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I am a little bit confusded with respect to the PnL of a delta-neutral portfolio.

We have $$d\Pi = \Theta dt + \frac{1}{2} \Gamma \Delta S^2$$

So, if our portfolio consists of 1 call options, and we dynamically short delta in the underlying + borrow/lend whenever needed via the bank account .... what is the theta and gamma of such a portfolio?

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  • $\begingroup$ Maybe this related question and answer helps: quant.stackexchange.com/questions/32974. While that one is considering only instantaneous jumps in the spot price, you have the additional $\Theta$ term. This corresponds to expanding the P&L of your portfolio to 2nd order in $\Delta S$ and 1st order in $\Delta t$. $\endgroup$
    – LocalVolatility
    Commented Apr 14, 2017 at 7:11
  • $\begingroup$ Are you sure about your equation? For a delta - neutral portfolio, you should have: $$ r \Pi = \Theta + \frac{1}{2} \sigma^2 S^2 \Gamma $$ What do you mean by "dynamically short delta"? If you are delta-hedged, this equation must be true. Therefore, since the value of your portfolio $\Pi$ cannot be negative, it means that when $\Theta$ is negative, $\Gamma$ is positive, and vice versa. $\endgroup$
    – JejeBelfort
    Commented Apr 18, 2017 at 8:15

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