Question

You have a Delta-neutral portfolio of options and underlying stocks with Gamma I1 and Vega I2. You can trade two options. The first option has Delta, Gamma and Vega, respectively, of .5, .6 and 1.5. The second option has Delta, Gamma and Vega, respectively, of .4, .7 and 2.5. Determine your hedging strategy to make your portfolio neutral for Delta, Gamma and Vega.

30.   How many numbers of the first option will you trade?

31.   How many units of the second option will you trade?

I1	I2
-4490	-7520

Answer 1

Solution :

We have two options and their respective values are given for delta gamma and vega

Option 1: Delta = .5, Gamma= 0.6, vega =1.5

Option 2: Delta = .4, Gamma= 0.7, vega =2.5

Portfolio is delta nuetral so Delta = 0 , Gamma = -4490, Vega = -7520

The stock has the delta of one so from these options we will try to make the portfolio gamma nad vega nuetral

Lets assume We trade X1 options of option 1 and X2 options of option 2

So these trades will be done in such a manner that these trades will equal the Gamma and Vega value of the portfolio

So : For Gamma , 0.6X1 + 0.7X2 = 4490...(.1)

So : For Vega , 1.2X1 + 2.5X2 = 7520.....(2)

Solving these two equations

Multiply the equation by 2 and then Substracting the equation

1.2X1 + 1.4X2 - 1.2X1 - 2.5X2 = 4490*2 - 7520 = 1460

-1.1 X2 = 1460

X2 = -1460/1.1 = -1327.27 = 1328

Putting the value of X2 in equation 1

0.6X1 + 0.7X2 = 4490

0.6X1 + 0.7 * -1327.27 = 4490

0.6X1 = 4490 + 929

X1 = 5419/0.6 = 9032

30.How many numbers of the first option will you trade?

X1 = 9032

31.   How many units of the second option will you trade?

X2 =1327