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We have $$ \begin{align} V(t) &= \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)] \\ &= \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B}\mathbb{1}_{S(T_2)>K} (S(T_2)-K))] \\ &= \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B}\mathbb{1}_{S(T_2)>K} S(T_2)]-K\mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B}\mathbb{1}_{S(T_2)>K}] \\ \end{align} $$ The second term ...


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