Combining Exercise-Date Bounds for an American Call with Discrete Dividends

Summary

The document asks how to strengthen a lower bound for an American call when a stock pays two discrete dividends before expiration. It starts from a basic bound using the stock price, the present value of dividends, and the discounted strike. It then adds candidate values obtained by considering exercise immediately before each dividend date. The proposed reasoning recognizes that exercising at different times changes both the dividends received and how long the strike payment can remain invested.

The suggested result is to take the maximum of the baseline terms and the values associated with exercise before the first and second dividend dates. The question specifically asks whether the immediate-exercise term, stock price minus strike, must also be retained; since it is one of the original valid lower-bound candidates, combining valid bounds by taking their maximum preserves it. No answer or numerical example is supplied, so the document presents a derivation question rather than a settled result. Its setup assumes a constant positive risk-free rate and specified dividend dates, and it does not discuss transaction costs or more general dividend and rate models.

Key ideas

• Each feasible exercise strategy can provide a lower-bound candidate for an American call's value.
• Exercising before a dividend date changes the dividends captured and the timing of the strike payment.
• The maximum of several valid lower bounds remains a valid lower bound.
• The immediate-exercise value should remain among the candidates when combining bounds.
• The question assumes two known discrete dividends and a constant positive risk-free rate.

Tags

# Improvement in lower bound of American call with discrete dividends


# Improvement in lower bound of American call with discrete dividends












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

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

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

- 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.