has Gatheral's ansatz E[x_t|x_T] = x_T*w_t/w_T, where w_t is expected total variance up to time t (p.32 in his book) actually been proved yet? (it holds for GBM with constant vol)

Not to my knowledge. Alan should know more about that. I always found it very artificial. I often asked to myself if it possible to find the implied vol surface equivalent local volatility model of a stochastic volatility model.I mean given a model dS=r*S*dt+sqrt(V)*S*dW and dV=a(V)*dt+b(V)*dW2 correlated find a local vol processdS=r*S+sigma(S,t)*dW1 which gives exatly the same implied vol surface at time t=0.I think that this question is closely related to your ansatz question.there is an interesting paper of Lee herehttp://www.cmap.polytechnique.fr/~rama/dea/rogerlee.pdfsee page 54

QuoteOriginally posted by: frenchXI often asked to myself if it possible to find the implied vol surface equivalent local volatility model of a stochastic volatility model.I mean given a model dS=r*S*dt+sqrt(V)*S*dW and dV=a(V)*dt+b(V)*dW2 correlated find a local vol processdS=r*S+sigma(S,t)*dW1 which gives exatly the same implied vol surface at time t=0.I think that this question is closely related to your ansatz question.there is an interesting paper of Lee herehttp://www.cmap.polytechnique.fr/~rama/dea/rogerlee.pdfsee page 54not sure if this is related to my question, can you explain a bit more? regarding your question though, doesn't gyongy's theorem (see also Piterbargs' paper on markovian projection) answer it? basically given a stochastic vol model it is always possible to find a local vol model with same terminal distribution, i.e. the same prices for european calls/puts.

Thank you very much for the nice references. I will certainly have a look at them. The question is will it works for all strikes and maturities. I will have a closer look to that. Thanks again for pointing!By memory if I remember well (I don't have the book under my eyes at the moment), the Gatheral formula you quoted is for calculating the local variance of the Heston model. He uses the paths of X while in fact only the path of the volatility process is needed. According to Derman-Kani, the local vol L(K,T)²=E(VT|XT=K) (which by memory is the starting point of Gatheral) but if you look at the formula 3.3 of the Lee paper I have posted, you have a formula without XT and without ansatz. I really believe that this formula is closed form and without approximation.You may also have a look here but I prefer the formula of Lee. http://www2.maths.ox.ac.uk/mcfg/OxfordP ... y.pdfThere is also a Malliavin calculus approach which gives a nice formula (easy to simulate by Monte Carlo). http://www.emis.de/journals/HOA/JAMSA/2005/3307.pdfsee 5.2

QuoteOriginally posted by: frollooshas Gatheral's ansatz E[x_t|x_T] = x_T*w_t/w_T, where w_t is expected total variance up to time t (p.32 in his book) actually been proved yet? (it holds for GBM with constant vol)He says it's only an approximation, so unless you have some evidence that it is true somewhere beyond GBM, I doubt anybodywould try to prove/disprove it. Also, it's pretty ill-defined, too. Does E[x_t|x_T] actually mean E[x_t|x_T, {v(s)}_(t,T)],i.e., conditional on knowing x_T and the entire volatility path from t to T? That or some similar filtration seems necessary just to make sense of the relation.My two cents.

on a slightly different topic, i was thinking of the following poor man's way to approximate probability distributions for T-->0 (for stoch vol models):given dx = -0.5*vdt + sqrt(v)[rho1*dW2 + rho2*dW2], and dv = a(v)dt + b(v)dW1.eliminate dW1 to obtain dx = -0.5*v*dt + sqrt(v)*rho1*(1/b(v))*[dv - a(v)*dt] + rho2*sqrt(v)*dW2. for small T make the following approximation: xT-x0 = -0.5*v0*T + sqrt(v0)*rho1*(1/b(v0))*[vT - v0 - a(v0)*T] + rho2*sqrt(v0)*W2TSince vT and W2T are independent the probability distribution of xT can be obtained with convolution. does this seem ok? if so, it can be extended quite straigforwardly to stoch vol + stoch int rates.

I think it should work but I fear that approximating the stochastic integral by only one differential increment will give a poor approximation with a very small radius of convergence.

yes it will work only for small times to maturity, and even then, as you point out, i am not sure how accurate it is (still need to test it with concrete examples). however, i think the accuracy can be improved by adding a correcting term such that in my approximation E[xT] = variance strike. in other words i need to add a correction term to the approximation so that i get the variance strike when i take expectations. what do you think?