The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 631 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 631

• ∫TfX( T,s)ds
  Fxt( , ) = Et e t Ψ(XT)Xt = x

We can now go back to the original problem of computing the term

structure from the stochastic differential equation of the short-rate pro-

cess. Recall that the term structure is given by the following conditional

expectation:

uis()ds

u = EQ

t e

Λ ∫t

t

If we apply the Feynman-Kac formula, we see that the term struc-

ture is a function

u

Λt = Fi(t,t)

of time tand of the short-rate itwhich solves the following PDE:

∂Fxt( , ) 1 ∂ ( , ( ,

2

Fxt) ∂Fxt)

--------------------+ ---σ , , ( ,

2

(xt)----------------------+ μ(xt)--------------------– xF x t) = 0

∂t (^2) ∂x^2 ∂x
with boundary conditions F(x,T) = 1.
Note explicitly that the solution of this equation does not determine
the dynamics of interest rates. In other words, given the short-term rate
itat time tthe function
u
Λt = Fi(t,t)
does not tell us what interest rate will be found at time s > t.It does tell,
however, the price at time sof a bond with face value 1 at maturity Tfor
every interest rate is. If the coefficients σ= σ(x), μ= μ(x) do not depend
on time explicitly, then one single function gives the entire term structure.
Note also that the above is true in general for any asset which does
not exhibit any intermediate payoff. Recall, in fact, the pricing formula:
St Et
Q
e
t – ru ud
T

∫

ST e

t – ru ud

s

∫

t

T

= +∫ dDs