# 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?

• Hi Ricky, welcome to quant.SE. I believe your question has been asked and answered, though not where you pointed. Check out the question above and, if your question differs, please specify how. Oct 18, 2011 at 11:50
• The question is answered here: [implied quant.stackexchange.com/questions/1621/… Oct 18, 2011 at 14:30
• @BrianB Thx it perfectly answers my question. Oct 18, 2011 at 15:17
• @TalFishman Thx it perfectly answers my question. Oct 18, 2011 at 15:17