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Problem 17
Problem 17.24. A financial institution has the following portfolio of over-the-counter options on sterling: Type Call Call Put Call Position -1,000 -500 -2,000 -500 Delta of Option 0.5 0.8 -0.40 0.70 Gamma of Option 2.2 0.6 1.3 1.8 Vega of Option 1.8 0.2 0.7 1.4 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 in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral? b. What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral? The delta of the portfolio is -1,000x0.50 - 500x0.80 – 2,000~(-0.40) - 500x0.70 = -450 The gamma of the portfolio is -1,000x 2.2 - 500x0.6 – 2,000~1.3 – 500x1.8=-6,000 The vega of the portfolio is -1,000x1.8 - 500x0.2 - 2,0000.7 - 500~1.4 = -4,000 a. A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of 4,000x1.5 = +6,000. The delta of the whole portfolio (including traded options) is then: 4,000x0.6-450 = 1,950 Hence, in addition to the 4,000 traded options, a short position of 1,950 in sterling is necessary so that the portfolio is both gamma and delta neutral. long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of 5,000x0.8 = +4,000. The delta of the whole portfolio (including traded options) is then 5,000x0.6 - 450 = 2,550 Hence, in addition to the 5,000 traded options, a short position of 2,550 in sterling is necessary so that the portfolio is both vega and delta neutral.
Expert Solution
Formula's Used :-
The Deta of Portfolio = Sum of all, Postion of each type of call multiply with Delta of each type of call
I.e.,in the 1st call type :- Postion is -1000 and Delta option is 0.5 then -1000*0.5, simillarl calculate of the types of calls in the portofolio and add them, the result will be the delta of Portfolio (1.e.,-450)
The Gamma of Portfolio = Sum of all, Postion of each type of call multiply with Gamma of each type of call
I.e.,in the 1st call type :- Postion is -1000 and Gamma option is 2.2 then -1000*2.2, simillarl calculate of the types of calls in the portofolio and add them, the result will be the Gamma of Portfolio.
The Vegaof Portfolio = Sum of all the, Postion of each type of call multiply with Vega of each type of call
I.e.,in the 1st call type :- Postion is -1000 and Vega option is 1.8 then -1000*1.8, simillarl calculate of the types of calls in the portofolio and add them, the result will be the Vega of Portfolio.
Formula used in Part -A:-
To find the delta of whole portfolio( including Traded Options) :-
= Delta of portfolio without Traded Options + Delta of portfolio of Traded Options
Delta of portfolio without Traded Options is already calculated above i.e., $-450.
Delta of Traded Options is given in the question as 0.6
Traded options is given in the question is 4,000
Now Delta of whole portfolio = Traded Positions (given in Question) * Delta of traded option + Delta of portofolio without Traded option.
=4000*0.6-450
=1950
Formula used in Part -B:-
To find the delta of whole portfolio( including Traded Options) :-
= Delta of portfolio without Traded Options + Delta of portfolio of Traded Options
Delta of portfolio without Traded Options is already calculated above i.e., $-450.
Delta of Traded Options is given in the question as 0.6
Traded options is given in the question is 5,000
Now Delta of whole portfolio = Traded Positions (given in Question) * Delta of traded option + Delta of portofolio without Traded option.
=5000*0.6-450
=2550
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