I would like to proof mathematically and intuitively that adding some quantity of underlying to a portfolio of option does not change the overall gamma. Can you help me?

  • $\begingroup$ Intuitively, and considering you are not talking about options on options, the underlying would have delta of 1 and gamma of 0 since it has no optionality. So no matter what quantity of the underlying you add to the portfolio, the change in delta for a change in the price of the underlying (gamma) will remain the same. $\endgroup$ – David Duarte Dec 27 '19 at 9:19
  • $\begingroup$ Could it be because the second derivative of the underlying with respect to the underlying is zero? $\endgroup$ – Magic is in the chain Dec 27 '19 at 12:23

overall gamma is second derivative of whole portfolio over underlying.
adding any function (such as underlying*constant) which second derivative is 0 does not alter overall second derivative.

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