Transform of payoff function $w_c=(\sqrt{y}-K)^+$ [closed]

Question

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

The payoff function of interest looks like

$w_c=(\sqrt{y}-K)^+$

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

The Fourier transform of this function takes the form:

$\hat{w_c}=\frac{\sqrt{\pi}}{2}\frac{1-\text{erf}(K\sqrt{-\phi})}{(\sqrt{-\phi})^3}$

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.

Answer 1

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.