The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 625 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 625

is called the discount function.^8

The term on the right side is the price at time t of a bond of face

value 1 maturing at u.

Forward Rates: Continuous Case

The forward rate f(t,u) is the short-term spot rate at time u contracted

at time t. To avoid arbitrage, the following relationship must hold:

logΛu + ∆u – logΛu ∂logΛu t

ft u ( , ) = lim – ----------------------------------------------t t = – ------------------

∆u → (^0) ∆u ∂u
In this deterministic setting, the above relationship yields: f(t,t) =
i(t). Given the short-rate function i(s), the term structure is completely
determined and vice versa.
In a stochastic environment, short-term interest rates form a sto-
chastic process is(ω). This means that for each state of the world there is
a path of spot interest rates. For each path and for each interval (t,u),
we can compute the discount function

• uis

e ∫t ()sd

Under a risk-neutral probability measure Q, the price at time t of a

bond of face value 1 maturing at time u is the expected value of

• uis

e ∫t ()sd

computed at time t:

u

u Q –∫is()sd

Λt = Et e t

The term structure function can be computed from the discount func-

tion as follows as follows:

(^8) Some authors call this function the term structure of interest rates. For example,
Darrell Duffie, Dynamic Asset Pricing Theory (Princeton, NJ: Princeton University
Press, Third Edition, 2001) and Steven Shreve, Stochastic Calculus and Finance
(Springer, forthcoming 2004).