Option Portfolios for Exposure to Higher Distribution Moments
Summary
The document asks whether options can create a position whose value responds only to the third moment of an implied stock distribution, while remaining insensitive to its first and second moments, and whether the idea extends to higher moments. The answer questions whether the intended target is the raw third moment or standardized central skewness, since these measure different features and cannot be treated as interchangeable.
As a point of comparison, it identifies a forward variance swap as a way to obtain exposure to future variance without direct exposure to the underlying’s price path before fixing begins. It suggests a forward sample skewness payoff as a possible analogue for third-moment exposure, then raises caveats about sampling frequency, non-independent returns, and the gap between terminal distribution properties and the single realized return path. It does not establish a replicating option portfolio or prove pure exposure; the proposed analogy remains tentative, and higher moments may not share variance’s special role in stochastic calculus.
Key ideas
- The question’s target must be clarified as a raw moment or a standardized central moment such as skewness.
- A forward variance swap is offered as a reference for exposure to future variance without initial delta exposure.
- A forward sample skewness payoff is suggested as a possible, but unproven, analogue for third-moment exposure.
- Sampling frequency and dependence among returns may affect whether sample higher moments are meaningful.
- The response does not derive a pure option replication for skewness or establish a general method for all higher moments.
Tags
Full text
# Get pure exposure to $\mathbb{E}[X^3]$
# Get pure exposure to $\mathbb{E}[X^3]$
Using option price and an interpolation method we can get for a particular expiry what is the implied density distribution of the stock by the market.
If we call this density $X$ then is it possible through a combination of options to get a pure exposure to the skew $\mathbb{E}[X^3]$?
So our position will gain money only if $\mathbb{E}[X^3]$ moves. If $\mathbb{E}[X]$ or $\mathbb{E}[X^2]$ move we don't make money nor loose money.
Is it possible to proove this result and is this generalizable to moment of order $n$ (we can get a position that has exposure only to $\mathbb{E}[X^n]$)
## Answer by Andrea (score 1)
https://quant.stackexchange.com/a/82024
I will provide my understanding of the question and a possible way to answer it.
When the OP mentions $\mathbb{E}[X^3]$, I wonder if he really means the skew, since it is otherwise unclear how one can separate moves in $\mathbb{E}[X^3]$ vs $\mathbb{E}[X^1]$ and $\mathbb{E}[X^2]$. What about standardised central moments?
Next, how would I answer the (simpler) question for $X^1$, $X^2$? Leaving aside that the former is irrelevant due to Risk Neutral drift, for the latter, I'd say the correct answer is a forward variance swap.
Why forward? Because, the request is to get exposure to properties of the distribution, not to actual market moves. We don't want any delta, so all I say is only valid before the first fixing.
What about exposure to $X^3$? Extrapolating from $X^2$, I think a forward sample skewness could be the right choice (see https://en.wikipedia.org/wiki/Skewness#Sample_skewness).
Payoff = $\frac{\frac{1}{n}\Sigma(x_i - \bar{x})^3}{\left(\frac{1}{n}\Sigma(x_i - \bar{x})^2\right)^{\frac{3}{2}}}$
where $x_i$ are returns.
And so on for higher powers (see https://en.wikipedia.org/wiki/Kurtosis#Sample_kurtosis).
A few points deserve to be mentioned:
- Quadratic variation has a special role in Ito calculus, no so higher variations: I suspect bad things will happen if the frequency of sampling increases (central limit theorem?)
- Do the above sample estimators make sense if the returns are not iid? (Levy models anyone?) Does it matter?
- How do properties of the "terminal distribution" compare to properties of the "returns" along the one and only path we are allowed to see? (Ergodic anyone?)
- But local vol will produce some delta even for forward variance swap? Sure, but, that is not the main point here.
- Can I use the Carr-Madan formula? Not sure what to do with it. You can definitely synthesize $X^3$ with calls and puts, but that does not provide the exposure required. Variance Swaps are equivalent to a log-portfolio: I don't see how this extends to higher moments.
That is my interpretation of the question.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.