A Good Introduction to GARCH and EWMA (Exponentialy Weighted Moving Average)

GARCH and EWMA

21 May 2010   by David Harper, CFA, FRM, CIPM

AIM:  Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility …
Including: GARCH APPROACH
Including: EXPONENTIAL SMOOTHING (EWMA)

Exponential smoothing (conditional parametric)

Modern methods place more weight on recent information. Both EWMA and GARCH place more weight on recent information. Further, as EWMA is a special case of GARCH, both EWMA and GARCH employ exponential smoothing.

GARCH (p, q) and in particular GARCH (1, 1)

GARCH (p, q) is a general autoregressive conditional heteroskedastic model. Key aspects include:

• Autoregressive (AR): tomorrow’s variance (or volatility) is a regressed function of today’s variance—it regresses on itself
• Conditional (C): tomorrow’s variance depends—is conditional on—the most recent variance. An unconditional variance would not depend on today’s variance
• Heteroskedastic (H): variances are not constant, they flux over time

GARCH regresses on “lagged” or historical terms. The lagged terms are either variance or squared returns. The generic GARCH (p, q) model regresses on (p) squared returns and (q) variances. Therefore, GARCH (1, 1) “lags” or regresses on last period’s squared return (i.e., just 1 return) and last period’s variance (i.e., just 1 variance).
GARCH (1, 1) given by the following equation.

The same GARCH (1, 1) formula can be given with Greek parameters: Hull writes the same GARCH equation as:

The first term (gVL) is important because VL is the long run average variance. Therefore, (gVL) is a product: it is the weighted long-run average variance. The GARCH (1, 1) model solves for the conditional variance as a function of three variables (previous variance, previous return^2, and long-run variance):


Persistence is a feature embedded in the GARCH model.

Tip:  In the above formulas, persistence is = (b + c) or (alpha-1+ beta). Persistence refers to how quickly (or slowly) the variance reverts or “decays” toward its long-run average. High persistence equates to slow decay and slow “regression toward the mean;” low persistence equates to rapid decay and quick “reversion to the mean.”

A persistence of 1.0 implies no mean reversion. A persistence of less than 1.0 implies “reversion to the mean,” where a lower persistence implies greater reversion to the mean.

Tip:  As above, the sum of the weights assigned to the lagged variance and lagged squared return is persistence (b+c = persistence). A high persistence (greater than zero but less than one) implies slow reversion to the mean. But if the weights assigned to the lagged variance and lagged squared return are greater than one, the model is non-stationary. If (b+c) is greater than 1 (if b+c > 1) the model is non-stationary and, according to Hull, unstable. In which case, EWMA is preferred.

Linda Allen says about GARCH (1, 1):

• GARCH is both “compact” (i.e., relatively simple) and remarkably accurate. GARCH models predominate in scholarly research. Many variations of the GARCH model have been attempted, but few have improved on the original.
• The drawback of the GARCH model is its nonlinearity [sic]

For example: Solve for long-run variance in GARCH (1,1)
Consider the GARCH (1, 1) equation below:

Assume that:

• the alpha parameter = 0.2,
• the beta parameter = 0.7, and

Note that omega is 0.2 but don’t mistake omega (0.2) for the long-run variance! Omega is the product of gamma and the long-run variance. So, if alpha + beta = 0.9, then gamma must be 0.1. Given that omega is 0.2, we know that the long-run variance must be 2.0 (0.2 ¸ 0.1 = 2.0).

GARCH(1,1): Mere notation difference between Hull and Allen

EWMA

EWMA is a special case of GARCH (1,1) and GARCH(1,1) is a generalized case of EWMA. The salient difference is that GARCH includes the additional term for mean reversion and EWMA lacks a mean reversion. Here is how we get from GARCH (1,1) to EWMA:

Then we let a = 0 and (b + c) =1, such that the above equation simplifies to:

This is now equivalent to the formula for exponentially weighted moving average (EWMA):

In EWMA, the lambda parameter now determines the “decay:” a lambda that is close to one (high lambda) exhibits slow decay.

The RiskMetricsTM Approach

RiskMetrics is a branded form of the exponentially weighted moving average (EWMA) approach:

The optimal (theoretical) lambda varies by asset class, but the overall optimal parameter used by RiskMetrics has been 0.94. In practice, RiskMetrics only uses one decay factor for all series:
· 0.94 for daily data
· 0.97 for monthly data (month defined as 25 trading days)
Technically, the daily and monthly models are inconsistent. However, they are both easy to use, they approximate the behavior of actual data quite well, and they are robust to misspecification.
Note: GARCH (1, 1), EWMA and RiskMetrics are each parametric and recursive.

Recursive EWMA

EWMA is (technically) an infinite series but the infinite series elegantly reduces to a recursive form:

Advantages and Disadvantages of MA (i.e., STDEV) vs. GARCH

GARCH estimations can provide estimations that are more accurate than MA
 

Graphical summary of the parametric methods that assign more weight to recent returns (GARCH & EWMA)

Summary Tips:

GARCH (1, 1) is generalized RiskMetrics; and, conversely, RiskMetrics is restricted case of GARCH (1,1) where a = 0 and (b + c) =1. GARCH (1, 1) is given by:

The three parameters are weights and therefore must sum to one:

Tip:  Be careful about the first term in the GARCH (1, 1) equation: omega (ω) = gamma(λ) * (average long-run variance). If you are asked for the variance, you may need to divide out the weight in order to compute the average variance.

Determine when and whether a GARCH or EWMA model should be used in volatility estimation
In practice, variance rates tend to be mean reverting; therefore, the GARCH (1, 1) model is theoretically superior (“more appealing than”) to the EWMA model. Remember, that’s the big difference: GARCH adds the parameter that weights the long-run average and therefore it incorporates mean reversion.

Tip:  GARCH (1, 1) is preferred unless the first parameter is negative (which is implied if alpha + beta > 1). In this case, GARCH (1,1) is unstable and EWMA is preferred.

Explain how the GARCH estimations can provide forecasts that are more accurate.
The moving average computes variance based on a trailing window of observations; e.g., the previous ten days, the previous 100 days.
There are two problems with moving average (MA):

• Ghosting feature: volatility shocks (sudden increases) are abruptly incorporated into the MA metric and then, when the trailing window passes, they are abruptly dropped from the calculation. Due to this the MA metric will shift in relation to the chosen window length
• Trend information is not incorporated

GARCH estimates improve on these weaknesses in two ways:

• More recent observations are assigned greater weights. This overcomes ghosting because a volatility shock will immediately impact the estimate but its influence will fade gradually as time passes
• A term is added to incorporate reversion to the mean

Explain how persistence is related to the reversion to the mean.
Given the GARCH (1, 1) equation:

Persistence is given by:

GARCH (1, 1) is unstable if the persistence > 1. A persistence of 1.0 indicates no mean reversion. A low persistence (e.g., 0.6) indicates rapid decay and high reversion to the mean.

Tip:  GARCH (1, 1) has three weights assigned to three factors. Persistence is the sum of the weights assigned to both the lagged variance and lagged squared return. The other weight is assigned to the long-run variance. If P = persistence and G = weight assigned to long-run variance, then P+G = 1.
Therefore, if P (persistence) is high, then G (mean reversion) is low: the persistent series is not strongly mean reverting; it exhibits “slow decay” toward the mean.
If P is low, then G must be high: the impersistent series does strongly mean revert; it exhibits “rapid decay” toward the mean.

The average, unconditional variance in the GARCH (1, 1) model is given by:

Explain how EWMA systematically discounts older data, and identify the RiskMetrics® daily and monthly decay factors.
The exponentially weighted moving average (EWMA) is given by:

The above formula is a recursive simplification of the “true” EWMA series which is given by:

In the EWMA series, each weight assigned to the squared returns is a constant ratio of the preceding weight. Specifically, lambda (l) is the ratio of between neighboring weights. In this way, older data is systematically discounted. The systematic discount can be gradual (slow) or abrupt, depending on lambda. If lambda is high (e.g., 0.99), then the discounting is very gradual. If lambda is low (e.g., 0.7), the discounting is more abrupt.
The RiskMetrics^TM decay factors:

• 0.94 for daily data
• 0.97 for monthly data (month defined as 25 trading days)

Explain why forecasting correlations can be more important than forecasting volatilities.
When measuring portfolio risk, correlations can be more important than individual instrument volatility/variance. Therefore, in regard to portfolio risk, a correlation forecast can be more important than individual volatility forecasts.
Use GARCH (1, 1) to forecast volatility
The expected future variance rate, in (t) periods forward, is given by:

For example, assume that a current volatility estimate (period n) is given by the following GARCH (1, 1) equation:

In this example, alpha is the weight (0.1) assigned to the previous squared return (the previous return was 4%), beta is the weight (0.7) assigned to the previous variance (0.0016).
What is the expected future volatility, in ten days (n + 10)?
First, solve for the long-run variance. It is not 0.00008; this term is the product of the variance and its weight. Since the weight must be 0.2 (= 1 - 0.1 -0.7), the long run variance = 0.0004.

Second, we need the current variance (period n). That is almost given to us above:

Now we can apply the formula to solve for the expected future variance rate:

This is the expected variance rate, so the expected volatility is approximately 2.24%. Notice how this works: the current volatility is about 3.69% and the long-run volatility is 2%. The 10-day forward projection “fades” the current rate nearer to the long-run rate.

Compare and contrast the use of historic simulation, multivariate density estimation, and hybrid methods for volatility forecasting.

Nonparametric Volatility Forecasting