Distinguishing Price Drift from Log-Price Drift in Geometric Brownian Motion

# What are $\mu$ and $a$ in $ \mu = a + \frac{\sigma^2}{2} $


# What are $\mu$ and $a$ in $ \mu = a + \frac{\sigma^2}{2} $












Considering GBM:

\begin{equation} S(t_i) = S_0 \exp(a \cdot t_i + \sigma \cdot W(t_i)) = S_0 \exp\left((\mu - \frac{\sigma^2}{2}) \cdot t_i + \sigma \cdot W(t_i)\right) \end{equation}

I am interested in this part: $$ \begin{equation} a = \mu - \frac{\sigma^2}{2} \end{equation} $$ or reformulated:

$$ \mu = a + \frac{\sigma^2}{2} $$

I experimented a bit with simulating price paths in Python. From there I gained the intuition that when we simulate say 100 stocks (all have same inital stock price $S_0$) with GBM over a period of 1 year and take the mean of the simulated prices (after 1 year) we end up with approximately $S_0 \cdot exp(a + \frac{\sigma^2}{2})$ rather than just $S_0 \cdot exp(a)$.

From this experiment, $a$ would denote whatever we assumed to be the rate of growth (e.g. the riskless rate). And consequently $\mu$ would denote the corresponding expected rate of change in the price after 1 year, i.e. the drift (e.g. risk-neutral drift). See also Pricing European Options with Monte Carlo.

But after having read "The drift of stock price becomes the risk-free interest rate" under RNP and looking through the blog-post of Oxymoron, I am not sure anymore whether my understanding is decent.

The same topic occured also in this post: Drift rate vs. Riskless rate in the Black-Scholes model. (see Chris Taylor's answer and the comment of Antoine Conze to it)

Any input is welcome! Would be great if you could explain in simple terms (as I am not too familiar with differential equations and stochastic calculus in general).

Edit: More generally spoken, can we consider $\mu$ to be related to percentage change, whereas $a$ denotes log returns?

## Answer by achirikhin (score 2, accepted)

https://quant.stackexchange.com/a/79313

The above stochastic exponential is merely a solution of

$dS/S = \mu dt + \sigma dW_t$

which is equivalent to, by Ito

$d \log{S} = ( {\mu - \sigma^2/2 }) dt + \sigma dW_t$,

which can be immediately integrated.

If $\sigma = 0$, then you get the elementary result from calculus

$dS/S = \mu dt \equiv d \log S = \mu dt$,

but if $\sigma \neq 0$, hence $S$ is stochastic, then transformation from $dS/S$ to $d \log S$ requires addition of the term to $dt$. This is the consequence of the definition of the Ito stochastic integral. In Stratonovich, for example, the non-stochastic formula holds.