numerator in dupire's formula(for Ivol to LocVol) is d/dt(sigma^2*t) ..if that is decresing wrt expiry then what is generally done. Is it a flaw in model or an arbitrage opportunity ? (A modeller would wait for the market(arbitrageurs) to trade in such a way to correct itself that numerator becomes positive, where as a arbitrageur would make exploiting trades and make numerator positive. )

Check in the literature (I think it is in Gatheral's book, otherwise in Mike Tehranchi's paper): The total implied variance, i.e. T*sigma(K,T)^2 must be an increasing function of T, for any K.

the original dupire lv formula is in terms of the derivatives of call prices, dCdT, dCdK and d2CdK2, that is, the infinitessimal calendare spread, call spread and butterfly spread. Positivity of these is a necessary and sufficient condition for preclusion of arbitrage in the vanilla surface (see "a note on sufficient conditions for no arbitrage", Carr & Madan)to get lv in terms of iv is simply applications of chain rule (dCdK=dCdK+dCdv x dvdK etc, see Gatheral ch1 for detail); if in using this you end up with -ve lv at any point, it implies one or more of your price derivatives has gone negative, i.e. an arbitrage. this may be a true arbitrage in the observable vanillas, or more likely you have interpolated iv badly between them to give rise to arbitrage.you can always simply floor your lv at zero; but then since it is in some sense a derivative of iv, when you integrate back to get iv (which is what you do implicitly when pricing options in lv framework) you will come back to something higher than your original iv, that is, you will overprice options. hopefully such an effect will be negligible.

i just can't believe something as popular(as far as i have heard) as local vol having such a basic flaw. sigma^2*T has to be increasing, otherwise what we are saying is that there is a time interval in future where uncertainty somehow decreases at such a pace that overall uncertainty beyond that time is negative, some black hole for uncertainty. and then beyond which things may again be normal.numerator in IV to LV conversion equation is d/dT{sigma^2*T} and if sigma^2*T is decreasing wrt T then i guess flooring might be done, but there is the problem that samyonez pointed, i.e. while converting LV back to IV i might get different IVs than i originally started with. In some other literature i saw people fit functional forms of LV like max(v_min,function(S,t)) there also i guess inserting this max and v_min is making information disappear somewhere and i would consider flooring and v_min to be dealing with same problem in almost same way.

I am not sure what you mean - if d sigma^2 T is null, the numerator should be null, too. NP.In practise the problem is your implied vol or rather its interpolation.

An decreasing total variance implies arbitrage opportunities using calendar spreadsYou can check chapter 6.4 of this doc http://www.lulu.com/content/e-book/a-ma ... s/7268112K