Question

A financial institution has the following portfolio of over-the-counter options on sterling:

Type	Position	Delta of Option	Gamma of Option	Vega of Option
Call	-2,000	0.5	2.2	1.8
Call	-1000	0.8	0.6	0.2
Put	-4,000	-0.40	1.3	0.7
Call	-1000	0.70	1.8	1.4

A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.

Is it possible to find a position in the traded option and in sterling that make the portfolio gamma neutral, vega neutral and delta neutral? Explain.        

Answer 1

The delta of the portfolio is

−1, 000 × 0.50 − 500 × 0.80 − 2,000 × (−0.40) − 500 × 0.70 = −450

The gamma of the portfolio is

−1, 000 × 2.2 − 500 × 0.6 − 2,000 × 1.3 − 500 × 1.8 = −6,000

The vega of the portfolio is

−1, 000 × 1.8 − 500 × 0.2 − 2,000 × 0.7 − 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, 000 × 1.5 = +6,000. The delta of the whole portfolio (including traded options) is then:

4, 000 × 0.6 − 450 = 1, 950

Hence, in addition to the 4,000 traded options, a short position in £1,950 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, 000 × 0.8 = +4,000. The delta of the whole portfolio (including traded options) is then

5, 000 × 0.6 − 450 = 2, 550

Hence, in addition to the 5,000 traded options, a short position in £2,550 is necessary so that the portfolio is both vega and delta neutral.