How to understand broken wing butterfly option strategies?

Question

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 1

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.