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Is there such a thing as an American butterfly spread?

For a European butterfly spread simply buying 1 put with strike price X+a, 1 put with strike price X-a and shorting 2 calls with strike price X, all with the same expiration date, would give you a butterfly spread. However if we now do the same with american options, then we could exercise various parts of the strategy before others.

Are "fully" American option strategies commonly traded?

Thanks in advance.

(Sorry if this isn't the blog I'm supposed to ask this in, it is not exactly Quantitative finance, but personal finance seemed even less relevant)

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When you say OTC options, I think you're talking about contracts that aren't exchange traded (non-standard), so no one knows the wide range of term sheets that have been executed. I do know that American style (early) payoffs exist in OTC term sheets.

Most OTC deals are not about speculation. They're usually about reducing risk, so initiating the exposures of all legs of a butterfly in one contract would be unusual. Deals are often initiated by a party who already has a delta-one exposure (two ATM calls would be a delta of 1.0 or two ATM puts would be a delta -1.0). And they want to put some limits on their exposure for which they're willing to pay in terms of the other greeks.

That said, if two parties agreed on pricing for a term sheet that included it then the whole butterfly exposure could be there. It's just that they would mostly likely composite it into a single agreement and term. And you will probably never see the word "butterfly" in the terms. It's just not very precise.

It would not be uncommon for you to look at the combination of risks after an OTC deal and find that one party's books now included something with the same risks as a butterfly.

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I have never seen those traded. But it is an interesting research topic.

The interest of the standard European butterfly spread is the decomposition into three vanilla European options. This decomposition is not exact anymore in the case of the American butterfly spread as specified: the linear combination of the three vanilla American contracts does not correspond to value of the butterfly American contract.

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    $\begingroup$ You say that is not a linear combination because once any of those legs are exercised, all the others cease to exist right? $\endgroup$ Commented Jan 11, 2023 at 0:44
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    $\begingroup$ That's correct @AndréBittencourt $\endgroup$
    – jherek
    Commented Jan 11, 2023 at 10:46
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This exactly composition I have never seen. As far as I know, multiple sequential decisions are not common in OTC trades, so an structure like this would probably be a "single american option", which I mean, once any of the legs is exercised the whole structure cease to exist, as mentioned by jherek here, would have a different (smaller) price than the combination of the options.

In practical terms, if you have an ISDA with a bank you can do it, but why would you do it? Could probably have an different structure that could suits you better and the bank is more used to trade (so, he would charge you less spread).

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