- Stochastic Interest Rates 253
Proof
Consider two bonds with maturitiesM≤Nand fix ann≤M. For each of the
two bonds we have
B(n, M;sn)=[p∗(n, M;sn)B(n+1,M;snu) + (1−p∗(n, M;sn))
×B(n+1,M;snd)] exp{−τr(n;sn)}, (11.7)
B(n, N;sn)=[p∗(n, N;sn)B(n+1,N;snu) + (1−p∗(n, N;sn))
×B(n+1,N;snd)] exp{−τr(n;sn)}. (11.8)Our goal is to show thatp∗(n, M;sn)=p∗(n, N;sn)inanystatesn.
We can replicate the prices of the bond maturing at timeMby means of the
bond with maturityNand the money market account. Hence we find numbers
x, ysuch that
B(n+1,M;snu) =xB(n+1,N;snu) +yA(n+1;sn),
B(n+1,M;snd) =xB(n+1,N;snd) +yA(n+1;sn).The No-Arbitrage Principle implies that equalities of this kind must also hold
at timen,
B(n, M;sn− 1 u) =xB(n, N;sn− 1 u) +yA(n;sn− 1 ),
B(n, M;sn− 1 d) =xB(n, N;sn− 1 d) +yA(n;sn− 1 ).Inserting the values of theM-bonds into (11.7) and using the formula for the
money market account, after some algebraic transformations we obtain
B(n, N;sn)=[p∗(n, M;sn)B(n+1,N;snu) + (1−p∗(n, M;sn))
×B(n+1,N;snd)] exp{−τr(n;sn)}.Thiscanbesolvedforp∗(n, M;sn). It turns out that the solution coincides
with the probabilityp∗(n, N;sn) implied by (11.8), as claimed.
Exercise 11.10
Spot an arbitrage opportunity if the bond prices are as in Figure 11.15.11.3 Interest Rate Derivative Securities..........................
The tools introduced above make it possible to price any derivative security
based on interest rates or, equivalently, on bond prices. Within the binomial tree