asset returns in (19.1) can
be derived in closed form (see Stein and Stein, 1991). The conditional mean and
variance of the SV1 model at timetfor a time periodT>tis given by:
Et(
VT)
=θ+(
Vt−θ)
e−k(T−t) (19.2)Vart(
VT)
=
σ^2
2 k(
1 −e−^2 k(T−t)). (19.3)
Unfortunately, the SV1 model is not fully consistent with the empirical proper-
ties of volatility. In particular, SV1 implies that volatility can take negative values
and that it is homoskedastic, which is evident from the fact that the conditional
variance of the process does not depend on the level of volatility. Hence SV1 is
not able to capture the positivity of volatility or the level effect. Consequently,
despite the analytical tractability offered by this specification, it is no longer in
common use.
SV2 was popularized after Heston (1993) and has been extensively used in a
variety of applications. It was proposed as an alternative to SV1 that constrained
volatility from taking negative values (see, for example, Bates, 1996b, 2000). Under
this volatility parameterization the distribution of stock prices in (19.1) is not
known in closed form but can be derived from the characteristic function (we
discuss this methodology in section 19.3). The conditional mean and variance of
the SV2 model are:
Et(
VT)
=θ+(
Vt−θ)
e−k(T−t) (19.4)Vart(
VT)
=Vt(
σ^2
k)(
e−k(T−t)−e−^2 k(T−t))
+(
σ^2
k)(
1 −e−k(T−t)) 2. (19.5)