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It is well known that in the Black Scholes model with implied vol $s$ and realized vol $\sigma$ the PnL from delta hedging a long position in an option is $$\tag{1} C(T,S_T)-\Pi_T=\frac{\sigma^2-s^2}{2}\int_0^TS_t^2\partial_x^2C(t,S_t)\,dt\,. $$ (see this post). The formula you use in Method 2 is similar but not exactly equal to this. The option price $C(t,...

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