# 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 Aug 26 '16 at 14:02