In time-series analysis, I keep running into graphs that look roughly like...
Recalling a trig computation from an ODE book which explored the acoustic phenomena of beats, I noticed the orange volatility line looks like an approximate envelope for the maxima of the blue data. This makes qualitative sense, since the data is stationary with mean $0$, thus a higher volatility correlates to a higher absolute value of the data.
Still, I'm curious whether there's a nice way to represent volatility and its reflection around the $x$-axis as an envelope for the extrema of the data. An implicit approach would be to just scale the data by a constant factor until it fits within the volatility envelope, or define an error as the amount by which the data breeches the envelope, but this feels arbitrary. Is there a natural way to transform the data or volatility to make the envelope pattern emerge? Alternatively, is there anything special about a time-series for which volatility and its reflection automatically form an envelope for the data?
Imagine a simplified abstract scenario where, every minute, you extract the value of a random variable distributed like a normal (Gaussian) distribution with mean $0$ and standard deviation $\sigma$. This is an elementary example of what is called a stochastic process. If you keep a chart where you record the outcomes of your extraction as a function of time, after a while you will get a jagged line similar to the one in your picture, one of many possible realizations of that stochastic process. The line will however look more “regular,” in the sense that always lands roughly in the same ballpark of the mean, without ebbs and flows. More precisely, if you draw two lines at height $+\sigma$ and $-\sigma$ about the mean value, assuming you have let the process run long enough, roughly 68% of the time the value of the random variable will have landed within that strip, and 32% of the time it will have landed outside; if you draw two further lines at $\pm 2\sigma$, roughly 95% of the realization will lie within that larger strip; and so on, with predictable, time-independent statistical behavior.
Now suppose you extract the random variable from a normal distribution with mean $0$ and standard deviation $\sigma_t$, allowing the latter parameter to depend on time. Then, the time series of one realization of that stochastic process will look a lot more like the one in your picture. Toward the beginning of the series, the standard deviation seems to be relatively small, whereas it increases toward the end, leading to a larger, well, deviation of real values from the mean around that time. If you graph $\pm\sigma_t$ against time, you get something similar to the yellow curve, instead of a simple band about the mean; yet, once again, about 68% of the data will have landed within the variable-sized strip. (Observe that these percentages are highly dependent on the fact that in our thought experiment extractions are performed from a normal distribution, instead of, say, a uniform distribution, a Weibull distribution, a Student $t$-distribution, ...)
The volatility of a financial asset is essentially the standard deviation of the stochastic process that regulates (the logarithm of) its market price over time. The situation I have described above is a thought experiment that hopefully clarifies the statistics behind the fact that increased volatility at a particular time is intrinsically connected with higher fluctuations in (the logarithm of) stock value. In real situations, you do not know the actual, intrinsic volatility behind the value of the asset: you have to rely on estimations based on time series (historic volatility) or comparisons between data and models (implied volatility). This seems to be the case with your picture: observe that bursts in volatility seem to lag behind spikes in returns.
So, assuming the stochastic process underlying returns is Gaussian at all times (a somewhat strong hypothesis to make, but a good starting point), to get the volatility curve to make a nice envelope around the returns, I'd try multiplying the latter by a constant factor of $2/3$.