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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?

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A butterfly (option) is an option strategy with the payoff structure like below (disregard the axis labels, just take note of the structure):

enter image description here

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.

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  • $\begingroup$ And could you explain me in the diagram what it means if the asset price is $100$ and the value is $5$? I don't get what the difference between asset price and value is. $\endgroup$
    – Summerday
    Commented Nov 19, 2023 at 7:52
  • $\begingroup$ Sorry I'm really new to this topic therefore I still have some questions. Can you explain a bit more how you get to the linear combination with put and call option and their values? what does this long and short mean we have only spoken about put and call and not about long/short put and call and also what is this 1x and 2x, how do you get this? $\endgroup$
    – Summerday
    Commented Nov 19, 2023 at 7:54

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