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:: ARCH and GARCH models for forecasting volatility ::

In the Black-Scholes model, the volatility of the underlying asset is the only non-directly observable variable.  For this reason, it is necessary to devise some method where by one can estimate (efficiently) and possibly anticipate the volatility.  In a previous article, we have already discussed a rather popular method of deducing the implied volatility from the Black-Scholes model and option prices.  This method has an advantage over direct estimations based on historical price changes, since it reflects how much volatility the market currently assumes within the Black-Scholes framework.  Quite often, the implied volatility is found to give rise to a skew, smile or frown, (as previously illustrated) depending upon the asset or market.

One problem with using the implied volatility is that whilst it takes into account the current view of the market, it does not give us any insight into possible future changes in volatility.  Given that the value of an option is primarily driven by the volatility, making predictions is a valuable tool from a practitioner's perspective.  To achieve this, we turn back our attention to historical price movements.

In the figure below, we have calculated the daily percentage volatility rate (per day) of the FTSE 100 over several years (data provided courtesy of Bloomberg).   [Note, for an index such as the FTSE 100, the volatility rates are generally less than that of a single stock option.]

We see that the volatility rates are higher over certain periods and lower in others.  The most striking feature is that periods of high volatility tend to cluster together. Therefore, one would expect the volatilities to be correlated to some extent.  The other noticeable feature is that the volatility tends to revert to some long-running average - a property commonly known as mean-reversion.  The flat red line in the figure above shows the average volatility over the entire period.  The mean-reversion nature of the volatilities helps ensures that the process remains statistically stationary.  [A discussion into stationary Vs non-stationary stochastic processes is beyond the scope of this article.  It is common to adopt a weaker definition of a stationary process where the first two order moments remain finite in time.]

Autocorrelation structures.  We take a slight detour to introduce the definition of the autocorrelation function.  The correlation function between two time series X and Y is given by the expression

where () and ()  are the mean and variance estimates of X (Y) respectively, and denotes the mean value of the expression inside the brackets.  The autocorrelation function is calculated by setting , where is some forward time lag of the time series X.  Hence, the autocorrelation function may be expressed as

The autocorrelation function is an average measure of the correlations that exist within a time series.  

To demonstrate the correlated nature of the volatilities, we have calculated the autocorrelation function of the daily volatility rates, illustrated in the figure below.

One immediately sees that the volatility autocorrelation decays extremely slowly with increasing time lag.  The form of this volatility autocorrelation has been empirically suggested to be either exponentially decaying, or exhibiting long-range memory (power-law decay).  We have performed a least squares estimate (red line) for an exponentially decaying function of the form

where a is a constant, as a rough comparison to the empirical data.  The exponential decay fit appears to be inadequate for characterising the form of the autocorrelation decay.

ARCH models

The autoregressive conditional heteroskedasticity model was introduced by Engle (1982) to model the volatility of UK inflation.  

As the name suggests, the model has the following properties:

1) Autoregression - Uses previous estimates of volatility to calculate subsequent (future) values.  Hence volatility values are closely related.
2) Heteroskedasticity - The probability distributions of the volatility varies with the current value.

In order to introduce ARCH processes, let us assume that we have a time series of asset price quotes for each time step i.  We calculate the fractional change in the price of the asset between time step i and i+1 using

Furthermore, we are required to determine the long-running historical volatility (e.g. over several years) denoted by .  In the first figure above, is illustrated by the flat red line.  We have seen that the volatility rates fluctuate about this mean long-running mean volatility, therefore, it seems reasonable to incorporate this quantity in the ARCH model.

Formally, an ARCH(m) process may be expressed mathematically as

where is the volatility at the time step, and are weighting factors that satisfy

Here m denotes the number of observations of used to determine .  The most common ARCH(m) process used to model asset price volatility dynamics is the ARCH(1) model where

or

using the above relation.

GARCH models

Bollerslev (1986) later proposed a more generalised form of the ARCH(m) model appropriately termed the GARCH(p,q) (General-ARCH) model.  The GARCH(p,q) model may be written as

The p and q denote the number of past observations of and , respectively, used to estimate .

The EWMA model

The Exponentially Weighted Moving Average model (EWMA) is a special case of the GARCH(1,1) model where .  Thus,

Since , we may express the EWMA model as

The EWMA model differs from ARCH and GARCH models since it does not mean-revert.  The preference between these different models is dependent upon many factors.  For example, the asset class, forcasting time frame under consideration, and the efficiency with which the weighting parameters may be calibrated to the time series.  Whilst the maximum likelihood estimators method may be the most obvious method to select for calibration with empirical data, more efficient algorithms have also been put forward.

Since these volatility forecasting models were introduced, there have been many alternatives/modifications proposed to these models to better their use in volatility forecasting.

Written by Alessio Farhadi.

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