# 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.

• Your notations are undefined. Please define the terms – Ezy Feb 6 '19 at 23:41

The fourier transform is $$$$\hat{w}_c= \int_{-\infty}^\infty (\sqrt{y}-K)^+ e^{-i\phi y}dy = \int_{K^2}^\infty (\sqrt{y}-K) e^{-i\phi y}dy$$$$
Now do a change of variable with $$t=\sqrt{i\phi y}$$ and solve the resulting integral.