The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 624 Wednesday, February 4, 2004 1:33 PM

624 The Mathematics of Financial Modeling and Investment Management

Spot Rates: Continuous Case

Assume for the moment that the evolution of short-term interest rates is

deterministic and it is known. Thus, at any time tthe function i(s) that

describes the short-term rate is known for every moment s≥t. Recall

that i(s) is the limit of the interest rate for an interval that tends to zero.

Earlier in this chapter we established that the value at time t 1 of capital

of a risk-free bond paying B(t 2 ) at time t 2 is given by

• ∫t^2 is()ds
  t
  ()= Bt^1
  Bt 1 () 2 e

The yield over any finite interval (t 1 ,t 2 ) is the constant equivalent

interest rate

t 2

Rt 1

over the same interval (t 1 ,t 2 ) which is given by the equation

t

• (t 2 – t 1 )Rt^22 is()ds
  Bt 1 ()e^1 = Bt 2


• ∫t

t

()= Bt^1

2 ()e

Given a short-term interest rate function i(t), we can therefore

define the term structure function Ru

t as the number which solves the

equation

uis

• (ut–)Ru t –∫t ()ds
  e = e

In a deterministic setting, we can write

u

Ru^1

t = ----------------∫is()ds

(ut– )t

This relationship does not hold in a stochastic environment, as we will

see shortly. From the above it is clear that Ru t is the yield of a risk-free

bond over the interval (t,u). The function

• uis

Λu= e∫t ()ds

t