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Butterfly Option Payoff from Calls and Puts at Three Strikes

Article Quant Q&A · Author: Sino
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Summary

The document gives a terminal payoff expression for a butterfly-like option position using calls and puts at three strike levels. It combines an upside call payoff above an upper strike with a downside put payoff below a lower strike, then subtracts the corresponding call and put payoffs at the middle, at-the-money strike. The positive-part notation makes each option contribute only when the underlying finishes on the relevant side of its strike.

This expression can be used to calculate the position’s payoff for a simulated terminal asset price in a Monte Carlo valuation. The exchange does not specify contract quantities, premiums, discounting, or the assumptions needed to price the position, and the exact payoff profile depends on the selected strikes and signs. It provides the payoff form only, rather than code or a complete valuation method.

Key ideas

  • The payoff combines an upper-strike call and lower-strike put with short positions at the middle strike.
  • Each option payoff is zero when the terminal underlying price is on the inactive side of its strike.
  • The expression can be evaluated across simulated terminal prices in a Monte Carlo calculation.
  • The payoff formula alone does not include premiums, discounting, or a complete option pricing model.

Tags

OptionsDerivatives pricing
Full text
# Payoff of a butterfly c++


# Payoff of a butterfly c++












I would like to price options (call, put,, butterfly) with monte-carlo method, but actually I need the expression of the butterflay payoff;

Could you ^please help me !

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/23243

A butterfly in general has a payoff of the form \begin{align*} (X_T-K_c)^+ + (K_p-X_T)^+-(X_T-K_{atm})^+-(K_{atm}-X_T)^+, \end{align*} where $X_T$ is the asset value at maturity $T$, while $K_c$, $K_p$, and $K_{atm}$ are strike levels.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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