Modeling Univariate Volatility

by Romain Berry
J.P. Morgan Investment Analytics & Consulting
romain.p.berry@jpmorgan.com

This article is the eighth in a series of articles exploring risk management for institutional investors.

What is volatility?

As we demonstrated in our previous articles, volatility is one of the main drivers of Value-at-Risk (VaR) models for market risk. Therefore, it is important to select the right volatility model to capture various properties such as the lag in time, the clustering and so forth. Volatility is simply the standard deviation of the returns of a distribution. If returns are independently and individually distributed (i.i.d.)¹, then the volatility is called the sample standard error² of the return distribution. However, since  returns of most financial assets are not normal, we need to estimate their volatility with some statistical analysis.

Financial time series exhibit volatility clustering, as can clearly be seen over various periods of time for the S&P 500 Index (1950 - 2010) in Exhibit 1.

Exhibit 1 – Returns of the S&P 500

Economies go through business cycles, which may be considered to be a consequence of the stochastic nature of the financial markets³. Volatility clustering models attempt to capture the volatility of the financial markets, which are sometimes low, sometimes high, over a given period of time. But within each state (and over a short time period), there is a strong chance that a day of high volatility will be followed by another day of high volatility. Therefore, we may estimate volatility conditionally to the observation of previous days. As a result, calculating the volatility of a return series the traditional way by

(1)

where:
σ² is the standard deviation
N is the sample size
r_i is the return at date i
is the mean of the returns

would raise two questions regarding the optimal size of N (and its perenniality in the future to keep consistent measures over time) and the fact that each observation carries the same weight (which is in contradiction with the volatility clustering assumption).  Unlike the mean, higher (central) moments⁴ of a return distribution exhibit recognizable patterns and can be helpful when forecasting future prices. Indeed, when prices increase sharply one day, it can be reasonably assumed that they will rise or fall sharply the following day (high volatility regime). Therefore, it seems feasible to forecast the magnitude of the price changes, but not the direction.

We introduce in this paper the two most common conditional volatility models used to estimate a univariate volatility financial time series: the Generalized Auto-Regressive Conditional Heteroscedasticity⁵ (GARCH) and the Exponentially Weighted Moving Average (EWMA) models. We also demonstrate how to estimate univariate volatility with either model and how to assess their prediction power.

Stochastic Volatility Models

The simplest and most commonly used GARCH model designed by Bollerslev⁶ is the GARCH (1,1) and is defined as

(2)

where:
α is the weight assigned to the lagged squared returns
β is the weight assigned to the lagged variances
ω is a constant equal to  γ x V_L where V_L is the long run variance rate and γ is its weight⁷.

This model estimates the volatility on a given day based on a linear combination of the squared returns and volatilities of the previous days plus a constant. Indeed the “(1,1)” term in GARCH (1,1) indicates that the current variance is based on the squared return and variance of the previous day (1 lag for each). We use the squared returns because they also exhibit strong recognizable patterns.

The EWMA model is also widely used to estimate and forecast volatility and is defined by

(3)

where:
λ is the weight assigned to the lagged variances.

It is easy to see that the EWMA is a particular case of a GARCH model where ω = 0, α = 1 – λ and β = λ. One of the main differences between these two models is that the GARCH model incorporates mean reversion⁸ while the EWMA does not.

Estimating The Parameters

In order to illustrate how the parameters are estimated, let us analyze the S&P 500 between January 1950 and January 2010.

The most common method to estimate the parameters is to take the Log Likelihood. It is the logarithm of the Maximum Likelihood (ML). ML employs trails and errors to determine the optimal parameters that maximize the likelihood of the data to occur. It requires first to estimate the distribution of returns. For illustration (only), let us assume it follows a normal distribution.

(4)

ML then takes the form of
 

	(5)

which log-likelihood trivially follows
 

(6)

As can be seen in Exhibit 2, we may set the mean to 0, and after dropping the constant multiplicative factors, our problem becomes
 

(7)

We will need to determine the parameters ω, α and β that maximize (7) by iterations.

Since the EWMA and GARCH models contain lagged squared returns and lagged volatilities, we need to estimate σ₁². Theoretically, it is sounder to include it in the parameters (α, β, ω) to be estimated. In practice, if the sample size is large enough (few hundred observations), we can specify σ₁²asthe unconditional variance⁹ of all observations of r_i. Since we have more than 15,000 observations, we go with the latter methodology.

Exhibit 2 and 3 show how to construct these two models for the S&P 500. I used the Solver functionality of Excel to estimate the parameters ω, α and β. The cell in yellow corresponds to σ₁² (since we can only start from January 5th, 1950) obtained by setting it to the unconditional variance of the sample (i.e., the long run variance).

Exhibit 2 – Methodology to estimate the parameters of a GARCH(1,1) model¹⁰

The estimation of the EWMA model is simpler since we set ω =0, α = 1 – λ, and β = λ.

Exhibit 3 – Methodology to estimate the parameters of an EWMA model¹⁰

The log-likelihood of both models are very close from each other (with an advantage to GARCH (1,1)) but diverge on the maximum volatility (6.46% for the GARCH (1,1) model compared to 5.45% with the EWMA model). Therefore, the next questions are which model to choose and why?

Selecting A Volatility Model

We noticed that the volatility calculated with the GARCH (1,1) model was higher than the one calculated with the EWMA model for 67% of the data. Most of these differences were quite reasonable – between -5% and 0% in 85% of the occurrences. Thus, the GARCH (1,1) model seemed a bit more reactive to new shocks than the EWMA model, which is a good feature for a model to have in a volatility clustering environment.

Exhibit 4 – GARCH(1,1) and EWMA for the S&P 500

We can perform a consistency test on these two models if we note that each model needs to capture adequately the volatility clustering. Therefore, the returns should have no significant autoregressive conditional heteroscedasticity after standardization by their conditional volatility¹¹.

A quick visual test would be to draw the correlogram¹² of the S&P 500 with each model and check how much autocorrelation has been effectively removed by each model (difference between a red bar and a green bar at the same time lag in Exhibit 5 and 6).

Exhibit 5 – S&P 500 correlogram with GARCH(1,1)

Exhibit 6 – S&P 500 correlogram with EWMA

From both graphs, we can clearly see that both models have dramatically removed the autocorrelation. But it seems that the GARCH (1,1) model has done a better job, especially in the first few lags (i-1, i-2, etc.) which are the most important because they are the closest to the ith day (today).

This is confirmed after running a Ljung-Box test¹³ over the first 30 lags at 99% confidence level.

LB statisticcritical_value             LB statistic(Autocorrelation )      LB statisticGARCH(1,1)              LB statisticEWMA

= 50.9 = 6,266.44 = 27.6281 = 52.0295

According to this test, both models have removed more than 99% of the autocorrelation but the GARCH (1,1) model did a slightly better job with a removal of 99.56% of the autocorrelation versus 99.17% for the EWMA model. We note though that we cannot reject at 99% confidence the hypothesis that the EWMA model has completely removed all the autocorrelation from the time series a contrario of the GARCH(1,1) model.

Other autocorrelation tests like the Box-Pierce could be applied to confirm that result.

Conclusion

We presented in this article two of the most commonly used stochastic models to estimate the volatility of financial returns. The GARCH model is to be preferred for short term horizons because it is mean reverted. Many different GARCH models have been designed since Engle and Bollerslev which are well worth considering because they take account of leverage, asymmetry, and other properties of financial time series not well captured by the two models presented here. They are IGARCH, GARCH-M, EGARCH, TGARCH, PGARCH, NGARCH, QGARCH, GJR-GARCH, AGARCH.

To view the next article, Multiple Asset Class Return Comparison, click here.

¹independently and individually distributed means that the observations of a random variable (the returns of an asset over a period of time, for instance) are independent of each other and have the same distribution. Thus, they all have the same mean and variance.
²The standard error of a time series refers to its standard deviation when the mean is unknown and needs to be estimated prior to calculating the standard deviation. The standard error (SE) is commonly calculated by
      SE (x) = σ / √N
      where:
      σ is the standard deviation of the population, and
      N is the size of the sample.
³E. Slutsky, “The Summation of Random Causes as a Source of Cyclical Processes”, Problems of Economic Conditions, Edition The Conjuncture Institute, Moscow, Vol. 3 No.1, 1927.
⁴A kth central moment of a distribution is a statistical measure that describes the nature of the distribution and is defined as

m_k = E(x – μ)^k

where:
E is the expectation operator
μ is the mean of the distribution

The second, third and fourth central moments are the variance, skewness and kurtosis of the distribution.
⁵Heteroscedasticity comes from the Greek “hetero” (different) and “skedasis” (dispersion), and means that the time series of a random variable has a time-varying variance.
⁶T. Bollerslev, “Generalized Auto-Regressive Conditional Heteroscedasticity”, Journal of Econometrics, 31, 1986, pages 307-327. The Bollerslev model is a generalized version of an earlier model, an ARCH model developed by R. Engle in "Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of U.K. Inflation," Econometrica, 50, 1982, pages 987-1008.
⁷Because the weights must equal to 1, we have α + β + γ = 1.
⁸Mean reversion implies the variance will tend over a long period of time to revert to its long-run average. In other words, the variance can be in turn high or low, but only over a temporary period of time. In the long run, it will tend toward its long-run average.
⁹The unconditional variance refers to the (equally weighted) variance of the original distribution of returns r_i.
¹⁰Adapted from Hull J., “Options, Futures, and Other Derivatives”, 5th Edition, Prentice Hall, 2003, page 380.
¹¹Andersen et al. “The Distribution of stock return volatility”, Working Paper No. W7933, Kellogg Graduate School of Management, Northwestern University, 1999.
¹²A correlogram is a graphical representation of the autocorrelations calculated at different time lags.
¹³This test calculates the Ljung-Box statistic defined as

where:
η_k is the autocorrelation for a lag of k and
w_k = (m-2)/(m-k)

The Ljung-Box statistic is asymptotically distributed as chi-squared with p degrees of freedom. Here, for k=30, the hypothesis of complete removal of the autocorrelation in the time series can be rejected with 99% confidence when the Ljung-Box statistic is greater than 50.9.