108156.pdf

242 Mathematics for Finance

that the words ‘up’ and ‘down’ lose their meaning here because the yield goes
down as the bond price goes up. Nevertheless, we keep the original indicators
u and d.

Example 11.3

We continue Example 11.1 and find the yields, bearing in mind thatτ= 121.
The results are collected in Figure 11.6.

Figure 11.6 Yields in Example 11.3

Exercise 11.2

Take the returns in Exercise 11.1 and find the yieldy(0,3). What is the

general relationship between the returns and yields to maturity? Can

you complete the missing returns without computing the bond prices?

Now consider the instantaneous forward rates. At the initial time 0 there
areNforward rates

f(0,0),f(0,1),f(0,2),...,f(0,N−1)

generated by the initial bond prices. Note that the first number is the short
rater(0) =f(0,0). For all subsequent steps the current bond prices imply the
forward rates. Formula (10.6) applied to random bond prices allows us to find
the random evolution of forward rates:

f(n, N;sn)=−

lnB(n, N+1;sn)−lnB(n, N;sn)
τ. (11.1)
At time 1 we have two possible sequences ofN−1 forward rates obtained from
two sequences of bond prices

f(1,1; u),f(1,2; u),...,f(1,N−1; u),

f(1,1; d),f(1,2; d),...,f(1,N−1; d).