# 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 $$r_{t}, r_{t+\delta}, r_{t+2\delta}, \ldots, r_T$$ (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?
• By rolling returns do you mean cumulative returns, $(\Pi (1+r)) -1$? – develarist Jul 20 '20 at 8:54
• What do you mean by multimodal? I understand a multimodal distribution to be one with two or more peaks. That seems to be a separate issue from heteroscedasticity (different values for the volatility). – Bob Jansen Jul 20 '20 at 14:50
• Thank you both for your comments. I have added some information to the body that hopefully clarifies your questions. Could you please let me know your thoughts? – Vivek Subramanian Jul 20 '20 at 21:02
• Maybe I am too narrow-minded but "volatility" in Finance is computed from non-overlapping returns. Volatility is the std deviation of independent increments. I am not sure what it would mean to compute it from rolling returns (which are not independent). – noob2 Jul 20 '20 at 21:34
• For what instrument are you observing this return distribution? Regarding your last comment: Maybe you can, I have never tried and I don't think many have. It seems more complicated with little benefit. Why would you? – Bob Jansen Jul 21 '20 at 7:11