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Why Option-Implied Log-Return Distributions Can Be Skewed

Article Quant Q&A · Author: Nick
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Summary

The document addresses whether an option-implied distribution for a stock’s log return must be symmetric because Brownian motion is symmetric. The response clarifies that symmetry is not a general property of option-implied distributions and points to models that can represent skewness. It names CGMY as an example of a model family with parameters governing distributional features such as skewness and kurtosis, and suggests a research reference on the implied volatility smirk.

The material is introductory and does not derive the distribution or explain how to infer it from option prices. It also does not describe CGMY parameter estimation, calibration, or empirical tests. The central distinction is that symmetric Brownian shocks alone do not establish that market-implied return distributions are symmetric; model assumptions and the pricing information reflected in options matter. The cited model and paper are pointers for further study rather than evidence developed in the document.

Key ideas

  • Option-implied log-return distributions are not necessarily symmetric.
  • Brownian motion’s symmetric increments do not guarantee that every model or market-implied distribution is symmetric.
  • CGMY is cited as a model family capable of representing skewness and kurtosis.
  • The document points to research on the implied volatility smirk but provides no derivation or empirical analysis.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Symmetry of option-implied probability density


# Symmetry of option-implied probability density












I was wondering whether the option implied probability density of the log returns:

$x = \ln\left(\frac{S}{S_0}\right)$ with S the value of a certain stock, is always symmetric ?

I was asking myself this question because we model the "randomness" in the log return with a Brownian motion which is symmetric around zero, which leads to a model of the form:

$dx_T = a(x,t)dt+b(x,t)dW_T$ with $W_T$ the Brownian motion. Where we simply have a drift where we superimpose a random walk.

In this kind of model there can't be a skewness, now I was wondering whether there were any models that take skewness into account and if it's already been seen in the distribution of the log returns?

## Answer by Nick (score 2, accepted)

https://quant.stackexchange.com/a/8465

No it's not always symmetric! There are a few models which take skewness into account. And it would seem weird to me that they'd build models to account for skewness if it wouldn't exist.

An example of these models is the CGMY model, the name comes from the parameters which are used to model the different moments like kurtosis and SKEWNESS. I don't know the details and I don't have a link since I've only seen it on internal papers at the university. But for as far as I've seen Google does just fine. If you look it up You'll find several CGMY densities where some have kurtosis and others skewness.

## Answer by plkn (score -1)

https://quant.stackexchange.com/a/29828

Google this: Zhang JinE., Yi Xiang, 2008, The implied volatility smirk, Quantitative Finance 8, p. 263–284.

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.