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I am reading Alternative Characterizations of American Put Options by Carr et al. It is stated there that:

Consider an American put option on the stock with strike price $K$ and maturity date $T$. Let $P_t$ denote the value of the American put at time $t \in [0,T]$. For each time $t \in [0,T]$, there exists a critical stock price, $B_t$ , below which the American put should be exercised early, i.e., $$if S_t \leq B_t, then P_t =max[0,K-S_t]\tag{2}$$ $$if S_t > B_t, then P_t >max[0,K-S_t]\tag{3}$$ The exercise boundary is the time path of critical stock prices, $B_t$ ,$t \in [0,T]$. This boundary is independent of the current stock price $S_0$ and is a smooth, nondecreasing function of time t terminating in the strike price, i.e. $B_T = K$. The put value is also a function, denoted $P(S,t)$, mapping its domain $D ≡ (S,t) ∈ [0,\infty)×[0,T]$ into the nonnegative real line. The exercise boundary,$B_t$ ,$t \in [0,T]$, divides this domain $D$ into a stopping region $S ≡ [0,B_t ]×[0,T]$ and a continuation region $C ≡ (B_t ,∞)×[0,T]$ (see Figure 1).

However, there is no Figure 1 in that article. I wonder what is the figure. In my opinion, the figure is like below Domain Figure

Is it correct? If it is not, then what is the correct figure? Thanks

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The exercise boundary $B_t$ for a finite maturity American put option is not a constant function of time as in your plot. As mentioned in the excerpt, $B_T = K$ at maturity. But for $t < T$, we have $B_t < K$ as you would never pre-maturely exercise to receive a zero payoff.

Below is a plot of the early exercise boundary that I once produced for a lecture. Note that I used a slightly different notation $E$ is the exercise price (the red line). The optimal exercise boundary is $S_t^*$ (the blue curve). The continuation region is in white and the stopping region in grey.

Early exercise boundary for American put options.

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  • $\begingroup$ thank you for the answer and all detail explanation. I understand it now, but what is $\hat{S}$ in your picture, Professor? $\endgroup$ – Monica Sendi Afa Sep 11 '16 at 9:06
  • $\begingroup$ Hi, I am not a professor unfortunately. $\hat{S}$ was notation related to something else in that lecture. I removed it from the plot to avoid confusion. $\endgroup$ – LocalVolatility Sep 11 '16 at 10:00
  • $\begingroup$ Ah, sorry, I assume you are a lecturer and it's been a habit for me to call a lecturer "Professor" even though the lecturer isn't a professor yet. But if you prefer me not to call you Professor, how can I call you then? As I believe LocalVolatility is not your real name (or is it?). Anyway, as you mentioned in your answer, the optimal exercise boundary is the blue curve. I want you to verify something. Is the $S^*$ in the vertical axis also a notation related to something else in that lecture? $\endgroup$ – Monica Sendi Afa Sep 12 '16 at 3:53
  • $\begingroup$ $S^*$ is the optimal exercise boundary (the blue curve) as a function of time. The dotted gray lines simply point to the value at one particular time point. $\endgroup$ – LocalVolatility Sep 12 '16 at 7:21
  • $\begingroup$ Alright, thank you! $\endgroup$ – Monica Sendi Afa Sep 12 '16 at 7:53

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