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??


2 Answers 2


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.

All of the above is based on Hull (2018) "Options, Futures and Other derivatives", i recommend chapters 14-15 for further explanation.

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    $\begingroup$ $(\mu - \frac{\sigma^2}{2} T)$ should be corrected to $(\mu - \frac{\sigma^2}{2})T$ in both formulas $\endgroup$
    – vas
    Aug 24, 2018 at 19:09
  1. BS assumes prices NOT returns are log-normally distributed. Why making that assumption? 1.log-normal is not perfect but OK to fit potential prices distribution. 2.The nature of log-normal distribution will force the left tail to be above zero. 3. There are definitely distributions work better than log-normal in terms of fitting stock price data, but that might involves a lot more work to do with uncertainties (parameterizations might fail).
  2. Log-normal distributions of prices implies normal distributions of returns. You can manifest it mathematically.

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