20-Term Structure Page 641 Wednesday, February 4, 2004 1:33 PM
Term Structure Modeling and Valuation of Bonds and Bond Options 641
f(t,t) = it
Integrating the first relationship we obtain
- u( ,
Λu = e∫t fts )ds
t
Now suppose that in the interval u∈(0,T] the forward rate obeys
the following SDE:
df= α(t,u)dt+ σ(t,u)dBt
Equivalently, this means that for each u∈(0,T] the following rela-
tionship holds:
t t
ftu ( , )= f( 0 ,u)+ ∫ α(su, )ds+ ∫σ(su, )dBˆs
0 0
Stochastic differentiation yields
u u
d–∫fts ( , )ds = ftt ( , )dt+ ∫dts f( , )ds
t t
u
= it()dt– ∫[α(ts, )dt+ σ(ts, )dBˆ t]ds
t
= it()dt– α*(tu, )dt+ σ*(tu, )dBˆt
where
u
α*(tu, )= ∫ α(ts, )ds
t
u
σ*(tu, )= ∫ σ(ts, )ds
t
Using Itô’s lemma, it can be demonstrated that the term structure
process obeys the following SDE: