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gateway/knowledge/trading/strategies/options/long-call-butterfly.md

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+---
+description: "A neutral low-cost debit strategy buying an OTM call at K1, selling two ATM calls at K2, and buying an ITM call at K3 with equidistant strikes, profiting if stock stays near K2."
+tags: [options, income, neutral, butterfly]
+---
+
+# Long Call Butterfly
+
+**Section**: 2.40 | **Asset Class**: Options | **Type**: Income
+
+## Overview
+The long call butterfly is a sideways strategy consisting of a long OTM call at K1, a short position in two ATM calls at K2, and a long ITM call at K3. The strikes are equidistant: K2 - K3 = K1 - K2 = 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 call option at strike K1 (OTM, upper wing)
+- Sell 2 call options at strike K2 (ATM, body)
+- Buy 1 call option at strike K3 (ITM, lower wing, K3 < K2 < K1)
+- All same expiry; K2 - K3 = K1 - K2 = kappa (equidistant)
+
+Net debit: D
+
+## Payoff Profile
+f_T = (S_T - K1)+ + (S_T - K3)+ - 2 × (S_T - K2)+ - D
+
+- Lower breakeven: S*_down = K3 + D
+- Upper breakeven: S*_up = 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
+
+### 2.40.1 Modified Call Butterfly
+A variation where the strikes are no longer equidistant; instead K1 - K2 < K2 - K3. This results in a sideways strategy with a bullish bias. We have:
+
+f_T = (S_T - K1)+ + (S_T - K3)+ - 2 × (S_T - K2)+ - D
+
+- Breakeven: S* = K3 + D  (single breakeven on the lower side)
+- Max profit: P_max = K2 - K3 - D  (at S_T = K2)
+- Max loss: L_max = D
+
+## 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.