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Homework answers / question archive / 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.
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