Apr 25

Persistence and long-run volatility

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant |

Vol_reversion1

We previously looked at two models used to forecast volatility, GARCH(1,1) and its special case EWMA; i.e., where the weight assigned to the long-run variance [gamma] is set to zero, such that the first term in GARCH drops out and we are left with an exponential model.

At first, we just want to understand the single-period idea of these models; e.g., what is an estimate of today's one-day volatility given yesterday's volatility? We might then be tempted to ask, what is our multi-period estimate of volatility; e.g., given a one-day volatility, what is the 10-day (or n-day) estimate of volatility?

We will soon become accustomed, in the FRM, to a key math concept: volatility scales with the square root of time. If the square root seems odd, remember that it's only because of a linear variance relationship: variance scales directly with (is a linear function of) time. So, n-period variance equals the single-period variance multiplied by n:

Vol_reversion_scalerule

However, this is true only on two conditions:

  1. The returns are uncorrelated,
  2. The returns have a common standard deviation

This is the familiar assumption of independent and identically-distributed random variables (i.i.d.).

Can we just apply the "square-root-of-time rule" to scale the volatility? No, because these models (GARCH and EWMA) are complicated by the fact that both models embed the idea of persistence:

Vol_reversion3

We can see straightaway that neither of these models meet the conditions. Let's take the GARCH(1,1). The sum of alpha and beta is persistence. Whatever is "left over" (1 - alpha - beta) is the weight assigned to the long run variance. That's the mean reversion. We might say, in English, the GARCH is "some persistence plus some mean reversion." The mean reversion is a gravitational pull that invalidates our square root scaling rule. As Jorian says,

"If the starting position [today's variance] is greater than the long-run value, the square-root-of-time rule will overestimate risk. If the starting position is less than the long-run value, the square-root-of-time rule will underestimate risk." Chapter 9, Value at Risk, Philippe Jorian

One final note, the mean-reverting aspect of GARCH(1,1) is one way to look at the violation. The other way to view the violation is by the serial correlation implied by persistence. If returns in a time series are not uncorrelated, then they are serially correlated. Serial correlation invalidates the square-root-of-time rule, too, but the implication varies on the direction:

  • Positive serial correlation implies "momentum:" the square-root-rule will underestimate multi-period volatility
  • Negative serial correlation implies a "mean-reverting" process: the square-root-rule will overestimate multi-period volatility

UPDATE: I just received this good question from Jose: "Why is the square root of periods used to annualize the standard deviation (instead of just the number of periods as statistics theory says)?"

Without going into the math, I have two suggestions:

1. Start with variance and remember that the standard deviation is the square root of the variance. Given an i.i.d random variable, the 2-period variance is 2 times the variance. The standard deviation is the square root of that, or the square root of 2 times the standard deviation. It may be helpful to start with the notion that the variance does act the way you are thinking!

2. But still, this is merely the mathematical implication of a very restrictive (but elegant) assumption about the behaviour of returns. Specifically, it's a Wiener process (a.k.a., Brownian motion). While it has elegant math, it's based on i.i.d. (see above for why "volatility clustering" immediately violates the assumption) and a very particular assumption about the behaviour of returns. Not necessarily reality. Especially when scaling over multiple periods. Given what we observe about return phenomena (e.g., volatility clustering, mean reversion, serial correlation), it's a brave thing to scale the periods at all...and some researchers will say you can't really do it beyond ten days or so.

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