READING 28: VOLATILITY

Volatility

        The volatility of a variable is the standard deviation of that variable’s continuously compounded return. The variance rate of a variable is the square of its standard deviation. Variance and standard deviation are computed using historical data. Risk managers may also compute implied volatility, which is the volatility that forces a model price (i.e., option pricing model) to equal the market price.

The Power Law

        The power law is an alternative approach to using probabilities from a normal distribution. It states that when X is large, the value of a variable V has the following property, where K and α are constants:
                 P(V > X) = K × X^–α

        where:
         V = the variable
         X = large value of V
         K and α = constants

Weighting Schemes

        Historical price data is used to generate return estimates, which are then used to estimate volatility. Traditional volatility estimation methods weight past information equally across time. Weighting schemes can be used to weight recent information more heavily than distant data.

The EWMA Model

        The exponentially weighted moving average (EWMA) model generates volatility estimates based on weightings of the last estimate of volatility and the latest current price change information. The objective is to account for previous volatility estimates, as well as to account for the latest return information.

        where:
         λ = weight on previous volatility estimate (λ between zero and one)

The GARCH(1,1) Model

        GARCH(1,1) models not only incorporate the most recent estimates of volatility and return, but also incorporate a long-run average level of variance.

         where:
         α = weighting on the previous period’s return
         β = weighting on the previous volatility estimate
         ω = weighted long-run variance = γV_L
         V_L = long-run average variance = ω/(1 – α – β)
          α + β + γ = 1
         α + β < 1 for stability so that γ is not negative
         GARCH(1,1) estimates of volatility have a better theoretical justification than the EWMA model. In the event that model parameter estimates indicate instability, however, EWMA volatility estimates may be used.

Mean Reversion

        In a GARCH(1,1) model, the sum of α + β is called the persistence. The persistence describes the rate at which the volatility will revert to its long-term value. A persistence equal to one means there is no mean reversion.

The weights in the EWMA and GARCH(1,1) models

        The EWMA is nothing other than a special case of a GARCH(1,1) volatility process, with ω = 0, α = 1 − λ, and β = λ. Similar to the EWMA model, β in the GARCH(1,1) equation represents the exponential decay rate of information. The GARCH(1,1) model adds to the information generated by the EWMA model in that it also assigns a weighting to the average long-run variance estimate.

Forecasting Future Volatility

        GARCH models do a very good job at modeling volatility clustering when periods of high volatility tend to be followed by other periods of high volatility and periods of low volatility tend to be followed by subsequent periods of low volatility.
        When forecasting future volatility, GARCH-generated volatility data does an excellent job in predicting the volatility term structure (i.e., differing volatilities for options given differing maturities). This modeling tool is quite frequently used by financial institutions when estimating exposure to various option positions.