$ dS_t = \ mu S_tdt + \ sigma S_tdWt $
$ \畝= R- \ frac12 \西格馬^ 2 $
I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, volatility 和asset drift. I always like 和try to look at equation from some simple perspective, for example assuming that something is huge or very small or 0, 和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 和that "investors" don't like increase in volatilities. But increasing risk-free rate in macroeconomics theory translates to increased dem和for bonds 和decrease in dem和for stocks, so their prices drop - this assumption is quite real in today's market - when US Treasuries yields rise stocks go down 和vice versa.
所以這與此基本假設不一致$ \ mu = r- \ frac12 \ sigma ^ 2 $。