284 Frequently Asked Questions In Quantitative Finance
simplest of these assumes a deterministic evolution of a
spot rate.
The spot rate and forward rates The interest rate we consider
will be what is known as ashort-term interest rateor
spot interest rater(t). This means that the rater(t)is
to apply at timet. Interest is compounded at this rate at
each moment in time butthis rate may change, generally
we assume it to be time dependent.
Forward rates are interest rates that are assumed to
apply over given periodsin the futureforallinstruments.
This contrasts with yields which are assumed to apply
from the present up to maturity, with a different yield
for each bond.
Let us suppose that we are in a perfect world in which
we have a continuous distribution of zero-coupon bonds
with all maturitiesT. Call the prices of these at timet,
Z(t;T). Note the use ofZfor zero-coupon.
Theimplied forward rateis the curve of a time-dependent
spot interest rate that is consistent with the market price
of instruments. If this rate isr(τ)attimeτthen it satisfies
Z(t;T)=e−
∫T
tr(τ)dτ.
On rearranging and differentiating this gives
r(T)=−
∂
∂T
(lnZ(t;T)).
This is the forward rate for timeTas it stands today,
timet. Tomorrow the whole curve (the dependence ofr
on the future) may change. For that reason we usually
denote the forward rate at timetapplying at timeTin
the future asF(t;T)where
F(t;T)=−
∂
∂T
(lnZ(t;T)).