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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?

  • $\begingroup$ 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. $\endgroup$ – Tal Fishman Oct 18 '11 at 11:50
  • $\begingroup$ The question is answered here: [implied quant.stackexchange.com/questions/1621/… $\endgroup$ – Brian B Oct 18 '11 at 14:30
  • $\begingroup$ @BrianB Thx it perfectly answers my question. $\endgroup$ – Ricky Bobby Oct 18 '11 at 15:17
  • $\begingroup$ @TalFishman Thx it perfectly answers my question. $\endgroup$ – Ricky Bobby Oct 18 '11 at 15:17