Continuous Time Finance Notes, Spring 2004 – Section 4,
Feb. 18, 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 begins our discussion of interest rate models. You should already have some
familiarity with basic terminology (e.g. bond prices, instantaneous forward rates), instru
ments (e.g. caps and captions, swaps and swaptions), and their valuation using Black’s
formula. If you need to review these topics, Hull is excellent; see also my Derivative Securi
ties lecture notes, sections 10 and 11 (warning: the notation there is slightly diﬀerent from
the present notes). Today we’ll get started by discussing relatively simple onefactor models
of the short rate (Vasicek, CoxIngersollRoss). Next time we’ll discuss the HullWhite (also
known as modiﬁed Vasicek) model, a richer onefactor short rate model that can be cali
brated to an arbitrary initial yield curve. After that we’ll turn to the HeathJarrowMorton
theory.
I’ve adopted Baxter & Rennie’s notation. However my pedagogical strategy is diﬀerent
from theirs. I’m starting with shortrate models, because they’re easier. Baxter & Rennie
start with HJM, specializing afterward to shortrate models, because it’s more eﬃcient. The
Vasicek model is by now quite standard; treatments can be found (with slightly diﬀerent
organization and viewpoints) in Brigo & Mercurio, Lamberton & Lapeyre, and Avellaneda
& Laurence. For a good survey of the “big picture,” I recommend reading chapter 1 of R.
Rebonato,
Modern pricing of interestrate derivatives: the Libor market model and beyond
(2nd edition, 2002, on reserve in the CIMS library).
*****************
Basic terminology.
The timevalue of money is expressed by the
discount factor
P
(
t, T
)=va
lueatt
ime
t
of a dollar received at time
T.
This is, by its very deﬁnition, the price at time
t
of a zerocoupon (riskfree) bond which
pays one dollar at time
T
. If interest rates are stochastic then
P
(
t, T
)w
i
l
lnotbeknown
until time
t
; its evolution in
t
is random, and can be described by an SDE. Note however
that
P
(
t, T
) is a function of
two
variables, the initiation time
t
and the maturity time
T
. The dependence on
T
reﬂects the
term structure
of interest rates; we expect
P
(
t, T
)
to be relatively smooth as a function of
T
, since interest is being averaged over the time
interval [
t, T
]. We usually take the convention that the present time is
t
=0–thuswhatis
observable now is
P
(0
,T
) for all
T>
0.
There are several equivalent ways to represent the timevalue of money. The
yield
R
(
t, T
)
is deﬁned by
P
(
t, T
)=
e

R
(
t,T
)(
T

t
)
;
it is the unique constant interest rate that would have the same eﬀect as
P
(
t, T
) under
continuous compounding. Evidently
R
(
t, T

log
P
(
t, T
)
T

t
.