Inferring a Future Stock Price Distribution from Implied Volatility

Summary

The document asks how an option implied volatility surface, which varies by strike and maturity, can inform the distribution of a stock price at a future date. It starts from geometric Brownian motion with constant volatility, under which the log price is normally distributed, then points out that actual future volatility is unknown and market implied volatility is not constant across strikes.

The text does not provide a solution or an empirical analysis. It frames the central issue: how to move from Black–Scholes implied volatilities to a risk neutral distribution, and how the volatility smile changes the distribution implied by the constant volatility model. Any inferred distribution would depend on option prices, model assumptions, and market conventions; the question itself does not address those details or establish that the implied distribution is the real world forecast distribution.

Key ideas

• A constant volatility geometric Brownian motion implies a normal distribution for log prices.
• Market implied volatility can vary with both option strike and maturity.
• The document asks how the volatility smile changes the terminal price distribution.
• It leaves the derivation and empirical validation unresolved.

Tags

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


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












> 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?

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