# Making portfolio Delta and Gamma neutral using 2 derivatives

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?

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.

• Can you calculate the hedge and see the outcome? – simsalabim Jan 12 '20 at 14:42
• the second solution looks correct - you can verify by just plugging in the computed weights into the two equations, i.e,: −19÷8×−1−7÷8×5=-2, and −19÷8×2−7÷8×−2=-3 – Magic is in the chain Jan 12 '20 at 14:54