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Given a strike price, current price, risk free rate, dividend yield and volatility, I have been asked to calculate: - a hedge which matches the value Delta and Gamma - a hedge which matches the value Delta and Vega

I have managed to calculate both of these, however I have also been asked whether it is possible to calculate a hedge which matches the value Delta, Gamma and Vega?

My gut instinct is no, but I am not entirely sure why, and I cannot find anything online to help me. If someone could help me understand why it is yes or no, it would be greatly appreciated!

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If you consider delta, gamma and vega as three variables, and you are able to construct a portfolio with any values, i.e. with three degrees of freedom:

$$ [\delta, \gamma, \theta]$$

And you have a space of products which allow you to construct a hedge for any such delta and vega then you must have at least these two degrees of freedom (in some basis):

$$ [\delta, \gamma, \theta] + x [1, 0, 0] + y [0, 0, 1] = [0, \gamma, 0] $$

Similarly if you can do it for delta and gamma then:

$$ [\delta, \gamma, \theta] + x [1, 0, 0] + z [0, 1, 0] = [0, 0, \theta] $$

So then by definition you must have the necessary degrees of freedom, i.e the combination of products to hedge the full suite:

$$ [\delta, \gamma, \theta] + a [1, 0, 0] + b [0, 1, 0] + c [0, 0, 1] = [0, 0, 0] $$

Failing that explanation you can always hedge your portfolio by doing exactly the same but opposite trades!!

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