244 Mathematics for Finance
Figure 11.8 Forward rates in Exercise 11.3We are now ready to describe the money market account. It starts with
A(0) = 1.The next value
A(1) = exp(τr(0))
is still deterministic. It becomes random at subsequent steps. At time 2 there
are two values depending on the states at time 1:
A(2; u) = exp(τ(r(0) +r(1; u)) =A(1) exp{τr(1; u)},
A(2; d) = exp(τ(r(0) +r(1; d)) =A(1) exp{τr(1; d)}.Next, for example,
A(3; ud) = exp(τ(r(0) +r(1; u) +r(2; ud)) =A(2; u) exp{τr(2; ud)}.In general,
A(n+1;sn− 1 u) =A(n;sn− 1 )exp{τr(n;sn− 1 u)},
A(n+1;sn− 1 d) =A(n;sn− 1 )exp{τr(n;sn− 1 d)}.Exercise 11.4
Find the evolution of the money market account if the forward rates are
the same as in Exercise 11.3.For bond investments the money market account plays the same role as
the risk-free component of investment strategies on the stock market in ear-
lier chapters. It is used to discount future cash flows when valuing bonds and
derivative securities, as will be shown below.