NO.PZ2023100703000108
问题如下:
A derivatives trader at an investment bank is considering how to hedge a relatively illiquid 7-year USD interest-rate swap the bank just entered into as the fixed-rate payer. The trader recognizes that any profit resulting from the bid-ask spread may be lost if the trade is hedged with another illiquid 7-year swap and considers using the more liquid 5-year and 10-year USD interest-rate swaps as a hedge. To evaluate this possible hedge, the trader runs a two-variable regression model using changes in the 5-year and 10-year swap rates to explain changes in the 7-year swap rate. The regression model, regression results, and information about the swaps are given below:
What are the correct notional amounts of 5-year and 10-year swaps needed to hedge a USD 100 million notional amount of 7-year swaps?
选项:
A.USD 23.76 million of 5-year swaps, and USD 65.81 million of 10-year swaps B.USD 24.71 million of 5-year swaps, and USD 65.36 million of 10-year swaps C.USD 34.03 million of 5-year swaps, and USD 47.74 million of 10-year swaps D.USD 68.85 million of 5-year swaps, and USD 36.52 million of 10-year swaps解释:
C is correct.
To determine the notional amount of
the 5-year and 10-year swaps required to hedge USD 100 million notional of a
7-year swap it must be understood that the
P&L of the position is given by:
-F7*(DV017/100)*Δyt7
– F5*(DV015/100)*Δyt5 – F10*(DV0110/100)*Δyt10
In a perfect hedge this P&L would
equal zero.Substituting the predicted change in the 7-year swap rate from the
regression equation into the P&L equation given above, and retaining only
the terms related to the changes in the 5-year and 10-year swap rates gives the
following equation for the P&L:
[-F7*(DV017/100)*β5
– F5*(DV015/100)]*Δyt5 + [-F7*(DV017/100)*β10
– F10*(DV0110/100)]*Δyt10
Now set the terms inside the brackets
equal to zero so the P&L is not dependent on changes in the 5-year and
10-year swap rates. This leaves:
F5 = -F7 * (DV017/DV015)
* β5
F10 = -F7 * (DV017/DV0110)
* β10
Now, using the information derived
from the regression and the DV01s, the notional amounts of the 5-year and
10-year swaps can be calculated.
F5 = -100* (0.084/0.061) *
0.2471 = -34.03, USD 34.03 million as the fixed-rate receiver
F10 = -100* (0.084/0.115) *
0.6536 = -47.74, USD 47.74 million as fixed-rate receiver
A is incorrect. The ratio of fixed
rates is used instead of the ratio of DV01s in the formula.
F5 = -100*
(0.02492/0.02591) * 0.2471 = -23.76, USD 23.76 million as the fixed-rate receiver
F10 = -100*
(0.02492/0.02475) * 0.6536 = -65.81, USD 65.81 million as fixed-rate receiver
B is incorrect. This answer choice
simply multiplies the Betas and the initial notional amount.
D is incorrect. This answer choice
uses the ratio of DV01s but not the betas, then divides by 2 since the position
is a butterfly.
F5 = -100* (0.084/0.061) /
2 = -68.85, USD 68.85 million as the fixed-rate receiver
这个解释里面DV01为什么都要除以100?