Greeks of self-financing portfolio

Question

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.

Thanks!

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)

Answer 2

Well if we take a call option the gamma is non-zero.

If we take the replicating portfolio for a call option, it consists of stock and bonds. Both of these have zero gamma.

So in the form asked, I think the result is false for second derivatives.

Answer 3

I think that the question in its current form is not really precise. The Greeks are quite easy to define for a given formula of a price: in that case they are simply partial derivatives, and of course linear (and in some case commute with integrals rather than just finite sums). However, they are crucially model-dependent - not only their expressions, but also their meanings. For example, Vega is sensitivity to the BS IV, which is rather strange to use in purely jump models, not to say that the definition of Vega through IV seems to be rather circular.

Nevertheless, in some sense your question has a positive answer. To any strategy $\varphi$ we can assign its value $V_t(\varphi) = S_t\cdot \varphi_t$, so let's talk about sensitivities of $V_t$ w.r.t. some reasonable variables. Consider a case of BS model, $S_0 =\mathrm e^{rt}$, and let $\varphi$ replicate the price of a European call. In that case $$ \varphi^0_t = \mathrm e^{-rt}(c - \Delta S), \quad \varphi_t^1 = \Delta $$ so $V_t(\varphi) = c$ and of course all the derivatives of $V$ are those of $c$. Yet again, you may see some circularity in this argument, but I do not know how to help here unless you clarify your question in more precise terms. By no means it's because of the way you have formulated it, more that it concerns the fact that Greeks mathematically are rather ambiguous as an object.

To summarize: as Mark pointed out in his answer, Gamma of $S$ and cash account is zero, hence in the dynamic portfolios it is crucial to check what are the Greeks of the coefficients $\varphi$ in your portfolio, not only those of the securities that enter it. With focus on Gamma, suppose you have a self-financing strategy which does not necessarily supposed to replicate anything, just the two stochastic processes $\varphi^0$ and $\varphi^1$ that represent the amount invested in cash account and stocks respectively. Let's say $\varphi^1$ is what you choose, so to satisfy the self-financing condition $\mathrm d\varphi^0_t = -\mathrm e^{-rt}S_t\mathrm d\varphi^1_t$. The way you choose $\varphi^1$ depending say on $S$ determines your Greeks as well, for example in case of a European call replicating strategy, $\varphi^1$ is its $\Delta$, and hence determines its Gamma. Perhaps, that also holds in general.