230 Mathematics for Finance
Remark 10.1
To determine the initial term structure we need the prices of zero-coupon bonds.
However, for longer maturities (typically over one year) only coupon bonds may
be traded, making it necessary to decompose coupon bonds into zero-coupon
bonds with various maturities. This can be done by applying formula (10.3)
repeatedly to find the yield with the longest maturity, given the bond price and
all the yields with shorter maturities. This procedure was recognised by the
U.S. Treasury, who in 1985 introduced a programme called STRIPS (Separate
Trading of Registered Interest and Principal Securities), allowing an investor
to keep the required cash payments (for certain bonds) by selling the rest (the
‘stripped’ bond) back to the Treasury.
Example 10.9
Suppose that a one-year zero-coupon bond with face value $100 is trading at
$91.80 and a two-year bond with $10 annual coupons and face value $100 is
trading at $103.95. This gives the following equations for the yields
91 .80 = 100e−y(0,1),
103 .95 = 10e−y(0,1)+ 110e−^2 y(0,2).From the first equation we obtainy(0,1)∼= 8 .56%. On substituting this into
the second equation, we findy(0,2)∼= 7 .45%. As a result, the price of the
‘stripped’ two-year bond, a zero-coupon bond maturing in two years with face
value $100, will be 100e−^2 y(0,2)∼= 86 .16 dollars. Given the price of a three-year
coupon bond, we could then evaluatey(0,3), and so on.
Going back to our general study of bonds, let us consider a deterministic
term structure (thus known in advance with certainty). The next proposition
indicates that this, in fact, is not realistic.
Proposition 10.2
If the term structure is deterministic, then the No-Arbitrage Principle implies
that
B(0,N)=B(0,n)B(n, N). (10.4)
Proof
IfB(0,N)<B(0,n)B(n, N),then: