The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 627 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 627

environment. This is because bonds of longer maturities are riskier than

bonds of shorter maturities. The martingale relationship holds only for

risk-neutral probabilities.

We can now go back to the forward rates. The expression

logΛ u + ∆ u– logΛ u ∂ logΛ u

( ,

t t

ft u ) = lim – ---------------------------------------------- = – -----------------

t -

u → ∆ (^0) ∆ u ∂ u
holds in a stochastic environment when the term structure is defined as
above.
We have now defined the basic terms and relationships that can be
used for bond valuation and for bond option valuation and we have
established a formula that relates the term structure to the short-rate
process. The next step is to specify the models of the short-term interest
rate process. The simplest assumption is that the short-term rate follows
an Itô process of the form
dr ˆ
t = μ( rt, t) dt + σ( rt, t) dBt
where dBˆ t is a standard Brownian motion under the equivalent martin-
gale measure.
As explained in Chapter 15 on arbitrage pricing, it is possible to
develop all calculations under the equivalent martingale measure and to
revert to the real probabilities only at the end of calculations. This pro-
cedure greatly simplifies computations. Under the equivalent martingale
measure all price processes St follow Itô processes with the same drift of
the form
dS ˆ
t = rtStdt + σ( rt, t) dBt
Note that the short-term interest rate process is not a price process
and therefore does not follow the previous equation. Models of the
short-term rate as the above are called one-factor model because they
model only one variable.
The Feynman-Kac Formula
Computing the term structure implies computing the expectation

• uis

Λ u Q ∫ () sd

t = Et e

t