1
$\begingroup$

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??

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ $(\mu - \frac{\sigma^2}{2} T)$ should be corrected to $(\mu - \frac{\sigma^2}{2})T$ in both formulas $\endgroup$ – vas Aug 24 '18 at 19:09
3
$\begingroup$
  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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.