Review of Quantitative Finance and Accounting, 19: 5–20, 2002
2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
A Simple Multi-Factor, Time-Dependent-Parameter
Model for the Term Structure of Interest Rates
T. L. TYLER YANG
Abstract. In this paper, we present a simple version of the Dufﬁe and Kan model (1996). Our model can perfectly
ﬁt the yield curve and the volatility curve and further provide true closed form solutions to the pure discount bond
price and its European contingent claims. Due to the speciﬁc factor structure in our model, the calibration exercise
is easy to implement. This advantage will improve the computational efﬁciency in pricing American style claims.
Keywords: term structure of interest rates, yield curve, volatility curve, factor model
JEL Classiﬁcation: G12, G13
The purpose of this paper is to develop a multi-factor, time-dependent-parameter model of
the term structure of interest rates that is easy to implement. It has true closed form solutions
for the pure discount bond and its European style claims. To price American style claims,
computational efﬁciency is signiﬁcantly improved by using of the closed form solution to
the underlying bond. Instead of generating a lattice up to the maturity of the underlying
bond, one only needs to build the lattice up to the expiration of the American claim.
In order to obtain true closed form solutions, we, following the suggestion by Dufﬁe and
Kan (1996) and Cox, Ingersoll and Ross (1985b), decompose a time-dependent-parameter
state variable into two independent variables, one of which contains only the volatility
parameter and the other of which contains only a deterministic drift. In doing so, the
problem caused by dependency among the reverting speed, the reverting level, and the
volatility parameters under one stochastic process is avoided. While the model we present
in the paper is based upon the extended Vasicek model, same principles can be applied to
the extended Cox-Ingersoll-Ross model.
The paper is organized as follows. The next section brieﬂy reviews equilibrium term
structure models. The main model is presented in Section 3. Section 4 discusses some prob-
lems with the model and how we should overcome them. Section 5 sketches the algorithm
Corresponding author: Ren-Raw Chen, Rutgers Business School, Rutgers University, Janice Levin Bldg.,
94 Rockafeller Road, Piscataway, NJ 08854, Tel.: (732) 445-4236. E-mail: firstname.lastname@example.org