American Put Payoff at Expiration and the Exercise Boundary

Summary

The document asks why an American put’s value approaches zero near expiration in a specified region above its optimal exercise boundary, even though its terminal payoff is the positive part of strike minus stock price. It introduces time remaining to maturity and quotes a boundary condition from a mathematical finance text, then seeks an intuitive interpretation of the zero-value condition.

The included answer interprets the condition as describing an out-of-the-money put: when the stock price exceeds the strike, the terminal payoff is zero, so the option becomes worthless at expiration. The post provides only a brief answer and no derivation of the American option boundary or its behavior over time. Its notation is also easy to misread because the condition’s region and the reply’s interpretation rely on the relationship between stock price and strike.

Key ideas

• An American put’s terminal payoff is the positive part of strike minus stock price.
• The document asks about a zero-value condition as time to maturity approaches zero above an exercise boundary.
• The answer explains zero terminal value through the put being out of the money.
• The exchange offers intuition but no derivation of the optimal exercise boundary.

Tags

# Terminal Condition for American Put Option


# Terminal Condition for American Put Option












In a recent book I read, the author mentioned the terminal condition

$$\mathop {\lim }\limits_{t \to T} V(S,t) = \max \left\{ {X - S,0} \right\}$$

This is intuitive to understand. Then he defines $$\tau \equiv T - t$$ and when $r>0$, the terminal condition above can be simplified as

$$\mathop {\lim }\limits_{\tau \to 0} V(S,\tau) = 0$$

in the range $${\Sigma _1} = \left\{ {(S,\tau )|B(\tau ) \le S < + \infty ,0 \le \tau \le T} \right\}$$

This is not so intuitive . How can the value of the option be equal to zero in this case?

@Update: The book is " Homotopy Analysis Method in Nonlinear Differential Equations" p. 432

$B(\tau ) $= optimal exercise boundary

## Answer by MatthewM (score 0)

https://quant.stackexchange.com/a/32050

It looks like r = S - X, or the strike price - stock price. Basically, if r > 0, the put is out of the money and becomes worthless as it goes to expiration.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.