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You need to use log of prices, because log of returns are normally distributed. So or where x is return- $$ x=-\frac{1}{\tau} ln(\frac{S_{t+\tau}}{S_{t}}) $$ The annualized standard deviation can be scaled as +/-$ n\frac{\sigma}{\sqrt{\tau}} $ where n is your multiple. You can either ignore or estimate drift. or look at it another way, S refers to the index ...



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