Why Dividend-Paying American Calls Have an Early Exercise Boundary

Summary

The document asks why an American call on a stock paying continuous dividends has a finite stock price above which early exercise is optimal. It contrasts the option’s value, which cannot fall below intrinsic value, with the possibility that the two values might approach one another without ever meeting. The central issue is the existence of a critical price separating continuation from exercise.

The text poses this as an unanswered question and points to a figure in unspecified notes, but gives no derivation, proof, model assumptions, or numerical evidence. It therefore identifies a useful conceptual problem in American option pricing without resolving it. The existence and location of an exercise boundary depend on the pricing framework and inputs, such as dividends, interest rates, and time to expiry; those details are not provided here.

Key ideas

• An American call’s value is at least its intrinsic value.
• The document asks why a continuous dividend yield creates a price region where early exercise is optimal.
• The critical stock price marks where continuation value meets intrinsic value.
• The text raises the boundary existence question but supplies no explanation or proof.

Tags

# 46606


# Why does an American option on a continuous dividend paying stock have a critical price above which it is optimal to exercise early?












An American call on a continuous dividend paying stock must be above its intrinsic value, i.e $c(t)\geq\max(S_t-K,0)$.

Why is there a critical price above which it is optimal to exercise (i.e. we have equality in the inequality above)? This is shown in Figure 5.1 in these notes but it is not really explained.

In other words, how can we rule out the situation where $c(t)>\max(S_t-K,0)$ for all prices $S_t$, so no critical price $S_t^*$ exists and the curves in that figure asymptote but never meet?

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