Estimating Volatilities and Correlations Chapter 21

Outline Standard Approach to Estimating Volatility Weighting Scheme ARCH(m) Model EWMA Model GARCH model Maximum Likelihood Methods

Standard Approach to Estimating Volatility Define sn as the volatility per day between day n-1 and day n, as estimated at end of day n-1 Define Si as the value of market variable at end of day i Define ui= ln(Si/Si-1)

Simplifications Usually Made Define ui as (Si −Si-1)/Si-1 Assume that the mean value of ui is zero Replace m−1 by m This gives

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

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

EWMA Model (1/3) In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time, and the decline rate is This leads to

EWMA Model (2/3)

EWMA Model (3/3)

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 RiskMetrics uses l = 0.94 for daily volatility forecasting

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

GARCH (1,1) Setting w = gV the GARCH (1,1) model is and

GARCH (p,q)

The weight The weights decline exponentially at rate β The GARCH(1,1) is similar to EWMA except that assign some weight to the long-run volatility

Mean Reversion The GARCH(1,1) model recognizes that over time the variance tends to get pulled back to a long-run avarage level The GARCH(1,1) is equivalent to a model where the variance V follows the stochastic process

Prove(1/2) moment function

Prove(2/2)

Example Suppose The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%

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

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

Example 1 We sample 10 stocks at random on a certain day and find that the price of one of them declined and the price of others remaind same or increased what is the best estimate of the probability of a price decline? The probability of the event happening on one particular trial and not on the others is We maximize this to obtain a maximum likelihood estimate. Result: p=0.1

Example 2(1/2) Estimate the variance of observations from a normal distribution with mean zero

Example 2(2/2)

Thank you for listening

back