American Put Options: Continuation Region and Exercise Boundary
Summary
The discussion explains why an American put is modeled with different conditions in its exercise and continuation regions. The holder chooses an exercise time to maximize expected discounted payoff, so the valuation is an optimal stopping problem. The option value must be at least its immediate exercise value; where it is strictly greater, exercising is suboptimal and the price follows the Black–Scholes differential equation. At the exercise boundary, the value meets the payoff, while the formulation can be expressed through a complementarity condition linking the differential equation and the value above intrinsic payoff.
The answers offer intuition from the put’s exercise payoff and the possibility of arbitrage if the option were worth less than immediate exercise. They do not provide a full derivation of the free-boundary problem or the high-contact condition the question also asks about. The explanation assumes the standard Black–Scholes setting and does not develop extensions for dividends, changing rates, or other contract features.
Key ideas
- American option valuation is an optimal stopping problem over permissible exercise times.
- An American put’s value cannot fall below its immediate exercise payoff.
- In the continuation region, where holding the option is more valuable than exercising, the value satisfies the Black–Scholes equation.
- The exercise boundary separates the continuation region from the region where the payoff determines value.
- A complementarity condition captures the relationship between the pricing equation and the exercise constraint.
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Full text
# Why does Black-Scholes equation hold on continuation region of American Option?
# Why does Black-Scholes equation hold on continuation region of American Option?
Explanation for Put Option:
$$ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0, $$
where
$\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\partial V}{\partial S} - r V$ holds for $S > S_f$, where $S_f$ is contact point.
Why does this equation hold for $S > S_f$? Could you give me link for proof?
Another question is: Why do we need high-contact condition?
Update:
Do I correctly understand that for American Put Option, if $S > S_{f}$, there is no sense to exercise at time $t<T$ (because it causes immediate loss: $-V+S-K<0$). So it behaves like European Option, hence $V^{Am}_{P}=V^{E}_{P}$ and it satisfies Black-Scholes Equation.
## Answer by Brian B (score 4)
https://quant.stackexchange.com/a/8145
The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is
$$ V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right] $$
where the maximum is taken over all stopping times (exercise strategies $\tau>0$ permissible in the contract).
With a PDE operator such as you have, the instantaneous equality can be expressed in linear complementarity form as
$$ \left(\frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V)\right)\cdot \left(V-g\right) = 0 $$
where $g$ represents early exercise value.
Note for convenience that (post exercise) the stock itself satisfies the BS PDE trivially.
## Answer by jens_bo (score 0)
https://quant.stackexchange.com/a/8148
The payoff when exercising the option is given by:
$$\max(K-S(t),0)$$
now assume there is a $V(S,t)<\max(K-S(t),0)$: there would be the opportunity for arbitrage. We could buy the asset for $S$ and the put option for $V$. Selling the asset for $K$ would lead to a risk free profit of $K-S-V$. Thus the value of the american put option must hold the additional constraint $$V(S,t)\geq \max(K-S(t),0)$$
As long as $V > K-S$ (or $S>S_f$) it is given by the BS PDE, otherwise the price is given by $K-S$. Most (if not all) textbook introductions to financial derivatives include more details on that and derivations of the BS PDE.
The second (more mysterious) constraint is a consequence of "optimal" behavior of the agents. Keywords to find more on that might be optimal stopping problem or game theory of options.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.