Intuitive Bounds for American Option Values
Summary
The document presents lower and upper bounds for American call and put values relative to their European counterparts. The lower bounds follow from exercise flexibility: an American option holder can choose to wait until maturity, so the American contract cannot be worth less than the corresponding European option. The stated upper bounds add terms related to dividends for calls and discounted interest on the strike for puts.
The answer gives an intuition for the upper bounds through the Snell envelope: an American option’s value is the envelope of its discounted exercise payoffs, and when that discounted payoff is a submartingale, early exercise is not optimal. This offers a way to think about the bounds rather than deriving all equality conditions or treating every dividend and rate case. The document does not provide a full proof, detailed assumptions, or a complete explanation of the European call’s forward-value lower bound raised in the question.
Key ideas
- An American option is worth at least its European counterpart because it retains the ability to exercise at maturity.
- The stated upper bounds relate the early-exercise premium to dividends for calls and interest on the strike for puts.
- The Snell envelope frames an American option as the value of choosing an exercise time.
- When discounted exercise payoffs form a submartingale, early exercise is not optimal under the stated framework.
- The discussion gives intuition but leaves equality cases and some related lower-bound questions unresolved.
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Full text
# Intuitively understand boundaries of American Call and Put
# Intuitively understand boundaries of American Call and Put
Denote American Call/Put $C_{am}/P_{am},$ European Call/Put $C_v/P_v,$ with constant risk-free interest rate $r,$ dividend yield rate $D,$ strike $K,$ maturity $T.$
1.We have the well know inequalities: $$C_v\leq C_{am} \leq C_v + S_t(1 - e^{-D(T-t)})$$ $$P_v\leq P_{am} \leq P_v + K(1 - e^{-r(T-t)})$$ Surely, we can build the portfolios to proof the inequalities, but is there any intuitive ways to demonstrate above inequalities? Or when does the `=` hold, for none-zero $D$ and $r$ since the proof by portfolio method is not clear to see the conditions of `=.`
For example, $S_t(1 - e^{-D(T-t)})$ is actually the sum of discounted dividend, that means
`American call will never be larger than the European call adding the dividend of its underlying asset.`
And we also have
`it's always optimal to exercise American call immediately before the ex-dividend` etc.
Maybe the conditions of `=` can effectively solve the problem.
2.Moreover, for the low boundary(value of forward) of European call $$\max(e^{-D(T- t)}S_t - e^{-r(T- t)}K,0)\leq C_v(t),$$ some book said it can be regard as the American call, I can not understand this statement?
## Answer by Antoine Conze (score 4, accepted)
https://quant.stackexchange.com/a/36095
The lower bounds are obvious since American options can be exercised at any time while European options can only be exercised at maturity.
The upper bounds are obtained from the property that an American option value is the Snell envelope of its discounted payoff, so that when the discounted payoff is a submartingale the American option should never be exercised early.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.