Prove that there exists a critical price for a American call option with continuous dividends

Question

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 1

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.