Hello all together, my question comes from the financial mathematics sector (pricing barrier options) but I am sure that you are the experts for this. I am trying to invert the following construct (equation 4.3/4.4 in http://laurent.nguyenngoc.free.fr/data/options-levy.pdf): Int(0,INF)exp(-q*T)dTInt(-INF,INF)exp(i*u*k)dkInt(0,INF)exp(-v*h)dhf(T,k,h) This is a combination of a laplace transform, a fourier transform and a laplace transform. I tried to find an algorithm which can inverse all these three transforms together but I was not able to find anything for this case. So now I think I have to invert separaetely the laplace transform, then the fourier transform and finally the second laplace transform sep by step. Do you agree? On the right hand side of the equation is a very ugly and long expression with complex numbers and fractions (can be seen via the link) where I need to have an algorihtm which can handle with such a structure. It would be great if anyone knows such an algorithm and could give me a hint. An answer would be a great support as this question did cost me a lot of nerves and time! Thanks very much in advance and best regards from Bavaria, Andrea

# inversion of joint laplace-fourier transform

Started by ●January 17, 2009

Reply by ●January 17, 20092009-01-17

Anderl wrote:> I am trying to invert the following construct (equation 4.3/4.4 > in http://laurent.nguyenngoc.free.fr/data/options-levy.pdf)Unless I misunderstand, 4.4 is the value of the integral 4.3, so the inverse transform of 4.4 would by definition be the UIC function (with exponential arguments) as defined just before proposition 3.2 on p. 6, no? Martin -- Quidquid latine scriptum est, altum videtur.

Reply by ●January 17, 20092009-01-17

>Anderl wrote: > >> I am trying to invert the following construct (equation 4.3/4.4 >> in http://laurent.nguyenngoc.free.fr/data/options-levy.pdf) > >Unless I misunderstand, 4.4 is the value of the integral 4.3, so the >inverse transform of 4.4 would by definition be the UIC function >(with exponential arguments) as defined just before proposition 3.2 >on p. 6, no? > > >Martin > >-- >Quidquid latine scriptum est, altum videtur. >Yes that's right. Just one annotation: There is a typo in 4.4 (the image function of the transforms), the right expression is: [S0*(Phi(q+r)-z)]/[z*(z-1)*(v+gamma)*(q+r-Psi(z))]*[1/(Phi(q+r)-z)- -1/(Phi(q+r)-z+v+gamma)] I hope this helps and I am looking forward to hear from you, Andrea

Reply by ●January 17, 20092009-01-17

Anderl wrote:> This is a combination of a laplace transform, a fourier transform > and a laplace transform. I tried to find an algorithm which can > inverse all these three transforms together but I was not able to > find anything for this case.If an FT-based numerical Laplace inversion method suits the problem (as ref. 8 in the paper seems to indicate) and the formulation is such that you can roll everything into a 3D FT then you might benefit from an explicitly multidimensional FFT code. I was going to give you a link to software that generates optimized factorizations of transform tensors but I seem to have lost it. However, there's a similar project at http://www.spiral.net/ that seems to handle n-D as well. Martin -- Quidquid latine scriptum est, altum videtur.

Reply by ●January 23, 20092009-01-23

Hi Martin, thanks for your reply. I am trying at the moment to invert the laplace transforms via the laplace inverse but am not sure about the result up to know. In the case that this won't work, you proposal would be my next alternative. Not so easy as it seems...>Anderl wrote: > >> This is a combination of a laplace transform, a fourier transform >> and a laplace transform. I tried to find an algorithm which can >> inverse all these three transforms together but I was not able to >> find anything for this case. > >If an FT-based numerical Laplace inversion method suits the problem >(as ref. 8 in the paper seems to indicate) and the formulation is >such that you can roll everything into a 3D FT then you might benefit >from an explicitly multidimensional FFT code. I was going to give you >a link to software that generates optimized factorizations of >transform tensors but I seem to have lost it. However, there's a >similar project at http://www.spiral.net/ that seems to handle n-D >as well. > > >Martin > >-- >Quidquid latine scriptum est, altum videtur. >