962 Continuous-Time Stochastic Volatility Models
equation (PDE):
1
2Vt
∂^2 φ
∂y^2 t+ρ√
V 2 σ(Vt;θ)
∂^2 φ
∂yt∂Vt+
1
2σ(Vt;θ)^2
∂^2 φ
∂Vt^2+(μ−
1
2Vt)
∂φ
∂yt+a(Vt;θ)
∂φ
∂yt−
∂φ
∂τ,(19.19)subject to the boundary conditionφ
(
iω 1 ,iω 2 ,yt,Vt,0,θ)
=eiω^1 yT+iω^2 VT.
When the stochastic volatility model belongs to the affine class, the solution
of the joint conditional characteristic function has a simple exponential form,
given by:
φ(
iω 1 ,iω 2 ,yt,Vt,τ;θ)
=eiω^1 yt+A(iω^1 ,iω^2 ,τ;θ)Vt+B(iω^1 ,iω^2 ,τ;θ). (19.20)The coefficientsA(iω 1 ,iω 2 ,τ;θ)andB(iω 1 ,iω 2 ,τ;θ)can be derived by solving
complex valued Ricatti equations. Duffie, Pan and Singleton (2000) show ana-
lytically how to derive the characteristic function for general affine diffusion-jump
diffusion stochastic volatility models. Knowledge of the characteristic function
also allows the derivation of non-central moments and cross-moments of affine
continuous-time stochastic models via simple differentiations.
If volatility is treated as observed then, in principle, the transition density can
be derived by Fourier inversion as:
p(yt+τ,Vt+τ|yt,Vt)=
1
( 2 π)^2∫∞−∞∫∞−∞φ(
iω 1 ,iω 2 ,yt,Vt,τ;θ)×e−iω^1 yt+τ−iω^2 Vt+τdω 1 dω 2. (19.21)Estimation with the characteristic function has been applied in latent stochas-
tic volatility models by Singleton (2001), Jiang and Knight (2002), Chacko and
Viceira (2003) and Bates (2006). Chacko and Viceira (2003) derive from (19.20)
the conditional characteristic function of the asset,φ
(
iω 1 ,0,yt,Vt,τ;θ)
, and then
integrate out the latent volatility to obtain, in closed-form solution, the character-
istic function conditional only on the current asset price,φ
(
iω 1 ,0,yt,τ;θ). They
then estimate various stochastic volatility models using Hansen’s (1982) method
of moments. However, the estimates are not fully efficient because the only con-
dition is on the current stock price and, as mentioned previously, when volatility
is integrated out the stock price on its own is no longer Markov. Jiang and Knight
(2002) use iterated expectations and derive the joint unconditional characteristic
function up tot−Lobservations,
φ(
iω 1 ,iω 2 ,...iωL,yt,0.τ,θ)
=E[
eiω^1 yt−^1 ,iω^2 yt−^2 ,...iωLyT−L]
.In practice, the joint unconditional characteristic function is derived for a relatively
smallL. Jiang and Knight (2002) estimate a stochastic volatility model using GMM
with non-central and cross-moments. Singleton (2001) and Bates (2006) achieve
full efficiency by conditioning on the full history of returns. Singleton proposes a
method that combines the simulated method of moments of Duffie and Singleton
(1993) with the characteristic function technology. Bates (2006) develops a direct
filtration-based maximum likelihood methodology using characteristic functions