Advances in Risk Management

96 AN ESSAY ON STOCHA ST IC VOLATILITY AND T HE YIELD CURVE

5.5.2 Algorithm

A standard setup of the Kalman filter is applicable to the linear state-space
model of the form:

yn=Znαn+dn+εn

αn=Tαn− 1 +cn+Rnηn

with var(εn)=Hnand var(ηn)=Qn. The first equation is the measurement
equation and the second equation is the transition equation. (εn) and (ηn)
are independent normal random variables with zero mean.
The Kalman filter for this approximate state-space model is then given by:

an/n− 1 =Tn(an− 1 ),

Pn/n− 1 =TˆnPn− 1 Tˆn′+RˆnQnRˆ′n,

Fn=ZˆnPn/n− 1 Zˆ′n+Hn,

an=an/n− 1 +Pn/n− 1 Zˆ′nF−n^1 (yn−Zn(an/n− 1 )),

Pn=Pn/n− 1 −Pn/n− 1 Zˆn′F−n^1 ZˆnPn/n− 1

To apply this algorithm, one should proceed to a discretization and a
linearization of the F&V short rate model which is discussed in next section.

5.5.3 Applying the Kalman filter to the F&V model

Recall that Fong and Vasicek model the short rate and its stochastic variance
with the following equations:

drt=k(μ−rt)dt+

√

vtdWt

dvt=λ(ν−vt)dt+τ

√

vtdWs

As we can see, the model respects the Kalman filter state-space form.
One can consider the first and the second equation as the measurement
equation and the transition equation respectively. But the model is still in its
continuous and non linear form. Before applying directly the algorithm of
extended Kalman filter, we try to put these two equations in their discrete
and linear form.^10

Discretization

An application of Ito formula to the first equation of the F&V model yields:

dekt(rt−μ)=ekt

√

vtdWt