How do we calculate the standard deviation of returns in a time step?

[courtrigrad]

Hello all

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

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

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

Thanks

 

[courtrigrad]

is it because there is a random term and a deterministic term?

 

[wais]

Hello,

To calculate the standard deviation of returns in a time step, we first need to understand the formula for returns. The formula for returns is the change in price divided by the initial price. In this case, we are using the formula \frac{S_{i+1}-S_{i}}{S_{i}} where S_{i+1} is the price at the end of the time step and S_{i} is the price at the beginning of the time step.

In order to calculate the standard deviation, we need to have a set of returns. This is where the formula \frac{S_{i+1}-S_{i}}{S_{i}} = \mu\delta t comes in. This formula is used to calculate the mean of the returns, which is represented by \mu . By multiplying the mean by the time step, we are scaling the mean to match the time step. This is why the formula for mean is \mu\delta t .

In order to calculate the standard deviation, we also need to know the variability of the returns. This is where the z-score comes in. The z-score is a measure of how many standard deviations a data point is from the mean. In this case, we are using the standardized normal variable \phi to represent the z-score. This is why the formula for returns is R_{i} = \mu\delta t + \sigma\phi\delta t ^{\frac{1}{2}} . By multiplying the z-score by the standard deviation, we are scaling the variability to match the time step. This is why the formula for standard deviation is \sigma\delta t^{\frac{1}{2}} .

In summary, the formula for returns is used to calculate the mean and variability of returns in a time step. By scaling the mean and variability to match the time step, we can calculate the standard deviation of returns in a time step. I hope this helps clarify the formulas for you.