Question

A trader decides to hedge her portfolio against large market moves by taking positions in the underlying asset and two options (see table below).

	Delta	Gamma	Vega
Portfolio	0.50	-1000	-500
Option 1	-0.03	10	2.5
Option 2	0.10	80	10

a) How many units of options 1 and 2, respectively, are needed to make the portfolio both gamma and vega neutral?

b) How many units of the underlying asset are needed to make the hedged portfolio from part (a) delta neutral (indicate long/short position)?

Answer 1

1.
Let the number of units of option a be x and option b be y

For gamma neutrality:
10x+80y-1000=0

For vega neutrality:
2.5x+10y-500=0

=>x=300, y=-25

2.
Let the units of underlying asste be z

For delta neutrality:
0.50+300*(-0.03)-25*0.10+z=0
=>z=11