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When American and European Puts Have Equal Value

Article Quant Q&A · Author: Idonknow
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Summary

The document asks when an American put and a European put on the same underlying can have equal value, motivated by checking an American put binomial-tree implementation. It notes that early exercise does not benefit a call on a non-dividend-paying stock, so European and American calls coincide in that setting, while an American put is generally worth at least as much as its European counterpart because it can be exercised early.

The response gives a sufficient setting for equality: a driftless underlying and an option value that is not discounted. It connects this to futures-style options with daily bilateral variation margin or to zero collateral interest, and invokes Jensen’s inequality. The answer is brief and does not derive the result, define a full pricing model, or establish that these are the only conditions. It therefore offers a useful benchmark for a particular model setup, but not a general validation test for every binomial implementation.

Key ideas

  • An American put cannot be worth less than the corresponding European put because early exercise is available.
  • The response identifies driftless underlying dynamics and no discounting as conditions for equal values.
  • Futures-style margining or zero collateral interest are given as examples of the no-discounting assumption.
  • The explanation cites Jensen’s inequality but does not provide a derivation or a complete set of equality conditions.

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OptionsDerivatives pricingFuturesStatistics
Full text
# Under what conditions will both European and American put options worth the same?


# Under what conditions will both European and American put options worth the same?












It is well-known that on a non-dividend paying stock, it is suboptimal to exercise an American call option earlier. In other words, both European and American call options on the same non-dividend paying stock worth the same. This can be proven using the Put-Call Parity.

However, I am not sure about European and American put options. I know that due to the ability that an American option can be exercised any time prior to maturity, it should worth at least as much as European put option. I am interested to know the conditions that give equality. In particular,

> Question: Under what conditions will both European and American put options worth the same?

My motivation behind asking this question is so that I can ensure that my binomial tree implementation to price American put option is correct.

When dividend is zero, my binomial output the same price for both European and American call options, which is a good sign. However, I have no alternative to check whethee the tree gives correct price for American put option or not.

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/54888

If the underlying is driftless (think futures) and the value of the option is not discounted (think future style options with daily bilateral variation margin or CSA's with zero collateral interest rates) then the value of an american put and a european put would be the same by Jensen's inequality.

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.