# Is it possible to construct a hedge that matches value Delta Gamma and Vega?

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!

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!!