Continuous Time Finance Notes, Spring 2004 – Section 8,
March 24, 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 by wrapping up our discussion of HJM – discussing its strengths and
weaknesses. Then we turn to the “Libor Market Model” – which though relatively new
is rapidly becoming the method of choice for many purposes. Our discussion of the Libor
Market Model follows Hull’s treatment (Section 24.3 of the 5th edition).
*****************
Some comments on the HJM approach to interest rate modelling:
HJM is a “framework” not a “model”
. When modeling equitybased derivatives, our
usual framework is to assume the underlying solves a diﬀusion process
ds
=
μ
(
s, t
)
dt
+
σ
(
s, t
)
dw
. The market is complete for any choice of the functions
μ
(
s, t
) and
σ
(
s, t
) (with
some minor restrictions, for example
σ
(
s, t
)
>
0). But what to choose for
μ
and
σ
? The
choice of
μ
is of course irrelevant for option pricing, because under the riskneutral measure
the underlying satisﬁes
ds
=
r dt
+
σ
(
s, t
)
dw
. But the choice of
σ
is crucial. We commonly
choose
σ
(
s, t
) =
σ
0
s
with
σ
0
constant, i.e. we commonly assume the underlying has
lognor
mal
dynamics. (This is not the only possibility; in a week or two we’ll discuss “local vol”
models, i.e. the idea of using the skew/smile of implied volatilities to infer an appropriate
choice of
σ
(
s, t
).)
The situation with HJM is analogous. For onefactor HJM, the framework is
d
t
f
(
t, T
) =
α
(
t, T
)
dt
+
σ
(
t, T
)
dw
.
Working under the riskneutral measure (i.e.
assuming
w
is a
Brownian motion in the riskneutral measure) the drift
α
is completely speciﬁed by
σ
. The
choice of
σ
is again crucial. It is no longer so obvious how to choose it; we saw how to
choose
σ
(
t, T
) to get either HoLee or HullWhite. But many other choices are possible.
Calibration to data is not easy, because when
σ
is not deterministic (e.g. if it depends
explicitly on
f
(
t, T
)) we have no exact pricing formulas.
It is natural to suppose
σ
is a function of
f
as well as
t
and
T
.
Indeed, if
σ
is a
deterministic function of
t
and
T
then
f
(
t, T
) is Gaussian, so there is a positive probability
that it is negative. We tolerated this in HullWhite, and one often tolerates it in HJM
for the same reason – namely analytical tractability of the model. However a model that
permits negative interest rates cannot be the last word. Amongst shortrate models the
simplest way to keep the short rate positive is to use a statedependent volatility: for
example the CoxIngersollRoss model assumes
dr
= (
θ
(
t
)

ar
)
dt
+
σ
0
√
r dw
(compare this
to HullWhite:
dr
= (
θ
(
t
)

ar
)
dt
+
σ
0
dw
). Our discussion of HJM (in particular: our
explanation how the volatility determines the drift) did
not
assume
σ
was deterministic. It
applies with no change if, for example,
σ
has the form