Kurtosis of a Straddle and the Role of Co-Kurtosis

Summary

The document addresses how to calculate the kurtosis of a straddle payoff formed by combining call and put outcomes. The response corrects an attempted combination rule: even if cross co-kurtosis terms are assumed to vanish, the weighted sum of the individual kurtoses must be divided by the fourth power of the combined standard deviation. That combined variance is needed in addition to the separate call and put variances.

It also gives the general fourth-moment expression, which includes three cross co-kurtosis terms with coefficients reflecting their multiplicities, alongside each leg’s own kurtosis contribution. The formula clarifies that the kurtosis of the sum depends on joint distribution features, not just the marginal kurtoses. The excerpt does not supply the underlying call and put payoff distributions or numerical inputs, so it offers a relationship for calculation rather than a computed straddle kurtosis.

Key ideas

• The kurtosis of a straddle is not generally the simple sum of call and put kurtosis values.
• The combined fourth central moment must be normalized by the combined variance squared.
• The full expression includes cross co-kurtosis terms between the call and put outcomes.
• Assuming zero co-kurtosis removes cross contributions but still requires the correct combined variance.

Tags

# Kurtosis of a straddle


# 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:

## Answer by ir7 (score 2, accepted)

https://quant.stackexchange.com/a/55167

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}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.