Description:
This is a sideways strategy consisting of a long position in an OTM put option with a strike price K1 , a short position in two ATM put options with a strike price K2 , and a long position in an ITM put option with a strike price K3 . The strikes are equidistant: K3 − K2 = K2 − K1 = κ. This is a relatively low cost net debit trade. The trader’s outlook is neutral. This is a capital gain strategy.
Modified Put Butterfly
This is a variation of the long put butterfly strategy where the strikes are no longer equidistant; instead we have K3 − K2 < K2 − K1 . This results in a sideways strategy with a bullish bias.
If H > 0 there is also S∗up = K3 − H

Understanding the Long Put Butterfly
The long put butterfly spread is a neutral strategy that aims to profit from minimal movements in the underlying asset’s price. The structure consists of buying one in-the-money (ITM) put option (Strike C), selling two at-the-money (ATM) put options (Strike B), and buying one out-of-the-money (OTM) put option (Strike A), all with the same expiration date. Since two options are sold, the strategy often results in a net debit or even a small net credit when established. The maximum profit is realized if the underlying price at expiration is exactly at Strike B. This strategy is most effective in a neutral market environment with minimal expected price movement.
The principle behind the long put butterfly spread is to profit from small price movements in the underlying asset. The greatest profit is achieved if the underlying price at expiration is exactly at the strike price of the two sold puts (Strike B). The strategy combines a bull put spread with a bear put spread, with the gain determined by the difference between the strike prices and the premiums involved.