The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 622 Wednesday, February 4, 2004 1:33 PM

622 The Mathematics of Financial Modeling and Investment Management

t

xt()= Φ()t x() 0 + ∫Φ–^1 ()sas()ds , 0 ≤t< ∞

0

where Φ(t), called the fundamental solution, solves the equation

dΦ

()Φ, 0 ≤t< ∞

dt

--------= At

In the case we are considering

xt()= Ct(), At()= it(), at()= ct(), ξ= 0

and

∫ 0 is()ds

t

Φ()t = e

and therefore

is()dst –siu

∫ ()du

Ct()= e cs

∫ 0

t

∫ ()e

0

ds

0

If we consider that

• ∫tis()ds
  P 0 = Ct()e^0

is the value at time 0 of the capital C(t), we again find the formula

t –siu

∫ ()du

P 0 = ∫cs()e^0 ds

0

that we had previously established in a more direct way.

If the coupon payments are a continuous cash-flow stream, the sen-

sitivity of their present value to changes in interest rates under the

assumption of constant interest rates are: