300 Mathematics for Finance
andf(0,N)<0 requires that (N+1)y(0,N+1)<Ny(0,N).The border
case is wheny(0,N+1)=NN+1y(0,N),which enables us to find a numerical
example. For instance, forN=8andy(0,8) = 9% a negative valuef(0,8)
will be obtained ify(0,9)<^89 ×9% = 8%.
10.20Suppose thatf(n, N) increases withN.We want to show that the same is
true for
y(n, N)=f(n, n)+f(n, n+1)+N−n···+f(n, N−1).
This follows from the fact that if a sequenceanincreases, then so does the
sequence of averagesSn=a^1 +···n+an.To see this multiply the target inequality
Sn+1>Snbyn(n+ 1) to get (after cancellations)nan+1>a 1 +···+an.The
latter is true, sincean+1>aifor alli=1,...,n.
10.21The values ofB(0,2),B(0,3),B(1,3) have no effect on the values of the money
market account.
10.22a) For an investment of $100 in zero-coupon bonds, divide the initial cash
by the price of the bondB(0,3) to get the number of bonds held, 102.82,
which gives final wealth of $102.82. The logarithmic return is 2.78%. b) For an
investment of $100 in single-period zero-coupon bonds, compute the number
of bonds maturing at time 1 as 100/B(0,1)∼= 100. 99 .Then, at time 1 find the
number of bonds maturing at time 2 in a similar way, 100. 99 /B(1,2)∼= 101. 54.
Finally, we arrive at 101. 54 /B(2,3)∼= 102 .51 bonds, each giving a dollar at
time 3. The logarithmic return is 2.48%. c) An investment of $100 in the money
market account, for which we receive 100A(3)∼= 102 .51 at time 3, produces
the same logarithmic return of 2.48% as in b).
Chapter 11
11.1We begin from the right, that is, from the face values of the bonds, first
computing the values ofB(2,3) in all states. These numbers together with
the known returns giveB(1,3; u) andB(1,3; d).These, in turn, determine the
missing returnsk(2,3; ud) = 0.20% andk(2,3; dd) = 0.16%.The same is done
for the first step, resulting ink(1,3; d) = 0.23%.The bond prices are given in
Figure S.10.Figure S.10 Bond prices in Solution 11.111.2Because of the additivity of the logarithmic returns,k(1,3; u) +k(2,3; uu) +
k(3,3; uuu) = 0.64% gives the return in the period of three weeks. To obtain
the yield we have to rescale it to the whole year by multiplying by 52/3, hence
y(0,3) = 11.09%.Note that we must havek(1,3; u)+k(2,3; ud)+k(3,3; udu) =