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 $p$ )?

• 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. Jan 23 '13 at 19:16

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

Variance swaps are arguably a little like that. As mentioned, power contracts too, but sometimes only by virtue of correlation/cross-gammas between volatilities and underlyings.

I haven't personally seen a contract with an exponent in it.

• There are probably currently none out there right now with the exact same pay off structure (not sure how you would set b and c and what rational you would apply thus I said variants may exist but probably not the exact same ones)

• Having said that you can request a bank to price you ANY derivative you like, they will quote you for sure. You better be very confident that you are able to price them yourself correctly or you will get ripped off royally.

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.