# Black Scholes and the Log Normal Distribution

Why does the Black Scholes Equation imply the returns are log-normally distributed??

How can we tell that the returns of the underlying asset wouldnt be normally distributed??

The Black-Scholes-Merton (1973) model implies that the prices of the underlying asset at maturity $S_T$ are log-normally distributed $$ln(S_T)\sim N\big[ln(S_0)+(\mu-\frac{\sigma^2}{2})T,\;\sigma^2T\big]$$ so that the logarithmic returns to maturity $ln(\frac{S_T}{S_0})$ are normally distributed $$ln(\frac{S_T}{S_0})\sim N\big[(\mu-\frac{\sigma^2}{2})T, \;\sigma^2T\big]$$ The reason for this follows from the assumption that the underlyings price follows a generalized wiener process, with constant drift and variance.
• $(\mu - \frac{\sigma^2}{2} T)$ should be corrected to $(\mu - \frac{\sigma^2}{2})T$ in both formulas – vas Aug 24 '18 at 19:09