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Notations are consistent with this answer.

Selling and delta hedging the option $V^i$ using the implied volatility $\sigma_i$ while the actual volatility of the underlying asset is $\sigma_r$. Then the portfolio pnl has $$d\Pi=dV^i-\Delta^i dS-r(V^i-\Delta^i S)dt\tag{*}$$

the underlying has $$\frac{dS}{S}=\mu dt+\sigma_rdW_t$$ apply Ito formula: $$dV^i=\Theta^idt+\Delta^i dS+\frac{1}{2}\Gamma^i\sigma_r^2S^2dt\tag{1}$$

by BS equation: $$\Theta^i+\frac{1}{2}\Gamma^i\sigma^2_iS^2=r(V^i-\Delta^i S)\tag{2}$$

then plug (1)(2) into (*), then we can get $$d\Pi= \frac{1}{2} \left( \sigma_r^2 - \sigma_i^2 \right) S^2 \Gamma^i dt$$

Question:

But I find it quite confusing, since (1) holds under the assumpsion that $$\frac{dS}{S}=\mu dt+\sigma_rdW_t$$

but (2) holds under the assumption that $$\frac{dS}{S}=\mu dt+\sigma_idW_t$$

why can we plug (1)(2) into (*) simutaneously when they have different assumption?

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    $\begingroup$ Rather than unravelling that derivation I recommend to study in detail this answer. $\endgroup$
    – Kurt G.
    Dec 20, 2021 at 13:43
  • $\begingroup$ @KurtG. I have the same question in your answer, the conditions on the equation dc and dPi are different, why you can substract them directly. $\endgroup$
    – Lookout
    Dec 21, 2021 at 4:04
  • $\begingroup$ If you have a detailed question to my answer it is better to write a detailed comment there. Not in this thread. Frankly, I do not see what is wrong with the conditions that lead to $dC$ and $d\Pi$ and why they are different. $dC$ is Ito's formula. $d\Pi$ is the well-known self financing condition. Eq. (1) is due to the BS formula that uses implied vol. The thing that seems to confuse you is that regardless how $C(t,x)$ is calculated, there is always an implied vol for the BS formula hitting that price. $\endgroup$
    – Kurt G.
    Dec 21, 2021 at 10:24

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