# Gamma PnL from Itô's Lemma derivation

The change in a call portfolio ($$f$$), derived from Itô's Lemma, is: \begin{align*} \left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=r\left( f-rS\frac{\partial f}{\partial S}\right)\mathrm{d} t, \\ \implies\frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} -rf&=0 \end{align*}

where $$\frac{\partial f}{\partial t}$$ denotes theta, $$\frac{\partial f}{\partial S}$$ denotes delta and $$\frac{\partial^2 f}{\partial S^2}$$ denotes gamma.

So gamma's PnL would be $$\frac{1}{2}\Gamma \sigma^2 \mathrm{d}S^2$$, where $$\mathrm{d}S^2$$ is the underlying price's change.

But why is gamma's PnL in reallity $$\frac{1}{2}\Gamma \mathrm{d}S^2$$, and not the previous formula? Why shouldn't volatility be included gamma's PnL?

$$\frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2 \approx \frac{1}{2} \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} dt$$

(for small $$dt$$, ignoring $$(dt)^2$$ terms )

$$\sigma$$ is embedded in $$dS = \mu S dt + \sigma S dW$$ and $$dS^2 = \mu^2 S^2 dt^2 + 2\mu \sigma S^2 dt dW + \sigma^2 S^2 dt \approx \sigma^2 S^2 dt$$

You picked up $$1/2\Gamma \sigma^2$$ from the PDE and for some (unknown) reason you multiplied it by $$dS^2$$. You can only multiply it by $$S^2$$ as in the PDE (to get PnL per unit of time) or by $$S^2 dt$$ like in the SDE (to get dollar PnL).

• Thanks for answering, but I don't get why, even for small $dt$, the volatility variable dissapears. Could you elaborate a little bit about it, please? Jun 30 '20 at 0:07
• @DUM03 Just added an explanation.
– ir7
Jun 30 '20 at 0:12
• Tks, things are becoming clearly. Why we have $≈ σ^2S^2 dt$, instead of $≈ σ^2S^2 dW^2$? It would be because $dW = e√dt$, that turns into $e^2*dt$, where "$e$" follows a Wiener process? If it is correct, what should we do with $e^2$? Thanks Jun 30 '20 at 0:47
• See this link for a formal explanation: math.stackexchange.com/questions/81865/wiener-process-db2-dt
– ir7
Jun 30 '20 at 0:53
• @DUM03 Great. Don't forget to accept the answer so I get the point bounty! :)
– ir7
Jun 30 '20 at 1:03