Why Straddle Payoff Moments Equal the Sum of Call and Put Moments
Summary
The document examines raw moments of a straddle payoff at expiry and asks whether the call and put components can be treated as independent. For a call payoff C and put payoff P at the same strike and maturity, the accepted explanation notes that both cannot be positive at once: when one has positive intrinsic value, the other is zero. Thus every mixed product of positive powers of C and P vanishes, and each power of the total straddle payoff equals the corresponding call power plus put power.
This payoff identity explains why the straddle’s raw moments can be obtained by adding the respective call and put raw moments, without assuming independence. It does not mean the options are independent; their payoffs are linked through the same underlying price. The response does not show the full moment derivation or validate the formulas in the question. A separate suggestion is to check lengthy algebra numerically against the original integral, which can help catch calculation errors.
Key ideas
- At a shared strike and expiry, call and put intrinsic values cannot both be positive.
- Every mixed product of positive powers of the call and put payoffs is therefore zero.
- Powers of the straddle payoff decompose into the matching powers of its call and put payoffs.
- This decomposition does not rely on independence between the options.
- Numerical evaluation of the defining integral can help verify a long analytical derivation.
Tags
Full text
# Higher moments of a straddle
# Higher moments of a straddle
Following the logic of Ben-Meir and Schiff (2012) and this question the first, second, third and fourth raw moments of a put are:
Similarity, for a call it is as follows:
where
and
`S` = spot price, `K` = strike price, `r` = risk-free rate, `T` = time to maturity and `sigma` is implied volatility.
I want to know what the third and fourth raw moments of a straddle are. A straddle consist of a call and a put If `S` > `K` at maturity. then the call option will have a value of `S` - `K`, and the put will have no value. Likewise if `S` < `K`, the call option will have no value, and the put will be worth `S` - `K`. This can be written as:
As a result the expected final value is equal to:
This can also be written as:
Which can be simplified to:
Following this logic for the other moments I get:
According to the theory about cumulants if two variables are independent, the `n-th`-order cumulant of their sum is equal to the sum of their `n-th`-order cumulants. Inspecting the final raw moments of the straddle it looks like this applies. However, a call and a put are not independent. When the value of a call increases/decreases, the value of a put decreases/increases, so the two option types are negatively correlated. This "fact" and the final results make me feel like I used the wrong assumptions.
Question: Are the the defined raw moments for a straddle correct or am I missing something?
## Answer by Gordon (score 3, accepted)
https://quant.stackexchange.com/a/55660
Let $C=(S-K)^+$ and $P=(K-S)^+$. Then it is clear, for any positive integers $i$ and $j$, \begin{align*} C^i P^j = 0. \end{align*} Consequently, for any positive integer $n$, \begin{align*} (C+P)^n = C^n + P^n. \end{align*} Your conclusion now follows immediately.
## Answer by Tom Gladd (score 0)
https://quant.stackexchange.com/a/55560
An easy way to check if you've made a mistake during a longish calculation like your derivation of the skew for a straddle is to numerically evaluate the original integral. It would be a one-liner in Mathematica.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.