hi, I was wondering if anyone had a good reference on third order mixed derivatives using numerical approx (finite difference). I know how to take third derivative this way, but not mixed derivative, for example f'''(x,y) twice with respect to x and once with respect to y.

Cuchulainn has a nice describtion on a splitting scheme that deals with the second order mixed term in his book. I dont know if it could be fed to third order.

CollectorCan you explain please? Third order accuracy? Can explain the context? miss big picture, like what do you want to do?(Something else, BTW I have started a thread in Payoff Catalogue, seen it?)cheers

Uyxx should be split, first in y at points j-1, j and j+1 and then in x, in i-1, i+1, thus Uyxx = (Uxx)yformula (check true by Taylor's formula, step length is h in both directions)(u(i+1, j+1) - 2u(i, j+1) + u(i-1,j+1) - (u(i+1, j-1) - 2u(i, j-1) +u(i-1, j-i)) / (2 * h * (h*h))Sanity check: do Uxy = (Ux)y to see how it worksYou now have a 9-point scheme. Don't usie with ADI (no good, yes Sammus?), use Soviet splitting and then at the 'eplicit' time level as discussed in my book.Hope this helps, let me know how get on.

excellent Cuchulainn, I am just comparing some analytical solutions with numerical. PS think you typed in the sign wrong, seems to work if I have(u(i+1, j+1) - 2u(i, j+1) + u(i-1,j+1) - u(i+1, j-1) + 2u(i, j-1) -u(i-1, j-i)) / (2 * h * (h*h))Thanks again!!!

No he hasnt - u just missed the parenthesis round (u(i+1, j-1) - 2u(i, j-1) +u(i-1, j-i))

ok ok I see

Thanks Daveangel.

> Thanks again!!!You're welcomeIs this part of some known PDE?

>Is this part of some known PDE?not as I am aware off, however some third derivatives are used in Taylor approximations of stochastic vol models, for example Hull and White-88 uses higher order mixed derivatives with respect to variance and spot (spot once and variance twice), in their paper they can use analytical greeks from Black-Scholes, but it could be one need numerical in other similar situations. PS: In their 1987 approximation no mixed derivatives are used, the reason I guess is they here assume zero correlation between spot and vol, in their 88 extension they add correlation and now mixed derivatives are necessary. Well I have not studied their paper in detail yet, I am using such greeks for some other crappy stuff...

Collector, it is better to specify the particular derivatives in question. What is it - Gamma Bleed or some Vega derivatives with respect to Stock, Time, Vola? It should be easy to construct scheme tailored to a specific risk factor.

> What is it - Gamma Bleed or some Vega derivatives with respect to Stock, TimeYes, that's the kind of question I wanted to ask (I see the world as a PDE).For the solution, spacial thanks to Sammus who made the link, good lateral thinking.

wow I am flattered >You now have a 9-point scheme. Don't usie with ADI (no good, yes Sammus?), use Soviet splitting and then at the 'eplicit' time level as discussed in my book.Yes, the Soviet splitting gives a better approximation to the mixed term than the ADI method.