I am working on a project where I price EU call options written on the VIX index.

The payoff function of interest looks like


where K is the strike price and y is the value of $VIX^2$

The Fourier transform of this function takes the form:


Here $\phi$ is the transform variable.

I would like to know more about how to get from $w_c$ to $\hat{w_c}$.

I have tried to look around in different transform tables, but with no luck.

Any help is much appreciated. Thanks.


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    $\begingroup$ Your notations are undefined. Please define the terms $\endgroup$ – Ezy Feb 6 at 23:41

The fourier transform is \begin{equation} \hat{w}_c= \int_{-\infty}^\infty (\sqrt{y}-K)^+ e^{-i\phi y}dy = \int_{K^2}^\infty (\sqrt{y}-K) e^{-i\phi y}dy \end{equation}

Now do a change of variable with $t=\sqrt{i\phi y}$ and solve the resulting integral.


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