176 EWMA Model In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time This leads to 2=n-1+(1-入)n Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.6 EWMA Model • In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time • This leads to 2 1 2 1 2 (1 ) sn = sn− + − un−
177 Attractions ofEwma 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 n=0.94 for daily volatility forecasting Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.7 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 = 0.94 for daily volatility forecasting
178 GARCH (1,1) In GarCH(1, 1)we assign some weight to the long- run average variance rate 2 YV1+Qln-1+βσ Since weights must sum to 1 y++β=1 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.8 GARCH (1,1) In GARCH (1,1) we assign some weight to the long-run average variance rate Since weights must sum to 1 + + b =1 2 1 2 1 2 + − +b s − s = n VL un n
179 GARCH (1,1) continued Setting @=y the GARCH (1, 1)model a+au n-1+βσ and Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.9 GARCH (1,1) continued Setting w = V the GARCH (1,1) model is and − −b w = 1 VL 2 1 2 1 2 sn = w+ un− + b sn−