Anyone seen these? Payoff is precisely the skew calc = 1 / (N-1) x SUM (Xi-Xbar)^3 / sigma^3Is there a risk-neutral hedge?Tks,Aaron

Hi Aaron-Do you mean Sigma^2 = varianceswap ?

Yes, you can have a look at a paper by Schoutens et al. called "Moment swaps" or something like that. Actually, for any moment, you have a hedge (dynamic of course) using all the lower moments.

I'm actually working on this subject, so if you have anyhting new or any suggestions, they'll be welcome.Thanks

amazing! has anyone executed a skew swap? i tried (mid 1990s) to work out a way of doing this such that you could replicate it with an option strategy similar to that of the var swap, but i failed.the closest i got was kind of interesting. the variance can be written as a series of one day power options, as the variance is sum( return^2), so that at any point you are looking at one live power option (straddle^2 struck at yesterday's close) plus a cliquet of one day power straddles. (odd that i have never heard anyone else describe the var swap that way.)so i thought i would substitute a "daily cliquet collar" struck at the previous day's close. to zeroth approximation, the collar payoff looks a bit like a graph of a cubic equation(!) what i ended up with was delta neutral, had zero vega, and had a giant skew exposure. we never dealt it.what was funny is that at the time, we had no skew exposure greeks in our books, so even if i had dealt this in huge size it would have shown up as zero for all the greeks on the risk management and trading systems! i never looked at kurtosis swaps... wonder if the 5th moment has a name? neurosis? Aaronbrask: I just found the paper : Moment Swaps I wonder if a var swap dealing desk would be suspicious if i asked to sell them a log-forward contract and buy a var swap contract, all with exactly the same terms??

you could buy a portfolio that paysf(S) = 1/2 max(S-K,0)^2 -1/2 max(K-S,0)^2i..e. a risk reversal of parabola contracts (which we could in turn replicate with vanillas), then by generalized Ito lemma, if u also dynamically trade -f'(S) units of stock, (ignoring interest rates)u will realize int_0^T sgn(S_t-K) \sigma^2_t dt dollars as P&Lwhich is essentially a skew trade. There's no model mispecification risk in doing this, so long as there's no jumps

I'd prefer a pentasis

I see your point : you trade the moving spread between the strike and the stock weighted by the instantaneous variance, which gives a measure of the symmetry of the final Payoff. But how do you analytically link that to the third order moment as described below ?

Moreover, who do you think could need such products (moment swaps with order higher than 2 ?) and who would possibly use them regarding their expensive price ?

can u tell me quickly how what the moment swap is,+ can the Math be formulated in continuous time, I hate discrete time stuff

in the past, the motivation for considering a skew swap was that one of the option books had a huge position in skew, and it was an ongoing business that looked like it would continue to grow. yet there was no obvious way to buy skew in any other package. the hope was that there would be hedge funds sophisticated enough to understand the risk, and take it on at a price far better than history would suggest. this would free up risk capital for the trading book so that they could grow the business, and the HF should in principle make money over time as well.but in the meantime, another trader there invented the "corridor variance swap". in this trade you would only include data if the spot price was within a corridor. if spot closed outside the corridor the points were ignored and not taken into account when calculating the swap payout. this trade is hedgable in a fairly simple way (options and daily delta hedge) and has a large skew element that was used to take risk off the books. hedge funds took the other side - happy ending!

MForde,You can have a look at the paper attached below. There's a paragraph on momet swap with order equal to three. It's in discrete time, but it's easy to write it in continuous time. You get the hedge using variance swaps.But what about using moments with order 4,5 or higher ?

Antonio: I must admit that the approach is slightly disappointing.The logic is like this: to hedge a var swap use the log forward. Error is order vol^3.to hedge a skew swap go long a log forward and short a var swap. Error is order vol^4!to hedge a kurtosis swap, go long a log forward, short a var swap and short a skew swap. Error is order vol^5.(!!)

What exactly is disappointing ?

I guess I had hoped that there was a different strategy involving only options and stock, similar to the var swap. I am not disappointed in the person doing the calculation, just disappointed in the answer!The philosophy of this strategy is that a log forward contains all the moment swaps, and so you never really synthesize a moment swap. You only ever synthesize a derivative which contains all the moment swaps up to inifinity, aside from those that you have hedged out! And so you simply round down all the higher order moments to zero. Thus, the log-forward strategy used to create a var swap, actually doesn't create a pure var swap. It creates an infinite series of moment swaps, each higher order moment swap being smaller by roughly a factor of sigma. This tells me that var swaps on 100% volatility underlyings probably are priced incorrectly, as sigma is of order 1, and the pricing mechanism is based on the log-forward contract. Thus if you go long a var swap on a 100% vol underlying, you are actually paying for var, skew, kurtosis, pentosis , and higher order swaps, but you are only getting a var swap.Hmmm....