The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 621 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 621

The first line of reasoning is the following. The cash received over the

infinitesimal interval (t,t + dt) is c(t)dt. Its value at time 0 is therefore

c(t)dte–it, if the short-term rate is constant, or, more in general,

• ∫tis() sd
  ct()dte^0

if the short-term rate is variable. The value at time 0 of the entire cash-

flow stream is the infinite sum of all these elementary elements, that is, it

is the integral

t

()e

• is

P 0 = ∫ cs sd

0

for the constant short-term rate, and:

t –siu

P ∫ ()ud

0 = cs()e

(^0) sd

∫

0

in the general case of variable (but known) short-term interest rates.

This present value has to be interpreted as the market price at which the

stream of continuous cash flows would trade if arbitrage is to be

avoided.

The second line of reasoning is more formal. Consider the cumu-

lated capital C(t) which is the cumulative cash flow plus the interest

earned. In the interval (t,t + dt), the capital increments by the cash c(t)dt

plus the interest i(t)C(t)dt earned on the capital C(t) in the elementary

period dt. We can therefore write the equation

dC = i(t)C(t)dt + c(t)dt

This is a linear differential equation of the type

dx

-------= At()xa t+ (), 0 ≤t < ∞

dt

with initial conditions x(0) = ξ. This is a one-dimensional case of the

general d-dimensional case discussed in Chapter 10. It can be demon-

strated that this equation has an absolutely continuous solution in the

domain 0 ≤t < ∞; this solution can be written in the following way: