6.1 Introduction 137
B(T,T)=B(
τ
= 0)=1.
The standard method to solve this PDE is to suggest an exponential solution struc-
ture, that is
B(
τ
;r
0
)=exp(a
1
(
τ
)+a
2
(
τ
)r
a
0
+ a
3
(
τ
)r
2b
0
). (6.4)
With this guess, a closed-form solution for the zero-coupon bond price is given
for all models except for model (4). With B(0,T ) having been known, the term
structure of interest rates is automatically available. Among the listed models, the
Vasicek model and the CIR model are termed as affine models of the term structure
and play a central role in modeling stochastic interest rates. Duffie and Kan (1996)
provide the necessary and sufficient conditions on this representation in a multi-
variate setting.
Not all models in Table (6.1) are reasonable with respect to the nature of interest
rates and only gain their significance in a historical context. The models (1) and (3)
have an unlimited variance if time goes to infinity whereas the model (4) is station-
ary only for 2
κ
>
σ
2
. Additionally, the models (1) and (3) do not perform the feature
of mean-reversion. The positive value of interest rate is guaranteed by all model ex-
cept for the Vasicek model. However, for most parameters that are consistent with
empirical values, the Vasicek model raises only a negligibly small probability of
negative interest rates. Model (6) is also called “double square root process” model
and has a closed-form solution for the zero-coupon bond price only for the special
case 4
κθ
=
σ
2
. In the following three sections, we will only concentrate on models
(1), (5) and (6), all of which are popular in interest rate theory, and incorporate them
into our option pricing models.
There are some further developments and extensions in modeling interest rates:
1. Time-inhomogeneous processes are proposed to specify interest rate dynamics
and lead to an inversion of the term structure of interest rates (Hull and White,
1990. 1993);
2. Starting with the current term structure of spot rates or future rates, Ho and Lee
(1986), Heath, Jarrow and Morton (1992) attempted to model directly the dynam-
ics of the term structure. A further development of this approach is Libor Market
Model (LMM), and also called the BGM model in honor of a contribution of
Brace, Gatarek and Musiela (1997). We will devote ourself in the last chapter to
address the smile modeling in LMM using the Fourier transform.
3. An interest rate process may be specified in a way that all parameters are matched
as good as possible to the given current term structure of interest rates. (Brown
and Dybvig, 1986).
These approaches are strongly associated with the term structure and not directly
with the dynamics of short interest rates. Hence they are difficult to be incorporated
into the valuation of asset (equity-like) options, and will not considered here.