Replicating a Butterfly Payoff with Calls or Puts
Summary
The document explains how to represent a piecewise-linear butterfly payoff as a portfolio of vanilla options. The payoff is flat outside two outer strikes, rises between the lower and middle strikes, and falls between the middle and upper strikes. The response gives two equivalent constructions: long one call at each outer strike and short two calls at the middle strike, or the corresponding put combination with outer-strike puts long and middle-strike puts short.
Writing the payoff as a combination of standard options makes it possible to price the whole strategy by combining the prices of its components, and can help identify related strategies. The explanation focuses on the payoff shape and replication; it does not work through a Black–Scholes valuation or discuss assumptions such as volatility inputs, expiry, or transaction costs. The stated equivalence is for the payoff described in the question.
Key ideas
- A butterfly payoff is flat beyond its outer strikes and forms a peak between them.
- The payoff can be replicated by long calls at the outer strikes and twice short calls at the middle strike.
- An equivalent construction uses long puts at the outer strikes and twice short puts at the middle strike.
- A replicated strategy can be valued by combining the prices of its constituent options.
Tags
Full text
# How can I price this option? # How can I price this option? > In the Black-Scholes model, I want to price the so called Butterfly option, where the payoff $P(x)$ is the following function: $P(x)=0$ if $0\leq x\leq 40$, $P(x)=x-40$ for $40\leq x\leq 60$, $P(x)=-x+80$ for $60\leq x\leq 80$ and $P(x)=0$ for $80\leq x\leq 100$ In the lecture they gave us the hint to write the payoff as a linear combination of put or call options. Somehow I need some explanations since I don't really get what I need to do. Could someone explain me what the Payoff is, and how to write it as linear combination of put or call options. I don't really get what it means to price an option. I mean I have a formula for pricing an option which is also on wikipedia, but I don't see how to use it. I know what put and call options are, for European option for example $(S_T-K)^+$ is a call option and $(K-S_T)^+$ is a put option. But I don't really get how to write it as linear combination and why this is useful. Can someone explain this to me with a bit more details? ## Answer by KaiSqDist (score 1, accepted) https://quant.stackexchange.com/a/77410 A butterfly (option) is an option strategy with the payoff structure like below (disregard the axis labels, just take note of the structure): There are 4 ranges you mentioned in your question: - $0 \le x < 40$: This is the leftmost flat line on the structure above. - $40 \le x \le 60$: This is the upwards sloping line after the leftmost flat line. - $60 \le x \le 80$: This is the downwards sloping line before the rightmost flat line. - $x > 80$: This is the rightmost flat line on the structure above. Your linear combination can be given in 2 ways (using all call or all put options): Long 1x Call at Strike 40, Short 2x Call at Strike 60 and Long 1x Call at Strike 80 OR Long 1x Put at Strike 80, Short 2x Put at Strike 60 and Long 1x Put at Strike 40. The reason why this is useful, is because being able to breakdown option strategies into their individual options allows you to deduce other option strategies such as spreads, straddles, strangles etc. I think there may be more to this but I'm not too big on option trading, just modelling.
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.