American Call Early Exercise Around Cash Dividends
Summary
The note explains a comparison used to assess whether exercising an American call on a dividend-paying stock is preferable to keeping the option. It contrasts the cash received from exercising just before an ex-dividend date with a lower bound on the option’s value after the stock price adjusts for the dividend. Rearranging the comparison yields a condition involving the dividend, strike, interest rate, and time remaining to maturity.
The key interpretive point is that this inequality is conditional: if it holds, the lower bound alone is at least as large as the early-exercise proceeds, so exercising at that date cannot be optimal. The inequality is not asserted to hold in every case. If the dividend is large relative to the right-hand side, the comparison does not rule out early exercise; the note also observes that the stock price can affect the case for exercising. The discussion is an explanation of one bound-based argument, not a complete valuation procedure.
Key ideas
- Compare immediate exercise proceeds with a lower bound on the option’s post-dividend value.
- The derived dividend condition is a sufficient reason not to exercise at the ex-dividend date when it holds.
- Rearranging an assumed inequality does not establish that the condition holds in every market situation.
- A large dividend or little time remaining can make the condition fail, leaving early exercise possible.
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Full text
# Early Exercise of American Options on dividend-stock
# Early Exercise of American Options on dividend-stock
I am reading the chapter 15 of Options, futures, and other derivatives by John Hull.
Specifically, 15.12 Dividends-American Call Options.
I am stuck while proving the fact that exercising an American options with dividend stock just before the last dividend date is optimal, rather than holding it til the maturity.
The investor will get $S(t_n)-K$ when he/she exercise before the ex-dividend date. After the ex-dividend date, the stock price would down to $S(t_n) - D_n$. And the lower bound of this option is known to be $S(t_n)-D_n-Ke^{-r(T-t_n)}$.
So, using proof by contradiction, the following is derived. $$S(t_n)-D_n-Ke^{-r(T-t_n)}\geqslant S(t_n)-K$$ $$-D_n-Ke^{-r(T-t_n)}\geqslant - K$$ $$D_n + Ke^{-r(T-t_n)}\leqslant K$$ $$D_n \leqslant K\left[1-e^{-r(T-t_n)}\right]$$
From the last equation, the book concluded that it cannot be optimal to exercise at time $t_n$(the ex-dividend date).
However, I don't understand how come we can conclude like this and the implication of right hand side of the last equation.
Can anyone help me to understand this proof?
## Answer by AKdemy (score 1, accepted)
https://quant.stackexchange.com/a/63687
$$S(t_n)-D_n-Ke^{-r(T-t_n)}\geqslant S(t_n)-K$$ means that `if` LHS (lower bound of option price) $\geqslant$ RHS (what you get if you exercise early), it cannot be optimal to exercise. This is an `assumption`, not a claim that it is true or must hold under any circumstances.
The rest is just reformulation to have $D$ on one side. Hence, iff $$D_n \leqslant RHS$$ it is not optimal to exercise at time $t_n$ (simply because the first equation says it is not and nothing changed). That said, it's also possible that $D_n > RHS$. Particularly if $T-t_n$ is small and / or the dividend is large. Starting with first equation, you can also argue that there exists a $S(t_n)$ that justifies early exercise.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.