216 Mathematics for Finance
10.1 Maturity-Independent Yields...............................
The present value of a zero-coupon unit bond determines an interest rate called
theyieldand denoted byy(0) to emphasise the fact that it is computed at
time 0:
B(0,N)=e−Nτy(0).
For a different running time instantnsuch that 0<n<Nthe implied yield
may in general be different fromy(0). For each suchnwe thus have a number
y(n) satisfying
B(n, N)=e−(N−n)τy(n).
Generally (and in most real cases), a bond with different maturityN will
imply a different yield. Nevertheless, in this section we consider the simplified
situation wheny(n) is independent ofN, that is, bonds with different maturities
generate the same yield. Independence of maturity will be relaxed later in
Section 10.2.
Proposition 10.1
If the yieldy(n)forsomen>0 were known at time 0, theny(0) =y(n)orelse
an arbitrage strategy could be found.
Proof
Suppose thaty(0)<y(n). (We need to know not onlyy(0) but alsoy(n)at
time 0 to decide whether or not this inequality holds.)
- Borrow a dollar for the period between 0 andn+ 1 and deposit it for the
period between 0 andn,bothattheratey(0). (The yield can be regarded
as the interest rate for deposits and loans.) - At timenwithdraw the deposit with interest, enτ y(0)in total, and invest
this sum for a single time step at the ratey(n).At timen+ 1 this brings
enτ y(0)+τy(n). The initial loan requires repayment of e(n+1)τy(0),leavinga
positive balance enτ y(0)(eτy(n)−eτy(0)),which is the arbitrage profit.
The reverse inequalityy(0)>y(n) can be dealt with in a similar manner.
Exercise 10.1
Letτ = 121. Find arbitrage if the yields are independent of maturity,
and unit bonds maturing at time 6 (half a year) are traded atB(0,6) =
0 .9320 dollars andB(3,6) = 0.9665 dollars, both prices being known at
time 0.