How Log Returns Evolve in the Black–Scholes Model
Summary
The document explains that a stock's log return in the Black–Scholes framework changes over time as a stochastic quantity. Starting from geometric Brownian motion for the stock price, it describes the log return as having a deterministic component tied to time and a random component driven by Brownian motion. Consequently, the return distribution's mean and dispersion both vary with the time horizon; the random component has standard deviation proportional to the square root of time.
A second answer highlights the risk-neutral pricing assumption: the stock's expected return under that measure equals the constant risk-free rate. The discussion is an introductory conceptual explanation rather than a derivation of option prices or an empirical study. The displayed stock-price solution in the accepted answer appears to use a positive one-half volatility-squared term, whereas the standard geometric Brownian motion solution has a negative one-half volatility-squared drift in the log price. Readers should verify that sign when applying the equations; the qualitative point that log returns are stochastic and their distribution changes with horizon remains central.
Key ideas
- In Black–Scholes, log return is a time-varying stochastic process rather than a fixed value.
- The return distribution's mean changes with the horizon, while its standard deviation grows with the square root of time.
- Under risk-neutral pricing, the stock's expected return is set to the risk-free rate.
- The accepted answer's displayed log-price drift sign should be checked against the standard geometric Brownian motion formula.
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# About the log return in the Black&Scholes model
# About the log return in the Black&Scholes model
I'm currently studying the Black&Scholes model and I'm not sure about the following thing: the log return, say r, doesn't evolve in time? I mean, dr/dt = 0, its derivative is zero? Does only its average evolve in time, that is, d average(r)/dt = something? Thank you.
## Answer by Jan Stuller (score 2, accepted)
https://quant.stackexchange.com/a/54724
Let me try to answer. In the Black-Scholes model, we have the following dynamics for a stock Price $S_t$:
$$S(t)=S(0)+\int^{t}_{0}r S(h)dh+\int^{t}_{0}\sigma S(h)dW(h)$$
The short-hand notation for the above would be:
$$dS_t= r S_t dt+\sigma S_tdW_t$$
The two equations are the same thing (just two different notations) and the solution to both is the log-normal process:
$$S_t = S_0exp{(rt+0.5\sigma^2t+ \sigma W(t)})$$
The log-return is defined as $ln\left(\frac{S_t}{S_0}\right)$, so we can easily see that:
$$ln\left(\frac{S_t}{S_0}\right)=rt+0.5\sigma^2t+ \sigma W(t)$$
You can see that the log-return is Normally distributed with mean $=rt+0.5\sigma^2t$ and standard deviation $=\sigma \sqrt(t)$ (why? Because by definition $\sigma W(t)$ is normally distributed with mean zero and standard deviation equal to $\sigma \sqrt(t)$) .
So the log-return itself evolves in time: it is a stochastic process that is normally distributed around its (time-dependent) mean and it has a (time-dependent) standard deviation. If you plot the log-return on x-y axes, with y being time and x being the log-return, you can picture it as a straight line with slope $rt$ where the Normal distribution of the log-return is tangentially centered on this straight line. As time goes on, the standard deviation of this Normal distribution around the line gets wider and wider.
## Answer by Bob Jansen (score 2)
https://quant.stackexchange.com/a/54720
The risk-free rate is assumed to be constant, see for example Wikipedia:
> (riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
Under the risk-neutral measure (where pricing using the Black-Scholes formula happens), the return on the stock is equal to the risk-free rate.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.