---
description: "A neutral low-cost debit strategy buying an OTM put at K1, selling two ATM puts at K2, and buying an ITM put at K3 with equidistant strikes, profiting if stock stays near K2."
tags: [options, income, neutral, butterfly]
---

# Long Put Butterfly

**Section**: 2.41 | **Asset Class**: Options | **Type**: Income

## Overview
The long put butterfly is a sideways strategy consisting of a long OTM put at K1, a short position in two ATM puts at K2, and a long ITM put at K3. The strikes are equidistant: K3 - K2 = K2 - K1 = kappa. This is a relatively low cost net debit trade. The trader's outlook is neutral. This is a capital gain strategy.

## Construction
- Buy 1 put option at strike K1 (OTM, lower wing)
- Sell 2 put options at strike K2 (ATM, body)
- Buy 1 put option at strike K3 (ITM, upper wing, K3 > K2 > K1)
- All same expiry; K3 - K2 = K2 - K1 = kappa (equidistant)

Net debit: D

## Payoff Profile
f_T = (K1 - S_T)+ + (K3 - S_T)+ - 2 × (K2 - S_T)+ - D

- Upper breakeven: S*_up = K3 - D
- Lower breakeven: S*_down = K1 + D
- Max profit: P_max = kappa - D  (achieved at S_T = K2)
- Max loss: L_max = D  (if S_T >= K3 or S_T <= K1)

## Key Conditions / Signals
- Neutral; expects stock to pin near K2 at expiry
- Low implied volatility after entry reduces theta bleed on short options
- Low cost structure makes it an efficient way to bet on stability

## Variations

### 2.41.1 Modified Put Butterfly
A variation where the strikes are no longer equidistant; instead K3 - K2 < K2 - K1. This results in a sideways strategy with a bullish bias. For H > 0 there is also S*_up = K3 - H. We have:

f_T = (K1 - S_T)+ + (K3 - S_T)+ - 2 × (K2 - S_T)+ - H

- Lower breakeven: S*_down = 2 × K2 - K3 + H
- Max profit: P_max = K3 - K2 - H  (at S_T = K2)
- Max loss: L_max = 2 × K2 - K1 - K3 + H

## Notes
The equidistant butterfly has two breakevens symmetric around K2. The maximum profit is the wing width kappa minus the debit. Low-cost entry makes the risk/reward ratio attractive for neutral views.