# 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?

• Why do you say there cannot be skewness in this model? You are allowing the volatility to depend on $x$, so how skewed/convex it is can be set in here... It depends on $b(x,t)$.
– will
Commented Aug 26, 2016 at 14:02

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.

• Could you give some examples? Commented Jul 14, 2013 at 13:51
• I know that there exist a CGMY-model which uses Levy distributions with a certain amount of skewness. I've also seen some data myself which might have skewness, but the dataset was not big enough to be conclusive. The fact that there exist models which take skewness into account, makes me believe that skewness is present in economic data for at least some stocks.
– Nick
Commented Jul 14, 2013 at 14:59
• Can you edit that into your answer, possibly with some references? Commented Jul 15, 2013 at 19:02
• Check, I hope it clears some things up ?
– Nick
Commented Jul 15, 2013 at 20:47

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