# Kurtosis of a straddle

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:

• No it is not. For the same reason you cannot assume the variance of a portfolio equals the sum of the of the individual variances. And it's even weird to talk about variance of a call/putinstead of variance of the return of a call/put. This may help: quant.stackexchange.com/questions/2273/… – phdstudent Jun 23 at 15:37

## 1 Answer

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}

• Thanks do you think it is reasonable to assume null cokurtosis terns? – HJA24 Jul 1 at 5:17