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Continuous Time Finance Notes, Spring 2004 – Section 5,
Feb. 25, 2004
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course Continuous Time Finance.
This section discusses the HullWhite model. The continuoustime analysis is not much
more diﬃcult than Vasicek – everything is still quite explicit. The basic paper is Rev. Fin.
Stud. 3, no. 4 (1990) 573592, downloadable via JSTOR; my treatment is much simpler
though because I keep the parameters
a
and
σ
constant rather than letting them (as well
as
θ
) vary with time.
The real importance of HullWhite is that while it’s rich enough to match any forward
curve, it’s also simple enough to be approximated by a (recombining, trinomial) tree. This
topic is covered very clearly in Sections 23.1123.12 of Hull (5th edition), so I won’t cover
it separately in these notes.
*****************
The HullWhite model
. Also sometimes known as “extended Vasicek,” this model as
sumes that the riskneutral process for the short rate has the form
dr
=(
θ
(
t
)

ar
)
dt
+
σdw
(1)
where
a
and
σ
are constant but
θ
is a function of
t
. (Actually the 1990 paper by Hull and
White also considers taking
a
=
a
(
t
)and
σ
=
σ
(
t
).) We’ll show that
(a) for a given choice of
θ
(
t
), the situation is a lot like Vasicek;
(b) there is a unique choice of
θ
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