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


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.

  • $\begingroup$ thank you for the answer and all detail explanation. I understand it now, but what is $\hat{S}$ in your picture, Professor? $\endgroup$ 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. $\endgroup$ 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? $\endgroup$ 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. $\endgroup$ Sep 12 '16 at 7:21
  • $\begingroup$ Alright, thank you! $\endgroup$ Sep 12 '16 at 7:53

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