Deriving Log-Return Volatility from the Geometric Brownian Motion Model
Summary
The document derives the familiar link between stock volatility and the variability of continuously compounded returns under the Black–Scholes geometric Brownian motion model. Starting from a proportional price-change equation with constant drift and volatility, it applies Itô’s lemma to the logarithm of the stock price. The resulting log price ratio consists of a deterministic drift term and a Brownian shock scaled by volatility. Since the Brownian increment is normal, the log return is normal, and its standard deviation grows with the square root of elapsed time; over a unit time interval, it equals the model’s volatility parameter.
This derivation explains why logarithms of price ratios are used for continuously compounded returns and how the volatility parameter is interpreted. It assumes constant coefficients and the stated diffusion model, so it does not establish the same relationship for real prices with changing volatility, jumps, or other dynamics. The answer also does not resolve the question about prepaid-forward volatility when dividends are paid.
Key ideas
- The geometric Brownian motion model describes proportional stock-price changes through drift and a Brownian shock.
- Applying Itô’s lemma to log price yields a normal log return with a deterministic drift component.
- Log-return standard deviation scales with the square root of elapsed time.
- Over a unit interval, the log-return standard deviation equals the model volatility parameter.
- The derivation assumes the model’s diffusion structure and does not address prepaid-forward volatility with dividends.
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# Answer by oliversm (score 3, accepted)
# What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?
I am self-studying for an actuarial exam, Models for Financial Economics.
It's stated as a given in my manual that $\sigma$ is the volatility of the stock, $\sqrt{\text{Var}(\ln(S_t/S_0))}$ and that the volatility of the continuously compounded return on a stock is given by $\sqrt{\text{Var}(\ln(S))}$.
Clearly $\sqrt{\text{Var(X)}}$ is volatility, but where does the $\ln(S_t/S_0)$ and $\ln(S)$ come from, respectively?
Also stated without proof is that if a stock pays continuous dividends, $\sqrt{\text{Var}(\ln(S_t/S_0))} = \sqrt{\text{Var}\big(F_{0, t}^p(S)\big)}$. i.e. the volatility of the stock is the same as the volatility of the prepaid forward on the stock.
I was hoping someone could provide reference to a derivation, or an explanation.
## Answer by oliversm (score 3, accepted)
https://quant.stackexchange.com/a/26431
The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the following stochastic differential equation (SDE) $$\dfrac{\textrm{d}S}{S} = \mu\:\textrm{d}t + \sigma\:\textrm{d}W_t$$ where $W_t$ is a standard Brownian motion (which is a random variable), and typically $ \mu$ and $\sigma$ are taken to be positive constants, (although this assumption can be dropped, the final result is more complicated).
A standard argument to solve this is to then apply Ito's Lemma to the process $\log(S)$ by considering $\textrm{d}\big(\log(S)\big)$. Although to formally appreciate the subtle details requires a course in stochastic calculus. The final result though is that the solution to the above SDE is $$ S = S_0\exp\left(\left(\mu - \dfrac{\sigma^2}{2}\right)t + \sigma W_t\right)\:. $$ Notice that again $S$ is still a random variable (technically a log-normal random variable). Then if we consider $\log\left(\dfrac{S}{S_0}\right)$ we see that $$ \log\left(\dfrac{S}{S_0}\right) = \left(\mu - \dfrac{\sigma^2}{2}\right)t + \sigma W_t $$ which is a normally distributed random variable with mean $\left(\mu - \dfrac{\sigma^2}{2}\right)t$ and standard deviation $\sigma t^{\frac{1}{2}}$. What is then typically done is a time scale is picked such that $t\to1$ and so the standard deviation of (aka the noise) evaluates to $\sigma$.
Otherwise I am not sure what your notation is for a stock which pays dividends, so can't comment on that.
Look up any introductory book on financial derivatives or stochastic calculus for financial applications and this should cover the issues in more detail.
I hope that helps.
## Answer by Kwitter (score 0)
https://quant.stackexchange.com/a/26430
You can see e^(rt) =St/S0, which is continuous and can generate r when t=1 time lag r= In(St/S0).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.