Quantitative Finance Asked by Vivek Subramanian on February 3, 2021
Take $x_1, x_2, ldots, x_T$ to be the price of a stock, indexed by $t=1, 2, ldots, T$. Define rate of return at time $t>W$ for a window size of $W$ to be
$$r_t = frac{x_t – x_{t-W}}{x_{t-W}}$$
Rolling returns for a shift $delta$ are thus given by the series $r_{t}, r_{t+delta}, r_{t+2delta}, ldots, r_T$ (for $t > W$). As shown below, the distribution of rolling returns could be multimodal, meaning that there may be more than one peak in the distribution.
What is the appropriate way to describe the volatility of the rate of return when determining risk? Is it:
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