Hi all,Is there a way to do complex analysis on the characteristic function in AJD models even when the characteristic function is not in closed form, and instead, specified via Riccatti type ODEs?In fact this is a problem to use the inverse Laplace transform approach. Many transforms are defined not in closed-form, but using Riccatti type ODEs, hence there is no way to do complex analysis to find out singularities and branch-cuts, hence there is no way to design contour when doing inverse Laplace transform. Any thoughts?Maybe for non-closed form characteristic functions, people just use FFT to do the inversion, but then there are other issues such as truncating error, grid size, and number of sampling points, etc. It's very hard to make it work in a calibration setting...What do you think?

hi this is not to respond to you, but to ask you a question. i am currently coding using the fft inversion technique however am having a problem with computing the grid size and the truncating error. do you hav eany insight on this or how to go about it. am using the paper by broadie and kaya

The solution to your ODE is guaranteed to be smooth, so you have none of the branch-cut problems you'd have with closed-form characteristic functions. Only problems are that your solution may blow up (due to the non-existence of a certain moment), but that can be checked by analysing your moment-generating function (be it numerically or by looking at whatever you know about it).

QuoteOriginally posted by: LordRThe solution to your ODE is guaranteed to be smooth, so you have none of the branch-cut problems you'd have with closed-form characteristic functions. Only problems are that your solution may blow up (due to the non-existence of a certain moment), but that can be checked by analysing your moment-generating function (be it numerically or by looking at whatever you know about it).I don't see your point -- I need a contour integral, and the integrand is based on the ODE solutions. The ODE solutions are smooth in time t, but as C.F. function we are treating them as a function of "v", where the "v" is the variable in the C.F. domain. And then we extend "v" to allow it be on the complex plane - hence turning a C.F. into a Laplace transform.Now we want to do inverse LT, we need contour integral, and the contour has to be carefully selected... And what do you mean by "but that can be checked by analysing your moment-generating function (be it numerically or by looking at whatever you know about it)."? How do you analyse a MGF that is in numerical form?

QuoteOriginally posted by: mmnjauhi this is not to respond to you, but to ask you a question. i am currently coding using the fft inversion technique however am having a problem with computing the grid size and the truncating error. do you hav eany insight on this or how to go about it. am using the paper by broadie and kayaIf I recall correctly, the formulaes there were not correct... I also haven't been able to get a correct result from their paper...

thanks, i am still working on the paper. have you managed to find the moments of the characteristic function

QuoteOriginally posted by: mmnjauthanks, i am still working on the paper. have you managed to find the moments of the characteristic functionI thought using finite differencing on C.F. will give moments... no?

as i understand to get moments of a characteristic function is basically i*E[x] and the second moment is i^2*E[x^2] am not sure if this will apply for this particular case. any insight? also do you have this code with you, i have managed to work with the euler disctretisation, but the exact simulation gives me very large numbers for the value of the integral of Vsds

when Phi is the characteristic function k times differentiable, the following holds for every n <= k : Then you get your formula for any moment.