Pricing a European Butterfly as a Portfolio of Calls

Summary

The document asks whether a butterfly option can be valued by pricing its component calls with Black–Scholes and combining those values. The answer explains the key principle: for European options with compatible terms, the value of a portfolio is the same linear combination of the component option values as the portfolio’s payoff. This follows from the linearity of expectation under risk-neutral valuation, so each leg can be priced separately and then combined using its position weight.

The example describes a multi-call payoff and gives an integral argument for adding component prices. Its displayed algebra appears to omit the negative weight on the middle call, and the wording of the strike ordering and payoff definition is not fully consistent. The lesson therefore supports linear pricing of a correctly specified option portfolio, but it should not be taken as a verified formula for the particular butterfly shown. The result applies to European options under the stated pricing framework; other exercise features or mismatched contract terms require additional care.

Key ideas

• A European option portfolio can be valued by combining the values of its component options with their portfolio weights.
• The portfolio payoff must be specified correctly before applying the linear pricing argument.
• Black–Scholes can price each European call leg when the model assumptions and contract terms apply.
• The displayed example has apparent inconsistencies in its weights and strike description.

Tags

# How to use Black-Scholes' formula for a butterfly option?


# How to use Black-Scholes' formula for a butterfly option?












I'm wondering if I can apply Black-Scholes formula to value a butterfly option, i.e: $$B(T)=V_\text{call}(S(T)-K,0)+V_\text{call}(S(T)-K',0)-2V_\text{call}(S(T)-K'',0)$$ with $K<K''<K'$, just evaluating each call with B-S formula and operating independently. I think so but I don't know how to explain it with a theoretical basis.

## Answer by onlyvix.blogspot.com (score 2)

https://quant.stackexchange.com/a/23116

As barrycarter stated in the comment - the value of a set of [European!] options is the sum of the values of the individual options. This is simply follows from integral of a sum being a sum of integrals.

$$butterfly\,option\,price = \\ \int_0^\infty butterfly\,payoff(S) dS = \\ \int_0^\infty (call\,payoff(S,K)+call\,payoff(S,K')+call\,payoff(S,K'')) dS = \\ \int_0^\infty call\,payoff(S,K) dS + \int_0^\infty call\,payoff(S,K') dS + \int_0^\infty call\,payoff(S,K'') dS = \\ call(K)+call(K')+call(K'') $$

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.