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trc
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Posts: 95
Joined: April 4th, 2002, 2:28 pm

E[X_t|Y_t]

June 23rd, 2006, 12:59 pm

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.
 
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GoGoFa
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Joined: March 3rd, 2005, 11:23 am

E[X_t|Y_t]

June 27th, 2006, 12:18 pm

If you can live with approximations there's a nice new paper by Vladimir Piterbarg on SSRN about this topic.
 
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Forde
Posts: 124
Joined: November 27th, 2002, 7:45 pm

E[X_t|Y_t]

June 27th, 2006, 1:19 pm

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
 
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piterbarg
Posts: 458
Joined: October 29th, 2002, 6:42 pm

E[X_t|Y_t]

June 27th, 2006, 1:37 pm

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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trc
Topic Author
Posts: 95
Joined: April 4th, 2002, 2:28 pm

E[X_t|Y_t]

June 27th, 2006, 2:41 pm

Tanks for all replies.
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