Why a Double Call May Not Be Exercised at the Earlier Date
Summary
The document analyzes a double call that gives its holder a choice between exercise at an earlier date with strike K1 and exercise at a later date with strike K2. It states a condition under which early exercise is suboptimal: the discounted earlier strike exceeds the discounted later strike. The argument compares holding the proceeds from early exercise with a strategy that shorts the stock at the earlier date, invests the sale proceeds, and retains the later call.
At the later date, the call payoff offsets the short stock exposure in a way that produces a payoff at least as large as the early-exercise alternative when the stated strike condition holds. Simply waiting and exercising later is not enough to establish dominance, because its payoff depends on the later stock price and cannot be directly compared with the earlier exercise payoff. The explanation is a payoff comparison; it does not discuss market frictions or broader contract details.
Key ideas
- The stated strike condition makes the later exercise strike sufficiently low relative to the earlier one after interest accrual.
- The dominance argument compares early exercise with investing proceeds while keeping a later call and shorting the stock.
- Waiting to exercise without the stock hedge does not guarantee a better payoff in every price scenario.
- The conclusion follows from the described payoff construction and stated condition.
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Full text
# Double Call Option
# Double Call Option
A double call option allows the holder to either exercise at time $T_{1}$ or time $T_{2}$, where $T_{2}$>$T_{1}$. With corresponding strike prices $K_{1}$ and $K_{2}$, it can be shown that it is never optimal to exercise at $T_{1}$ if $K_{1}e^{-rT_{1}}>K_{2}e^{-rT_{2}}$. This is shown by the idea that, at $T_{1}$ you have two options (if $S_{T_{1}}>K_{1}$):
- Exercise and put $S_{T_{1}}>K_{1}$ in the bank
- Sell the stock short, put $S_{T_{1}}$ in the bank and wait until $T_{2}$ to exercise the call
I understand how this proves the required inequality, but I don't understand why these are the two options - why are the options not just to either exercise at $T_{1}$ or exercise at $T_{2}$?
## Answer by nluckn (score 2)
https://quant.stackexchange.com/a/53503
1st option: Your payoff in $T_2$ is $$(S_{T_1} - K_1)e^{r (T_2 - T_1)}.$$ 2nd option: Your payoff in $T_2$ is $$S_{T_1}e^{r (T_2 - T_1)} + \max (S_{T_2} - K_2,0) - S_{T_2} \geq S_{T_1}e^{r (T_2 - T_1)} + S_{T_2} - K_2 - S_{T_2}$$ and the RHS is equal to $$S_{T_1}e^{r (T_2 - T_1)} - K_2.$$ The term $S_{T_1}e^{r (T_2 - T_1)}$ is in the payoff of the 1st option and in the payoff which is dominated by the payoff of the 2nd option. As $$K_1 e^{-r T_1} > K_2 e^{-r T_2}$$ implies $$K_2 < K_1 e^{r (T_2 - T_1)},$$ the 2nd option, i.e. not executing in $T_1$, is always better.
Your suggestion for the 2nd option is to just exercise in $T_2$. Your payoff in $T_2$ would then be $$\max (S_{T_2} - K_2,0).$$ Compare this to the payoff in $T_2$ of the 1st option. As $S_{T_1}$ and $S_{T_2}$ are random and therefore unknown, you can't tell which of the two options is better. The short selling in the 2nd option is a construction to introduce $S_{T_1}$ in the payoff of that option and get rid of $S_{T_2}$ in the dominated payoff.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.