Assume you have a two dimensional diffusion processd(X_t,Y_t)=b(t,X_t,Y_t)dt+A(t,X_t,Y_t)dW_tis there an elegant way to compute the functionH(t,y)=E[X_t | Y_t=y]?I'll accept a two dimensional differential equation but not a three dimensional one.The problem comes up through Dupire's equation where we need the coefficient H(t,K)=E[\sigma_t^2 | S_t=K].The objective is to compute this efficiently and then solve Dupire's (two dimensional) equation for call prices ina variety of stochastic volatility models. These models are diffusions as above with variables X_t the asset price and Y_t=\sigma_t^2 the variance of returns.Then we can write down a three dimensional differential equation for the call price directly from the diffusion equation.However a three dimensional diffeq. is too slow to solve.

If you can live with approximations there's a nice new paper by Vladimir Piterbarg on SSRN about this topic.

and if u can't live with approximations, check out the Atlan paper on arxiv.org,or go the short-maturity limit and use the BBF02 formula linking implied and local vol, and pick a model like SABR/Heston for which we knowshort maturity implied in closed form.Also "Smile at the uncertainty" by Brigo,Mercurio+co computes Dupire96 conditional localized variance in closed form.Best-M

QuoteOriginally posted by: GoGoFaIf you can live with approximations there's a nice new paper by Vladimir Piterbarg on SSRN about this topic.In case anyone is wondering it is this oneV

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