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Question

Suppose a stock pays 2 discrete dividends $d_1, d_2$ at times $t_1, t_2$ respectively, where $ t < t_1 < t_2 < T.$ Assume the risk-free rate, $r$, is a positive constant. Given that

  1. The lower and upper bounds for an American call, $C_t$, is trivially determined as $$\max{ \left( S_t - D_t - Ke^{-r(T-t)}, S_t - K, 0 \right)} \leq C_t \leq S_t.$$

  2. Consider a strategy that exercises the option at the instant before time $t_1$, we have $$S_t - Ke^{-r(t_1 -t)} \leq C_t.$$

  3. Consider a strategy that exercises the option at the instant before time $t_2$, we have $$S_t - d_1e^{-r(t_1 -t)} - Ke^{-r(t_2 -t)} \leq C_t.$$

Combining the three inequalities above, what is the improved lower bound for an American call in this situation?

My attempt

I understand that American call with discrete dividends should at most be exercised at the instant before the ex-dividend dates for optimal payoff, so I obtained the lower bound as

$$\max{\left(S_t - Ke^{-r (t_1 - t)}, S_t - d_1e^{-r (t_1 - t)} - Ke^{-r (t_2 - t)}, S_t - D_t- Ke^{-r (t_1 - t)}, 0\right)} \leq C_t $$ where $D_t = d_1e^{-r (t_1 - t)} + d_2e^{-r (t_2 - t)}.$

However, I am not sure if I should include $S_t - K$ into the max function which was given in the original question.

Any intuitive explanation will be highly appreciated!

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