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If the underlying returns R, then the squared power perp returns 2R + R^2 (in price terms, ignoring funding). We can compare that to a 2x leveraged position in perpetuals. We want to equate the _expected_ returns of the power perp and 2x levered perp to avoid risk-neutral arb,
So get 2R + R^2 - F’ = 2R - 2F where F’ is the funding on the power perp and F is the funding on the perp. Taking expectations gives σ^2 - F’ = -2F which can be rearranged in two interesting ways,
1. F’ = 2F + σ^2 so the funding on the power perp is at least 2x the funding on the perp, and typically much more. Plugging in some realistic numbers I would expect a funding rate of 0.2-0.4% per day for the power perp — that’s huge!
2. σ^2 = F’ - 2F, this is more interesting since both F and F’ are observable and you can use them to derive an at-the-money implied vol for the underlying. Could use this in a yield strategy (sell power perps and hedge) or to do vol arb of power perps vs options.
I think you could go further — the higher order power perps depend on R^3, R^4 etc and I haven’t bothered to do the math but I’m pretty sure you could get an implied skewness/kurtosis from their traded prices which gives you most of a vol surface.
It’s be cool if an exchange listed these and they became popular, it seems like a much more straightforward way to get leverage and volatility exposure than options.
Someone should definitely list these because I can almost guarantee they will trade at negative implied vols in a big selloff and it will be really fun to come in and scoop them up cheap vs a 2x short in perpetuals.
why is it
2R + R^2 - F’ = 2R - 2F here, not
2R + R^2 - F’ = 2R - F
I don't get why we need a 2* before funding of the perp 
We're levering the perp 2x times, so we incur 2 times the funding F. What I am not getting is how we magically introduce sigma to the equation. Which mathematical rule are we applying here when "taking the expectation"?
How do we derive σ^2 - F’ = -2F from 2R + R^2 - F’ = 2R - 2F? I see how to get to R^2 - F’ = -2F. Thanks for the help!
So 2R on both sides can be cancelled out. Then we can take E[] (expectation) for both side. I guess E[F] = F whereas E[R^2] = E[(R-0)^2] = E[(R-meanR)^2] = sigma^2 (assume meanR=0)
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