Why Deep In-the-Money American Calls May Have an Exercise Threshold
Summary
The document sketches why an American call on a dividend-paying stock may become optimal to exercise early once the stock price is sufficiently high. Exercising captures future dividends, while waiting preserves the possibility of avoiding exercise if the stock later falls below the strike and delays payment of the strike price. The proposed argument compares these benefits and costs as the stock price increases.
Under a constant dividend yield, the answer argues that the value of dividends gained from exercise grows without bound as the stock price tends to infinity, while the downside protection from waiting shrinks toward zero and the strike-payment cost does not depend on the stock price. It concludes that exercise must be preferable beyond some price. This is only a sketch: it does not specify a full stochastic model, formalize the value comparison, or prove the claim for general market dynamics. It therefore does not resolve the question’s request for a model-free continuous-time proof.
Key ideas
- Early exercise of a dividend-paying call captures dividends that would otherwise be missed.
- Waiting retains the option to avoid paying the strike if the stock falls below it.
- The strike-payment cost from exercising early is treated as independent of the stock price.
- The sketch argues that the dividend benefit eventually dominates the other costs at sufficiently high prices.
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Full text
# Prove that there exists a critical price for a American call option with continuous dividends
# Prove that there exists a critical price for a American call option with continuous dividends
For a American call option on a stock with continuous dividend yield, show that there exists a critical price, that is a price $S^*_t$ such that if the stock price is above this at time $t$, then it is optimal to early exercise.
Is anyone aware of a proof of this fact in continuous time, ideally in a model-free way (i.e. not needing to assume the Black-Scholes model, that is not assuming the Black-Scholes PDE)?
The usual explanation is this: using the Black-Scholes equation, if a European call is very deep in the money, then its value is $c_t \sim e^{-qT}S_t -e^{-rT}K$ which will be below its intrinsic value $S-K$, but the holder of an American call would never let the call value fall below its intrinsic value, so they will early exercise. But why? Why won't they let the call fall below its intrinsic value?
And secondly, this assumes the Black-Scholes model. What if the Black-Scholes model doesn't hold. Can purely no arbitrage arguments be used to show that there exists a critical price?
A citation would be sufficient.
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/76130
Sketch proof: what do you get from early exercise : the PV of future dividends. (A). What do you give up from early exercise : the benefit of potentially not exercising if S were to fall below K in the future (B) and you have to pay the strike price K earlier, which costs $Ke^{-r(T-t)}$. (C)
Now as S -> infinity , A tends to infinity assuming constant dividend yield. However B-> 0 , and C is not dependent on S.
Hence above critical value of S we must have A>B+C. Hence optimal to exercise early at that point.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.