Kurtosis of a straddle

Question

I want to determine the kurtosis of a straddle. My question is closely related with the following topic here. According to the following paper of Ben-Meir and Schiff (2012) the expected value of a call is equal to

where

The variance of the call is

Following the standard definition of kurtosis I can write:

Similar, I can write the same for the puts:

Is it correct to assume that:

Answer 1

Even if you assume null cokurtosis terms, your equality is still off:

\begin{align} \operatorname{Kurt}[X+Y] = {1 \over \sigma_{X+Y}^4} \big( & \sigma_X^4\operatorname{Kurt}[X] + \sigma_Y^4\operatorname{Kurt}[Y] \big). \end{align}

Note that you need $\sigma_{X+Y}^2$. You already have $\sigma_X^2$ and $\sigma_Y^2$ (computed in the paper).

Full formula is:

\begin{align} \operatorname{Kurt}[X+Y] = {1 \over \sigma_{X+Y}^4} \big( & \sigma_X^4\operatorname{Kurt}[X] + 4\sigma_X^3\sigma_Y\operatorname{Cokurt}[X,X,X,Y] \\ & {} + 6\sigma_X^2\sigma_Y^2\operatorname{Cokurt}[X,X,Y,Y] \\[6pt] & {} + 4\sigma_X\sigma_Y^3\operatorname{Cokurt}[X,Y,Y,Y] + \sigma_Y^4\operatorname{Kurt}[Y] \big). \end{align}