To consider the pricing of derivatives written on the spot rate, we construct the recombining

binomial tree model for $6-$month spot rates as in Figure $1$:

$t=0,t=0\mathrm{.}5,t=1\mathrm{.}0,t=1\mathrm{.}5,t=2\mathrm{.}0$

Figure $1$: Recombining binomial tree model for $6-$month spot rates hat$(r)(0\mathrm{.}5{)}_t.$ Here, $q_t$ is the

risk$-$neutral probability of the upward event at the calendar time $t.$

Calculate the price of the constant$-$maturity Treasury $($CMT$)$ swap written on the $6-$month

spot rate.

The risk$-$neutral probability at each period is assumed to be common among the states. For

example, under the risk$-$neutral probability,

$q_{1\mathrm{.}5}=P^{**}\{(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}5}=2\mathrm{.}3\%|(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}0}=2\mathrm{.}2\%\}$

$=P^{**}\{(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}5}=2\mathrm{.}1\%|(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}0}=2\mathrm{.}0\%\}$

$=P^{**}\{(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}5}=1\mathrm{.}9\%|(hat(r))(0\mathrm{.}5{)}_{1\mathrm{.}0}=1\mathrm{.}8\%\},$

where hat$(r)(0\mathrm{.}5{)}_t$ is the $6-$month spot rate at the calendar time $t.$

Remark: The cashflow of the CMT swap contract at $t$ is given by

$1,000,000\frac{(hat(r))(0\mathrm{.}5{)}_t-2\%}{2}.$

For example, if the $6-$month spot rate at $t=2\mathrm{.}0$ is $2\mathrm{.}2\%,$ the cashflow is

$2\%.$