# Are there any derivatives which pay amount $a(p-b)^{2}-c$ where $p$ is the price of underling asset?

Are there any derivatives which pay amount $a(p-b)^{2}-c$ where $p$ is the price of underling asset ? (or in the case of options $max(0,a(p-b)^{2}-c)$) I'm not very strict here but I only want to know if there are any derivatives which profile of profits is quadratic function of price of underling assets (quadratic on some subset of set of all $p$ )?

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They're sometimes called "power contracts" and though I've known them to be coded into the pricing libraries of at least 2 big investment banks, I've never seen one on the books. –  Brian B Jan 23 '13 at 19:16
@Brian B very interesting, thanks –  Qbik Jan 23 '13 at 21:57

You can ask for a quote from a bank as I am sure they will create it for you. If you want to create this kind of payoff yourself, you can use the following paper from Peter Carr where he introduces the spanning formula for replicating any twice differentiable payoff.

http://www.math.nyu.edu/research/carrp/papers/pdf/twrdsfig.pdf

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It's pretty straightforward to replicate (in theory!) by just buying a call struck at b, and another one struck at b+1, and another one struck at b+2, and another one struck at b+3, and so on, and then buy a put struck at b, and another one struck at b-1, and another one struck at b-2, etc., and then scaling the whole thing and adding c.

You can easily construct any European payoff, i.e. any payoff that is some function f(S(T)) for a fixed T, if you have call options (expiring at T) at any strike available.

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