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Hello all

Let's say we have [tex] \frac{1}{\sqrt{2\pi}} e^{\frac{-1}{2\phi^2}} [/tex] where [tex] \phi [/tex] is a standarized normal variable. Let [tex] R_{i} = \frac{S_{i+1} - S_{i}}{S_{i}} [/tex]Also let's say we have a time step [tex] \delta t [/tex] and the mean of the returns scaled with the timestep.Then mean = [tex] \mu\delta t [/tex]

Then why does [tex] \frac{S_{i+1}-S_{i}}{S_{i}} = \mu\delta t [/tex]? Isn't this supposed to be a z-score? Also suppose we want to know how the standard deviation scales with the timestep [tex] \delta t [/tex] The sample standard deviation is [tex] \sqrt{\frac{1}{M-1}\sum^M_{i=1}(R_{i}-R)^2} [/tex] How do we use this to get standard deviation = [tex]\sigma\delta t^{\frac{1}{2}} [/tex]

Also why does [tex] R_{i} = \mu\delta t + \sigma\phi\delta t ^{\frac{1}{2}} [/tex]?

Thanks

Let's say we have [tex] \frac{1}{\sqrt{2\pi}} e^{\frac{-1}{2\phi^2}} [/tex] where [tex] \phi [/tex] is a standarized normal variable. Let [tex] R_{i} = \frac{S_{i+1} - S_{i}}{S_{i}} [/tex]Also let's say we have a time step [tex] \delta t [/tex] and the mean of the returns scaled with the timestep.Then mean = [tex] \mu\delta t [/tex]

Then why does [tex] \frac{S_{i+1}-S_{i}}{S_{i}} = \mu\delta t [/tex]? Isn't this supposed to be a z-score? Also suppose we want to know how the standard deviation scales with the timestep [tex] \delta t [/tex] The sample standard deviation is [tex] \sqrt{\frac{1}{M-1}\sum^M_{i=1}(R_{i}-R)^2} [/tex] How do we use this to get standard deviation = [tex]\sigma\delta t^{\frac{1}{2}} [/tex]

Also why does [tex] R_{i} = \mu\delta t + \sigma\phi\delta t ^{\frac{1}{2}} [/tex]?

Thanks

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