I don't understand this passage from Shreve, Stochastic Calculus for Finance II. The points I want to clarify :
1 ) How does the volatility tells us which paths are possible ?
2 ) How does the change of measure from the Real to the Risk Neutral one put more probability on the paths with lower returns ?
this is from page 217 :
"How does the change of measure from the Real to the Risk Neutral one put more probability on the paths with lower returns ?"
Are you familiar with how polling firms like Gallup adjust their sample to make them more representative of the general population? If they have interviewed 1000 people and find that they have fewer poor people than in the general population, they will overweight the opinions of the lower income subjects that they do have (i.e. give them a weight higher that $\frac{1}{1000}$). Through this ex-post re-weighting, you can change a sample that you already have to give it the statistical properties you want (in this case to lower the mean income of the sample).
Similarly, if you have a sample of 1,000,000 real price paths (having return $\mu$) generated by Monte Carlo simulation, you can overselect the lower return price paths and underselect the high return ones to bring the average return down to the risk free rate $r_f<\mu$.
This is what the famous "change of measure" and the Girsanov process does. It "reweights the probabilities" of various paths. See illustration here https://en.wikipedia.org/wiki/Girsanov_theorem
