# Volatility of multimodal distribution of returns

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:

1. a set of 2-tuples, where the first value of each 2-tuple is the index of the mode (e.g., 1, 2, 3, etc.) and the second value is the volatility of the mode,
2. the average of the volatilities of each mode, weighted by the probability of being in each mode,
3. the volatility of a single unimodal distribution fit over the entire dataset, or
4. something else?

Quantitative Finance Asked by Vivek Subramanian on February 3, 2021

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