# Getting the actual distribution of a stock price at time T using implied volatility [duplicate]

Possible Duplicate:
How to derive the implied probability distribution from B-S volatilities?

Let's assume a stock price S, with volatility $\sigma$ constant, no dividend, and risk free interest rate $r$ we have :

$dS=\mu Sdt +\sigma SdW$

And therefore using ito's lemma we get that $\ln(S)$ follows the following normal distribution:

$\ln(S) \rightarrow \phi((r -\frac {\sigma^2}2)T,\sigma \sqrt(T))$

But in real life the volatility is not a constant, and he future volatility is actualy unknown.

Now using black scholes equation I can calculate the implied volatility and I use it as an approximation of the expected volatility in the future. The implied vol will depend on the maturity T and the strike K (the current stock price and r are known).

How can I get the actual distribution of S (or ln(S)) using the value implied volatility $\sigma(T,K)$ ? Or in another words, how does the volatility smile modify the expected distribution of S ?

Not sure my question is clear, If you need any more clarification tell me.

I read a question with a similar title but the actual question is different : How to calculate future distribution of price using volatility?