Question

Consider a portfolio dependent on the price of a single asset that is delta neutral, with a gamma of -6000 and a vega of -9600. Suppose that a traded option (called Option 1) with a delta of 0.3, a gamma of 0.5 and a vega of 1.0 is available.

1. How could the portfolio be made delta and gamma neutral?
2. How could the portfolio be made delta and vega neutral?
3. If another traded option (called Option 2) with a delta of 0.7, a gamma of 1.2 and a vega of 1.6 is available. How could the portfolio be made delta, gamma and vega neutral?

Answer 1

	Delta	Gamma	Vega
Portfolio	0	-6000	-9600
Option 1	0.3	0.5	1
Option 2	0.7	1.2	1.6

(a) The portfolio could be made Gamma neutral by including in the portfolio a long position of 12000 (=6000/0.5) of Option 1.

This would increase delta to 3600 (= 12000*0.3) and require that 3600 units of the asset be sold to maintain delta neutrality.

The vega of the portfolio would change to 2400 (= −9600 +12000*1).

(b) The portfolio could be made vega neutral by including in the portfolio a long position of 9600 (=9600/1) of Option 1.

This would increase delta to 2880 (= 9600*0.3) and require that 2880 units of the asset be sold to maintain delta neutrality.

The gamma of the portfolio would change to -1200 (= −6000 +9600*0.5).

(c) To make the portfolio gamma and vega neutral, both Option 1 and Option 2 can be used.

Let w1, w2be the numbers of traded Option 1, traded Option 2 that are added to the portfolio. Then we require that

−6000 + 0.5w1+ 1.2w2= 0

−9600 + 1.0w1+ 1.6w2= 0

Solving the two equations simultaneously gives w1= 4800, w2=3000.

Therefore, the portfolio can be made gamma and vega neutral by including 4800 of Option 1 and 3000 of Option 2.

The delta of the portfolio after the addition of Option 1 and Option 2 is:

[0 + 0.3*4800 + 0.7*3000] = 3540

Hence, 3540 units of the asset need to be sold to maintain delta neutrality.