What's Volatility?

Think you know what volatility is? Think again:

A stock (or a fund) has an average return of 0%. It moves on average 1% a day in absolute value; the average up move is 1% and the average down move is 1%. Assume that we live in the Gaussian world in which the returns (or daily percentage moves) can be safely modeled using a Normal Distribution. Assume that a year has 256 business days. The following questions concern the standard deviation of returns (i.e., of the percentage moves), the "sigma" that is used for volatility in financial applications. What is the daily sigma? What is the yearly sigma?

See this paper for the answers. This probably doesn't have much real world implication though. Once presented the full time serie of stock prices, instead of this "on average" stuff, I can't imagine any one would make similar mistake.