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Think of this in terms of Taylor series. Let's say the option price today is $C\left(S,t\right)$ where S is the underlying price and t time. Let's say the underlying price changes by $\Delta S$ in a time interval $\Delta t$, so your P/L will be: $\mathrm{P/L}=C\left(S+\Delta S,t+\Delta t\right)-C\left(S,t\right) $ Use Taylor series to first order in t and ...

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