# inversion of joint laplace-fourier transform

Started by January 17, 2009
```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

```
```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.
```
```>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

```
```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.
```
```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.
>
```