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).

  • $\begingroup$ 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? $\endgroup$ – DUM03 Jun 30 '20 at 0:07
  • $\begingroup$ @DUM03 Just added an explanation. $\endgroup$ – ir7 Jun 30 '20 at 0:12
  • $\begingroup$ 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 $\endgroup$ – DUM03 Jun 30 '20 at 0:47
  • 1
    $\begingroup$ See this link for a formal explanation: math.stackexchange.com/questions/81865/wiener-process-db2-dt $\endgroup$ – ir7 Jun 30 '20 at 0:53
  • 1
    $\begingroup$ @DUM03 Great. Don't forget to accept the answer so I get the point bounty! :) $\endgroup$ – ir7 Jun 30 '20 at 1:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.