Black–Scholes and Log-Normal Asset Prices
Summary
The document explains why the Black–Scholes–Merton model implies log-normally distributed asset prices at maturity and normally distributed log returns. It traces this result to the model’s assumption that price changes follow a generalized Wiener process with constant drift and variance: taking logarithms converts the resulting price distribution into a normal distribution.
The discussion distinguishes the distribution assumed for prices from the implied distribution of returns. A log-normal price model also keeps prices above zero, which can make it a practical approximation. The answers acknowledge that other distributions may fit observed stock prices better, but may require more complex parameter choices. The explanation describes a model implication rather than evidence that real market prices always follow this distribution; its fit and assumptions should be evaluated for the asset and setting in question.
Key ideas
- Black–Scholes–Merton assumes the underlying price follows a process with constant drift and variance.
- The model implies log-normal prices and normally distributed log returns at maturity.
- The log-normal price distribution places prices above zero.
- Other distributions may fit market data better, though potentially with greater modeling complexity.
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# Black Scholes and the Log Normal Distribution
# 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??
## Answer by jthg (score 11, accepted)
https://quant.stackexchange.com/a/39751
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.
## Answer by Hui (score 4)
https://quant.stackexchange.com/a/39769
- 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).
- Log-normal distributions of prices implies normal distributions of returns. You can manifest it mathematically.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.