The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 640 Wednesday, February 4, 2004 1:33 PM

640 The Mathematics of Financial Modeling and Investment Management

Note that the first term under the expectation sign is the expectation

under risk-neutral probabilities of the formula for the present value of a

continuous cash-flow stream that we established earlier in this chapter:

t –siu

V ∫ () ud

0 = cs()e

(^0) sd

∫

0

where c(s) = h(is,s) and the initial time is 0.

The Feynman-Kac formula can be extended to this case. In fact it

can be demonstrated that the function F obeys the following PDE:

∂Fx t( , ) 1 ∂^2 Fx t( , ) ( ,

,

∂Fx t)

--------------------+ ---σ , ( , ( ,

2

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

∂t (^2) ∂x^2 ∂x
with boundary conditions F(x,τ) = g(x,τ). If h(x,t) = 0, g(x,τ) ≡1, we find
the bond valuation formula of the previous section.
THE HEATH-JARROW-MORTON MODEL OF THE
TERM STRUCTURE
In the previous sections we derived the term structure from a short-term
rate process which might depend, in turn, on a number of factors. How-
ever, this is not the only possible choice. In 1992, David Heath, Robert
Jarrow, and Andrew Morton introduced a methodology that recovers
the term structure (i.e., bond prices) from the forward rates.^20 The key
issue with this methodology is to ensure the absence of arbitrage.
Recall that the forward rate f(t,u) is the short-term spot rate at time
u contracted at time t. In a deterministic environment (that is, assuming
that the forward rates are known) to avoid arbitrage, the following rela-
tionships must hold:
∂(logΛu
t )
ft u( , )= – -----------------------
∂u
(^20) David Heath, Robert A. Jarrow, and Andrew J. Morton, “Bond Pricing and the
Term Structure of Interest Rates: A New Methodology for Contingent Claim Valu-
ation,” Econometrica (1992), pp. 77–105.