$dS_t=\mu S_tdt+\sigma S_tdWt$
I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, volatility and asset drift. I always like and try to look at equation from some simple perspective, for example assuming that something is huge or very small or 0, and trying to watch how it impacts other variables. This is good approach to remember some dependencies.
So looking at this later equation, first thing to note is the negative sign of volatility. This is OK when trying to explain why VIX is index of fear and that "investors" don't like increase in volatilities. But increasing risk-free rate in macroeconomics theory translates to increased demand for bonds and decrease in demand for stocks, so their prices drop - this assumption is quite real in today's market - when US Treasuries yields rise stocks go down and vice versa.
So this is not in agreement with this also fundamental assumption $\mu=r-\frac12\sigma^2$.
How do you interpret this fact?