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Broken-Wing Put Butterfly Pricing and Break-Even Conditions

Article Quant Q&A · Author: Edward Wang
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Summary

This note examines a broken-wing put butterfly made from long puts at the outer strikes and two short puts at the middle strike, with unequal wing widths. Its entry cost depends on the prices of all three strike levels: moving the low strike farther from the middle generally reduces its cost, while moving the higher strike deeper in the money raises its cost. The structure can therefore open for a credit or a debit; the strike spacing alone does not settle the question.

The response links the number of break-even points to the credit collected and the width of the riskier wing. When the credit exceeds that wing's width, the described position has no upside risk and one break-even; otherwise a debit position may have two. It does not resolve the question about theta, and the explanation gives no numerical example or option-pricing framework. Actual premiums, expiry, and market conditions still determine the trade's entry value and payoff.

Key ideas

  • The entry value depends on the premiums of all option legs, including how far each outer strike lies from the middle strike.
  • A lower low-strike put reduces its cost, while a deeper in-the-money upper-strike put costs more.
  • The position may be entered for either a net credit or a net debit.
  • A credit exceeding the wider risk wing can eliminate upside risk and leave one break-even point.
  • The response does not explain the strategy's theta behavior.

Tags

OptionsDerivatives pricingRisk management
Full text
# How to understand broken wing butterfly option strategies?


# How to understand broken wing butterfly option strategies?












I feel very confused about the greeks analysis for the broken wing butterfly strategy.

Let's say for the stock ABC, we enter into a such strategy: we long a put option with strike $k_1$ and another put option with strike $k_3$，and at the same time short two put options with strike $k_2$, where $k_2-k_1>k_3-k_2$.

Then what I read tells me that this position is possibly a net debit or a net credit when starting with. Intuitively, if $k_1$ is too low, then the strategy should be a net credit. Intuitively, if the stock price is above $k_3$, the $\Theta$ should be positive since I profit from time decay. At expiration all puts will become worthless. But if $k_1$ is not that low, I can imagine it would be a net debit position when entering, then the $\Theta$ would be negative since I will lose money from time decay.

How to understand such behavior of this strategy. And another confusion is, if I enter the strategy with net debit, I will have two break even price; but if I enter with net credit, I will have only one. This also seems strange to me. So what determines if I enter with debit or credit, and how to understand the behavior of this strategy? Thank you so much!

## Answer by Bob Baerker (score 2)

https://quant.stackexchange.com/a/57399

> Intuitively, if $k_1$ is too low, then the strategy should be a net credit.

That is an incorrect conclusion. The lower that $k_1$ is, the larger the credit will be or the lower the debit cost will be. Bear in mind that $k_3$ also affects cost. The deeper ITM it is, the more it costs, and vice versa..

> ... if I enter the strategy with net debit, I will have two break even price; but if I enter with net credit, I will have only one. This also seems strange to me. So what determines if I enter with debit or credit, and how to understand the behavior of this strategy?

The distance of $k_1$ and $k_3$ from $k_2$ determines the cost of the respective legs. The further away from $k_2$ they are, the less $k_1$ costs and the more $k_3$ costs. The more that $k_3$ is ITM, the greater its cost, eventually exceeding $k_2 - k_1$, resulting in a debit spread.

IOW, if the spread credit is greater than $k_3 - k_2$, there is no upside risk and there is only one break-even.

I can't help with your theta question since I only use delta in my hedging. The rest of them are Greek to me.

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