$ While in the "do the job situation" you liquidate the portfolio at $t_1$ realising its PnL (allow me to simplify the notation a bit) $begingroup$ For an alternative with price $C$, the P$&$L, with respect to variations in the fundamental asset price $S$ and volatility $sigma$, is provided by https://www.youtube.com/watch?v=qMmsQ4kKgY4