Has anyone worked on this subject?Gatheral, taking inspirations from some interesting results of Roger Lee, wrote a parametric function to be used to interpolate BS implied total variance (sigma * T^2). One you try this, typically you have arbitrage opportunities but this function behaves correctly for large/small strikes. Then one should just check that there is no aribtrage for small absolute moneyness. My problem is that I can't find a satisfactory set for the parameters value such that I don't have between-strikes-arbitrage.any idea?

Lee's papers tell u what the upper and lower bounds on the vol skew are to preclude such "between-strike arbing"

QuoteOriginally posted by: FordeLee's papers tell u what the upper and lower bounds on the vol skew are to preclude such "between-strike arbing"Yes, Gatheral's idea is based on these interesting results. So a necessary condition is that the asypthotes grow linearly and with slope less than 2. This is not sufficient around moneyness=0.By the way, given Lee's theorems, it would be interesting to study vol stoch models moments to deduce the asympthotic shape of their smiles.

in Lee's very nice paper "Implied Vol: Statics, Dynamics + probabilistic Interpretation" section 3.1he derives L/UBound on vol skew at ANY strike, + discusses special case of K=ATM

QuoteOriginally posted by: Fordein Lee's very nice paper "Implied Vol: Statics, Dynamics + probabilistic Interpretation" section 3.1he derives L/UBound on vol skew at ANY strike, + discusses special case of K=ATMYes, those are the bounds I whish to satisfy for any strike, given some bounds on the parameters of Gatheral's smile. Certainly they are satisfied for moneyness suff. small or large (when the total variance smile is linear and grows slower than 2abs(log(K))) .

if Jim doesn't get back to u, I'll email the params he used tonite (don't have details on me)

QuoteOriginally posted by: boschianQuoteOriginally posted by: FordeBy the way, given Lee's theorems, it would be interesting to study vol stoch models moments to deduce the asympthotic shape of their smiles.We have done something like that here-Vladimir

unfortunately, I don't have time to check validity of Gatheral's b(1+|rho|)<4/T condition, but r u saying thateven when this is satisfied, ur finding call/put spread arbs?

Hi Stefano, I've just calibrated SVI to usdjpy and eurusd smiles, + it works perfectly for the former, but is a few bp out on the implieds for the former. Hav u tried for similar underlyings? maybe my smile input data is a bit bogus?cheers,Martin

Hi, I tried with implied volatility on DJES50. Good results for most of the days (less than 1bp errors). (I don't manage to attach my error plot). Some difficulties on reproducing smiles of this kind:very short maturities: implied volatility is very smiledshort - mid maturities: implied vol is just skewedThis is because I calibrate on each maturity individually, starting from the longest.As a matter of fact, in order to prevent calendar arbitrage, when I calibrate on the shortest maturity, I can't permit to the right wing of the smile to grow too much, otherwise first maturity implied variance will cross implied variance of the following maturity. It seems like, for some days, the shortest maturity variance smile should change convexity. This is not allowed by Gatheral formula.

I have not tried to use Gatheral's method, but it should not be too difficultto fit (for each expiry separately) _if_ you can guess the minimum of the smile.It should give you a (too smooth) approximation in the traded range (and it maybe not very stable in parameters which determine the limiting cases).If you do not see the minimum you have no information on 'the other wing' andwhere it starts, so you may wish to assume something on both. Symmetry is onething, but starting (=location of the vertex) ... This typically happens forlonger expiries. So you will not need the other wing and moreover only a verysmall part of the variance is traded, so estimating an (expiry wise) approx isalways a little bit questionable (you have almost a line [given by 2 params]and wish to determine 5 params ...), may be his 'no-time-arbitrage-condition'can help here. So i would try to start at 3 - 6 month with 2 triple whitches and work backand forward, possible ignore extreme positions (=very small vega).But i have only some pdf-handout from a talk and not the article, may be igot not all the things.To upload your files: zip them first, then within contributing to a threadchoose to browse&upload your file and then use 'attach' (for which you have to markthe file in an obvious drop down list) to make it public. Edited to upload the pdf (i hope, that is ok).

At the end, I think that the Gatheral sufficient condition on parameters that prevent between-strikes arbitrage, can be obtained in the case sigma = 0. In fact, the derivate of implied variance wrt monenyess are decreasing in sigma and non aribtrage is obtained when this derivate is bounded above and below by some functions (see Roger Lee paper). I will try and let you know (if someone is interested).

sigma should never be 0 (and increase with time) i think (because thevertex should become 'smoother'). I played with the function, find aMaple session enclosed.

Yes, sigma makes the curve smoother. And the derivative of SVI(x) w.r.t. x increases as sigma decreases. Because I need to upper bound this derivative, I was thinking of finding a sufficient condition posing sigma = 0.