Replicating Quadratic Payoffs with Options Across Strikes
Summary
The document asks whether a derivative can have a payoff that is quadratic in the underlying asset’s price, including a version floored at zero. The responses explain that a bespoke payoff could be requested from a bank, while emphasizing that the buyer would need to understand its pricing. They also describe theoretical replication using European options expiring at the same date.
The key replication idea is that a sufficiently broad set of calls and puts across strikes can span a payoff that is a function of the terminal asset price. One response points to a spanning formula for twice differentiable payoffs; another sketches combining options at a sequence of strikes, then scaling and adjusting the position. Variance swaps and power contracts are mentioned as related structures, not exact matches. The discussion is theoretical and does not derive hedge weights, prices, or practical trading costs. Replication depends on the availability of suitable strikes and maturities, and the answers do not assess liquidity, discrete strike coverage, or hedging risk.
Key ideas
- A customized derivative can be designed with a payoff that depends quadratically on the underlying price.
- European calls and puts across strikes can theoretically replicate terminal payoffs that are functions of the underlying price.
- A spanning formula provides a general approach for replicating sufficiently smooth payoffs.
- Variance swaps and power contracts may have related features but are not identified as exact matches.
- Practical replication depends on option availability, pricing, and hedging considerations.
Tags
Full text
# 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?
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 $p$ )?
## Answer by Andrew (score 4, accepted)
https://quant.stackexchange.com/a/7084
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
## Answer by Fab (score 2)
https://quant.stackexchange.com/a/10828
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.
## Answer by experquisite (score 1)
https://quant.stackexchange.com/a/7090
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.
## Answer by Matt Wolf (score 1)
https://quant.stackexchange.com/a/7091
- 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.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.