Excel
02 Dec 2008
Apr
11
Approaches to current volatility
by David Harper, CFA, FRM, CIPM
The FRM program starts with volatility, as an introduction to Quantitative Analysis. Often, we want to know the unknowable: what is the current volatility for asset returns (the yellow oval below)?
1. Market Implied
One approach is to reverse-engineer the implied (implicit) volatility from the market price. But we need a market price, in order to solve for only one variable. Armed with Black-Scholes, and given volatility, we can solve for price. Or instead given price, we can tease out the volatility. But not both simultaneously. Option traders do this all the time. Given a traded price, they solve for the volatility that, if plugged into an option pricing model (OPM), would produce a "model value" that is equal to the "market value." It's the volatility "implied" by the market price.
2. Unconditional Parametric
If we can't get the market to imply a volatility, the common approach is to refer to history. We hope and assume that past is prologue, more or less. The simplest approach, so unrealistic as to be unhelpful, is to assume volatility is unconditional (blue rectangle above). By unconditional, we mean something like, "the average volatility is 20% with a standard deviation of 10%, regardless." It is unconditional because today's volatility does not depend (i.e., is not conditional on) yesterday's volatility.
3a. Conditional: Unweighted
Conditional volatility is more realistic. The basic conditional volatility (lower left in above diagram) is the unweighted variance. We pick a series of historical asset returns (e.g., 250 trading days) and we compute the variance of the series (and the square root of the variance will be the standard deviation, or volatility). This turns out to be too democratic: distant returns get the same weight as yesterday's return. (Note: if we take a simple average of a series, it is implicitly the same thing as assigning equal weights to all observations).
3b. Conditional: Weighted
So, we get to conditional weighted volatility schemes (lower-right in the above diagram). Most everyone uses these. It is helpful, per the excellent John Hull reading (Chapter 19), to start with ARCH(m). ARCH(m) refers to a variance that is based on a function of (m) lagging observations and a single long-run average variance. In English, for example,
"the variance will be a weighted average of the last 200 observations but with a tendency to revert to its long-run average of 20% (i.e., it is mean-reverting)"
GARCH(1,1) refers to a variance that is based on only the single most recent (one) variance and the single most recent squared-return. And also a term for the long-run variance. So GARCH(1,1) has three terms: average variance, yesterday's variance and yesterday's squared return. Each of these is weighted by, respectively, gamma, alpha, and beta. The weights must sum to one:
Note per the purple diagrams above: the exponentially weighted moving average (EWMA) is a special (or particular) case of the GARCH(1,1). Here is more on EWMA. If you take GARCH(1,1) and set gamma (the weight assigned to the average variance) equal to zero, then this term drops out. Make sure you understand what this means: if you drop the gamma-weighted term, that means the series is no longer mean-reverting! Then, if we force the other two weight to equal one, we've got the EWMA.
Finally, RiskMetricsTM is merely a branded flavor of EWMA, where the average weight assigned to the most recent variance is about 94% (it varies based on asset class).
Comments
Good illustration. One question. Is there a typo in the GARCH (1,1) formula under the conditional weighted section (in blue box)? The alpha and beta seems to be mis-placed with variance and return.
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