need some explanation for 17.24

The Greek Letters Problem 17.24. A financial institution has the following portfolio of over-the-counter options on sterling Type Position Delta ofoption Gamma of option Vega o Option Call -1.000 0.5 -500 08 Call 0.6 0.2 13 Put 2,000 -0.40 18 Call -500 0.70 at is ce 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 -l,000x0.50-500x0.80-2,000x (-0.40)-500x0.70 -450 The gamma of the portfolio is 1,000x2.2-500x0.6 -2,000x1.3-500 1.8 -6,000 The vega of the portfolio is 1,000 x 1.8-500x0.2-2,000x0.7 -500x1.4 -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: 4,000x 0.6 450 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. b. A 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 5,000 x 0.6-450 2,550 Hence, in addition to the 5,000 traded options, a short position of 2,550 in sterling is ecessary so that the portfolio is both vega and delta neutral. Problem 17.25 Consider again the situation in Problem 17.24. Suppose that a second traded option with a of 0.5, and a vega of 0.6 is available. How could the portfolio be ma delta of 0. a gamma