- Stochastic Interest Rates 249
These fractions, defining thecoupon rate, are obtained by converting the short
rate to an equivalent discrete compounding rate. In practice, whenτis one day,
the coupon rate will be the overnight LIBOR rate.
Proposition 11.1
A coupon bond maturing at timeNwith random coupons
Ck(sk− 1 )=(exp{τr(k−1;sk− 1 )}−1)F (11.4)for 0<k≤Nis tradingat par. (That is, the priceP(0) is equal to the face
valueF.)
Proof
Fix timeN−1 and a statesN− 1 .In this state the valueP(N−1;sN− 1 )
of the bond is F+CN(sN− 1 ) discounted at the short rate, which gives
P(N−1;sN− 1 )=F if the coupon is expressed by (11.4). Proceeding back-
wards through the tree and applying the same argument for each state, we
finally arrive atP(0) =F.
Exercise 11.6
Find the coupons of a bond trading at par and maturing at time 2,given
the yields as in Example 11.5, see Figure 11.11.11.2.1 Risk-Neutral Probabilities
In Chapter 3 we have learnt that the stock priceS(n)attimenis equal to the
expectation under the risk-neutral probability of the stock priceS(n+1) at
timen+ 1 discounted to timen. The situation is similar in the binomial model
of interest rates.
The discount factors are determined by the money market account, or, in
other words, by the short rates. In general, they are random, being of the form
exp{−τr(n;sn)}.
Suppose that statesnhas occurred at timen. The short rate determining
the time value of money for the next step is now known with certainty. Consider
a bond maturing at timeNwithn<N− 1 .We are given the bond price
B(n, N;sn) and two possible values at the next step,B(n+1,N;snu) andB(n+
1 ,N;snd).These values represent a random variable, which will be denoted by
B(n+1,N;sn·).Ifn=N− 1 ,then the bond matures at the next stepN,