问一道题：NO.PZ2016070201000092

问题如下：

A constant maturity Treasury (CMT) swap's payoff will be ($1,000,000/2) x ( $y_{cmt}$ - 9%)every six months. There is a 70% probability of an increase in the 6-month spot rate and a 60% probability of an increase in the I-year spot rate. The rate change in all cases is 0.50% per period, and the initial $y_{cmt}$ is 9%). What is the value of this CMT swap?

选项：

$2,325.

$2,229.

$2,429.

$905.

解释：

The payoff in each period is ($1,000,000/2) x $y_{CMT}$yCMT​ - 9%). For example, the 1-year payoff of $5,000 in the figure below is calculated as ($1,000,000/2) x (10% -9%) = $5,000. The other numbers in the year one cells are calculated similarly.

In six months, the payoff if interest rates increase to 9.50% is ($1,000,000/2) x (9.5% - 9.0%) = $2,500. Note that the price in this cell equals the present value of the probability weighted 1 -year values plus the 6-month payoff:

$V_{6month,U}=\frac{(\mathrm{\$}5,000\times 0.6)+(\mathrm{\$}0\times 0.4)}{1+\frac{0.095}{2}}+\mathrm{\$}2,500=\mathrm{\$}5,363.96$V6month,U​=1+20.095​($5,000×0.6)+($0×0.4)​+$2,500=$5,363.96

The other cell value in six months is calculated similarly and results in a loss of $4,418.47.

The value of the CMT swap today is the present value of the probability weighted 6-month values:

$V_0=\frac{(\mathrm{\$}5,363.96\times 0.7)+(-\mathrm{\$}4,418\times 0.3)}{1+\frac{0.09}{2}}=\mathrm{\$}2,324.62$V0​=1+20.09​($5,363.96×0.7)+(−$4,418×0.3)​=$2,324.62

Thus the correct response is A. The other answers are incorrect because they do not correctly discount the future values or omit the 6-month payoff from the 6-month values.