Making portfolio Delta and Gamma neutral using 2 derivatives

Question

We have an option portfolio with delta =2 and gamma 3 and we want to making this portfolio delta and gamma neutral using two derivatives D1 and D2:

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|        |Delta | Gamma|
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| Option | 2    | 3    |
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| D1     | -1   | 2    |
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| D2     | 5    | -2   |
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I have tried two ways for solving this and they both give different answers:

1)

$w_{D1}*\Delta_{D1} + w_{D2}*\Delta_{D2} = -2$ $w_{D1}*\Gamma_{D1} + w_{D2}*\Gamma_{D2} = -3$

With answers: $w_{D1}$ = -4/9 and $w_{D2}$ = -1/9

2)

$2 -1w_{D1} + 5w_{D2} = 0$;

$3 + 2w_{D1} + -2w_{D2} = 0$

With answers: $w_{D1}$ = -19/8 and $w_{D2}$ = -7/8

Can someone tell me where I do go wrong and give an interpretation of the results? Which technique should be used?

Answer 1

The two formulations seem to be exactly the same. If I take the equations from the first method:

$w_{D1}*\Delta_{D1} + w_{D2}*\Delta_{D2} = -2$

$w_{D1}*\Gamma_{D1} + w_{D2}*\Gamma_{D2} = -3$

And substitute for delta and gamma of the two options:

$-w_{D1}+ 5 w_{D2}= -2$

$2w_{D1} -2w_{D2} = -3$

which after shifting the constants to the left becomes exactly the same set as in method 2:

$2-w_{D1}+ 5 w_{D2}= 0$

$3+2w_{D1} -2w_{D2} = 0$

Maybe there is a typo in the solution method you used when solving the first set of equations.