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The derivation in the book appears wrong. However, the results make sense as the option price at time $t$ should not be impacted by prior dividend payments. It may be out-of topic, I would like to provide some justification of the Musiela-Rutkowski formula. Let $\{H_t \mid t >0\}$, where \begin{align*} H_t = \sum_{0 < T_i \leq t} q_i, \end{align*} ...

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