American Option Value, Exercise Payoff, and Remaining Time Value

Summary

The exchange explains the value relationship for an American option immediately before and after exercise. Once exercised, the option contract ends, so its post-exercise value is zero. The holder instead receives the exercise payoff: for a call or put, this is represented by the positive part of the relevant intrinsic-value expression, with the sign convention depending on the option type.

Before exercise, the option is worth at least its immediate exercise payoff because the holder can choose to exercise. If the pre-exercise value exceeds that payoff, the difference is remaining time value, and exercising would forfeit it. The response illustrates the principle with a payoff expression but does not fully address put versus call notation, dividends, or special exercise features. Its conclusion concerns the value comparison at an exercise decision, not a complete valuation method for American options.

Key ideas

• An exercised option ceases to exist and has no remaining contract value.
• Exercise delivers the option's intrinsic payoff to the holder.
• The pre-exercise value is at least as large as the immediate exercise payoff.
• A premium above intrinsic value represents time value that would be lost by exercising.

Tags

# Value of American option after exercise


# Value of American option after exercise












Suppose $V^+(S,t;K)$ is the value of a American option with strike $K$ before the exercise, and $V^-(S,t;K)$ is the value after exercise. Then how to understand the inequality $$V^+(S,t;K)\geq V^-(S,t;K) + (S - K)^+$$

## Answer by amdopt (score 2)

https://quant.stackexchange.com/a/34358

After exercise, the option ceases to exist. It has no value.

$V^−(S,t;K)=0$

$V^+(S,t;K)=(S−K)^+$

If an option was exercised and

$V^+(S,t;K)>(S−K)$

then there was still time value in the option and it should not have been exercised.

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.