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Estimating Volatilities and Correlations Chapter 21

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Standard Approach to Estimating Volatility Define n as the volatility per day on day n, as estimated at end of day n -1 Define S i as the value of market variable at end of day i Define u i = ln( S i /S i-1 ), an unbiased estimate is

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Simplifications Usually Made Define u i as (S i -S i-1 )/S i-1 Assume that the mean value of u i is zero Replace m-1 by m This gives

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Weighting Scheme Instead of assigning equal weights to the observations we can set

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ARCH(m) Model In an ARCH(m) model we also assign some weight to the long-run variance rate, V:

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EWMA Model In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time.

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Especially, if

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Attractions of EWMA Relatively little data needs to be stored We need only remember the current estimate of the variance rate and the most recent observation on the market variable Tracks volatility changes JP Morgan use = 0.94 for daily volatility forecasting

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GARCH (1,1) In GARCH (1,1) we assign some weight to the long-run average variance rate Since weights must sum to 1

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GARCH (1,1) continued Setting V the GARCH (1,1) model is and

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Example Suppose the long-run variance rate is so that the long-run volatility per day is 1.4%

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Example continued Suppose that the current estimate of the volatility is 1.6% per day and the most recent proportional change in the market variable is 1%. The new variance rate is The new volatility is 1.53% per day

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GARCH (p,q)

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Mean Reversion The GARCH(1,1) is equivalent to a model where the variance V follows the stochastic process GARCH(1,1) incorporates mean reversion, EWMA does not. When is negative, GARCH(1,1) is not stable, and we should use EWMA.

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Other Models We can design GARCH models so that the weight given to u i 2 depends on whether u i is positive or negative.

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Maximum Likelihood Methods In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring

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Example 1 We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens The probability of the outcome is We maximize this to obtain a maximum likelihood estimate: p=0.1

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Example 2 （ Suppose the variance is constant) Estimate the variance of observations from a normal distribution with mean zero

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Application to GARCH We choose parameters that maximize

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Excel Application (Table 21.1, page 477) Start with trial values of , , and Update variances Calculate Use solver to search for values of , , and that maximize this objective function Important note: set up spreadsheet so that you are searching for three numbers that are the same order of magnitude (See page 478)

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Variance Targeting One way of implementing GARCH(1,1) that increases stability is by using variance targeting We set the long-run average volatility equal to the sample variance Only two other parameters then have to be estimated.

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How Good is the Model? We compare the autocorrelation of the u i ’s with the autocorrelation of the u i / i The Ljung-Box statistic tests for autocorrelation

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Forecasting Future Volatility A few lines of algebra shows that The variance rate for an option expiring on day m is

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Forecasting Future Volatility continued (equation 19.4, page 473)

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Volatility Term Structures (Table 21.4) The GARCH (1,1) suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities Impact of 1% change in instantaneous volatility for Japanese yen example: Option Life (days) Volatility increase (%)

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Correlations Define u i =(U i -U i-1 )/U i-1 and v i =(V i -V i-1 )/V i-1 Also u,n : daily vol of U calculated on day n -1 v,n : daily vol of V calculated on day n -1 cov n : covariance calculated on day n -1

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Correlations continued Under GARCH (1,1) cov n = + u n-1 v n-1 + cov n -1

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Positive semi-definite Condition A variance-covariance matrix, is internally consistent if the positive semi-definite condition for all vectors w is satisfied.

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Example The variance covariance matrix is not internally consistent

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