A financial institution has the following portfolio of over-the- counter derivatives on pounds sterling: Type Position Call Put Call Forward -1,000 -2,000 -500 1,000 Delta of Gamma of Option/Forward Option/Forward 0.5 -0.4 0.7 -1 2.2 1.3 1.8 0 Vega of Option/Forward 1.8 0.7 1.4 0 A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position is the traded option and in pounds sterling would make the portfolio both gamma neutral and delta neutral? Solution: The delta of the portfolio is -1,000 x 0.5-2000 x (-0.4) - 500 x (0.7)+1,000x (-1)= -1050 The gamma of the portfolio is-1,000 x 2.2-2,000 x 1.3-500 x 1.8+1000 x 0 = - 5,700 The vega of the portfolio is -1,000 x 1.8-2,000 x 0.7-500 x 1.4 +1,000 x 0= -3,900 A long position in 3,900 traded options will give a gamma-neutral portfolio since the long position has a gamma of 3,900 x 1.5=+5,850. The delta of the whole portfolio (including traded options) is then: 3,900 x 0.6-1050=1,290. Hence, in addition to the 3,900 traded options, a short position of 1,950 in sterling is necessary so that the portfolio is both gamma and delta neutral. (b) Estimate what happens to the value of the delta and gamma neutral portfolio when there is a shock to the foreign exchange market causing pounds sterling drop by 5% and its volatility increase from 11% to 12%. Note that vega is in the unit of per %. You can ignore the elapsed time in this case (At = 0).
A financial institution has the following portfolio of over-the- counter derivatives on pounds sterling: Type Position Call Put Call Forward -1,000 -2,000 -500 1,000 Delta of Gamma of Option/Forward Option/Forward 0.5 -0.4 0.7 -1 2.2 1.3 1.8 0 Vega of Option/Forward 1.8 0.7 1.4 0 A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position is the traded option and in pounds sterling would make the portfolio both gamma neutral and delta neutral? Solution: The delta of the portfolio is -1,000 x 0.5-2000 x (-0.4) - 500 x (0.7)+1,000x (-1)= -1050 The gamma of the portfolio is-1,000 x 2.2-2,000 x 1.3-500 x 1.8+1000 x 0 = - 5,700 The vega of the portfolio is -1,000 x 1.8-2,000 x 0.7-500 x 1.4 +1,000 x 0= -3,900 A long position in 3,900 traded options will give a gamma-neutral portfolio since the long position has a gamma of 3,900 x 1.5=+5,850. The delta of the whole portfolio (including traded options) is then: 3,900 x 0.6-1050=1,290. Hence, in addition to the 3,900 traded options, a short position of 1,950 in sterling is necessary so that the portfolio is both gamma and delta neutral. (b) Estimate what happens to the value of the delta and gamma neutral portfolio when there is a shock to the foreign exchange market causing pounds sterling drop by 5% and its volatility increase from 11% to 12%. Note that vega is in the unit of per %. You can ignore the elapsed time in this case (At = 0).
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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Transcribed Image Text:A financial institution has thefollowing portfolio of ove
over-the-
counter derivatives on pounds sterling:
Туре
Position
Delta of
Gamma of
Vega of
-1,000
2,000
-500
1,000
Option Forward Option Forward Option/Forward
2.2
1.3
1.8
Call
0.5
1.8
Put
Call
-0.4
0.7
0.7
1.4
Forward
-1
A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.
(a) What position is the traded option and in pounds sterling would make the portfolio
both gamma neutral and delta neutral?
Solution:
The delta of the portfolio is -1,000 x 0.5 - 2000 x (-0.4) - 500 x (0.7) + 1,000 x (-1) =
-1050
The gamma of the portfolio is-1,000 x 2.2 –2,000 x 1.3 - 500 x 1.8+ 1000 x 0 = -
5,700
The vega of the portfolio is -1,000 x 1.8 – 2,000 x 0.7 - 500 x 1.4 +1,000 x 0= -3,900
A long position in 3,900 traded options will give a gamma-neutral portfolio since the
long position has a gamma of 3,900 x 1.5 = +5,850. The delta of the whole portfolio
(including traded options) is then: 3,900 x 0.6 - 1050 = 1,290. Hence, in addition to the
3,900 traded options, a short position of 1,950 in sterling is necessary so that the portfolio
is both gamma and delta neutral.
(b) Estimate what happens to the value of the delta and gamma neutral portfolio when
there is a shock to theforeign exchange market causing pounds sterling drop by 5%
and its volatility increase from 11% to 12%. Note that vega is in the unit of per %.
You can ignore the elapsed time in this case (At = 0).
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