# Interpretation of drift parameter $\mu$ in GBM

Currently studying Ito's calculus. Looking on the GBM model: $\frac{d S_t}{S_t} = μ dt + \sigma d B_t$ we end up on the expected stock price at time t: $E[S_t]=s_0 e^{\mu t}$.What does actually $\mu$ represents?

My initial thought is the opportunity cost of Capital for investements with the same characteristics (risk) and time horizon,that produces NPV=0.Also, $\mu$ is the instantenious expect return for dt $\rightarrow$ 0.

Finally, I wonder whether there is a Connection between GBM and martingale processes.

• When $\mu=0$ GBM is a martingale, otherwise not. – noob2 Dec 21 '17 at 20:53
• You said it yourself: $\mu$ is the instantaneous rate of expected return. If one believes that stock price follows the GBM $dS = S[\mu dt + \sigma dW_t]$ where $\mu$ is quoted in annualized terms (say, 5\% per annum), then the expected return on the stock in a time period $\Delta t$ is $\mu \Delta t$. – user217285 Dec 21 '17 at 23:03