华东师范大学：《金融工程》英文版 Chapter 15 Estimating Volatilities and Correlation

15.1 Estimating Volatilities and Correlations Chapter 15 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 15.1 Estimating Volatilities and Correlations Chapter 15

152 Standard Approach to Estimating volatility (Equation 15.1) Define on as the volatility per day between day n-1 and day n, as estimated at end of day n-1 Define s as the value of market variable at end of day i Define u =In(S, /Si-1) m-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 15.2 Standard Approach to Estimating Volatility (Equation 15.1) • 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 ) s n n i i m n i i m m u u u m u 2 2 1 1 1 1 1 = − − = − = − =   ( )

153 Simplifications Usually made (Equation 15.4) Define u; as(S Si-1VSi-1 Assume that the mean value of u is zero Replace m-I by m This gives MLe) n 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 15.3 Simplifications Usually Made (Equation 15.4) • Define ui as (Si -Si-1 )/Si-1 • Assume that the mean value of ui is zero • Replace m-1 by m This gives (MLE) sn n i i m m u 2 2 1 1 = = −

154 Weighting scheme Instead of assigning equal weights to the observations we can set au where 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 15.4 Weighting Scheme Instead of assigning equal weights to the observations we can set s   n i n i i m i i m u 2 2 1 1 1 = = = − =   where

15.5 ARCH(m Model In an ARCH(m) model we also assign some weight to the long-run variance rate, V 7+ c u where 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 15.5 ARCH(m) Model In an ARCH(m) model we also assign some weight to the long-run variance rate, V: s     n i n i i m i i m V u 2 2 1 1 1 = + + = = − =   where

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

15.10 Example Suppose 0.000002+0.1312,+0.86o the long-run variance rate is V=0.0002 So that the long-run volatility per day is 1.4% 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 15.10 Example • Suppose • the long-run variance rate is V=0.0002 so that the long-run volatility per day is 1.4% sn un sn 2 1 2 1 2 = 0 000002 + 013 − + 086 − . .