286 Frequently Asked Questions In Quantitative Finance
Now move on to the second bond having maturity date
T 2. We know the rate to apply between now and time
T 1 , but at what interest rate must we discount between
datesT 1 andT 2 to match the theoretical and market
prices of the second bond? The answer isr 2 which
solves the equation
Z 2 M=e−r^1 (T^1 −t)e−r^2 (T^2 −T^1 ),
i.e.
r 2 =−
ln
(
ZM 2 /ZM 1
)
T 2 −T 1
.
By this method ofbootstrappingwe can build up the
forward rate curve. Note how the forward rates are
applied between two dates, for which period I have
assumed they are constant.
This method can easily be extended to accommodate
coupon-bearing bonds. Again rank the bonds by their
maturities, but now we have the added complexity that
we may only have one market value to represent the
sum of several cashflows. Thus one often has to make
some assumptions to get the right number of equations
for the number of unknowns.
To price non-linear instruments, options, we need a
model that captures the randomness in rates.
Black 1976
Market practice with fixed-income derivatives is often
to treat them as if there is an underlying asset that
is lognormal. This is the methodology proposed by
Black (1976).
Bond options A simple example of Black ’76 would be a
European option on a bond, as long as the maturity of
