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

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.

closed as unclear what you're asking by LocalVolatility, Daneel Olivaw, Alex C, skoestlmeier, HelinFeb 14 at 22:08

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Your notations are undefined. Please define the terms – 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.