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I would like to learn more about the Greeks of portfolios of options:

In textbooks and websites, I commonly encounter the unqualified claim that "The Greek measure of a portfolio is the sum of the Greeks of the individual portfolio components". This statement is obviously true for constant portfolios (due to the linearity of mathematical derivatives), but I am pretty sure it cannot always be true for non-constant, non-self-financing portfolios.

What I am unsure about, is whether the claim is always true for non-constant, self-financing portfolios. For self-financing portfolios, dV = h·dS (where V is the value of the portfolio and S the vector of the values of portfolio components), so I expect that that the claim should be true for first derivatives such as delta and theta. But what about second derivatives such as gamma?

I would appreciate it very much if anyone could provide me with a proof that each of the Greeks of a self-financing portfolio is indeed the sum of the Greeks of the components. Or if the proof is a bit lengthy, if you could refer me to a book (or article or website) where the proof is discussed in proper detail.


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1 Answer 1

If you look at the Black-Scholes demonstration using self-financing portfolio, it uses the assumption of continuous hedging.

Because you continuously hedge and the function is differentiable, it contains the gamma as the delta is hedged continuously. (I think)

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