156 EWMA Model (Equation 15.7) In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time This leads to( a special case of (15. 4)with 4+=1,0<2≤1) =an21+(1-A) Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 15.6 EWMA Model (Equation 15.7) • 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 (a special case of (15.4) with i+1= i ,0<<1) sn sn un 2 1 2 1 2 = − + 1− − ( )
157 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 n=0. 94 for daily volatility forecasting Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 15.7 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 • JP Morgan use = 0.94 for daily volatility forecasting
158 GARCH (11) (Equation 15.8) In GARCH (1, 1)We assign some weight to the long-run average variance rate rk+aonI +u Since weights must sum to 1 γ++β=1 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 15.8 GARCH (1,1) (Equation 15.8) In GARCH (1,1) we assign some weight to the long-run average variance rate Since weights must sum to 1 + + b =1 sn V sn bun 2 1 2 1 2 = + − + −
159 GARCH(,1)(continued) Setting o=yV, the GarCH (1, 1)model +aO1+ and a-B Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 15.9 GARCH (1,1) (continued) Setting w = V, the GARCH (1,1) model is and sn w sn bun 2 1 2 1 2 = + − + − V = − − w 1 b