20-Term Structure Page 623 Wednesday, February 4, 2004 1:33 PM
Term Structure Modeling and Valuation of Bonds and Bond Options 623
t
()e –is
∂P∂
0∫cs sd t
∂[cs()e –ist------ = ------------------------------------- = ∫--------------------------- ()e –is
]sd = –∫sc s sd
∂i ∂i 0 ∂i 0The above formula parallels the discrete-time formula that was estab-
lished in Chapter 4.^7THE TERM STRUCTURE OF INTEREST RATES IN
CONTINUOUS TIMEOur ultimate objective is to establish a stochastic theory of bond pricing
and of bond option pricing. To do so, we will reformulate term struc-
ture theory in a continuous-time, continuous-state environment. We will
subsequently develop examples on how processes can be discretized,
thus going back to a discrete-state, discrete-time environment. The sto-
chastic description of interest rates is challenging from the point of view
of both mathematics and economic theory. We discussed the economic
theories of interest rates earlier in this chapter.
Mathematical difficulties stem from the fact that one should con-
sider not just one interest rate but the entire term structure of interest
rates that was defined earlier. This is, in principle, a (difficult) problem
of infinite dimensionality. Though attempts have been made in the aca-
demic literature to describe the stochastic behavior of a curve without
any restriction, in practice models currently in use make simplifications
so that the movement of the term structure curve is constrained to that
of one or a small number of factors.
The term structure of interest rates is a function U(t,s) of two vari-
ables t,s that represents the yield computed at time t of a zero-coupon
risk-free bond with maturity s. The yield on a zero-coupon bond is
called the spot rate. In calculating the spot rate in developed bond mar-
kets, the yields on government bonds are used. Government bonds are
typically coupon-paying instruments. However, we have seen in this
chapter how to obtain, from arbitrage arguments, the theoretical spot
rates from a set of yields of coupon-paying bonds. The term structure of
interest rates is a mathematical construct as only a finite number of spot
rates can be observed. A continuous curve needs to be reconstructed
from these discrete points.(^7) See footnote 7 in Chapter 4, p. 114. Note that in Chapter 4, V is used rather than
P to denote market price.