《金融期货与期权》（英文版） Chapter 17 Estimating Volatilities and Correlations

17.1 Estimating Volatilities and correlations Chapter 17 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.1 Estimating Volatilities and Correlations Chapter 17

Standard approach to 17.2 Estimating Volatility Define on as the volatility per day between day n-1 and day n, as estimated at end of day Define s as the value of market variable at end of day i Define u; =In(s si-d Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.2 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 ) s n n i i m n i i m m u u u m u 2 2 1 1 1 1 1 = − − = − = − =   ( )

173 Simplifications usually made Define u; as(Sisi-vsi-I Assume that the mean value of u: is zero Replace m-1 by m This gIves_21、m2 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.3 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 sn n i i m m u 2 2 1 1 = = −

174 Weighting Scheme Instead of assigning equal weights to the observations we can set where ∑α Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.4 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

17.5 ARCH(m Model In an aRCH(m) model we also assign some weight to the long- run variance rate V L 2 L where +∑ i=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 17.5 ARCH(m) Model In an ARCH(m) model we also assign some weight to the long-run variance rate, VL :   = = −  +  = s =  +  m i i m i n VL i un i 1 1 2 2 1 where

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−

17.10 Example Suppose 2=0.000002+0.13u2,+0.862 The long-run variance rate is 0.0002 So that the long- run volatility per day is 1.4% Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 17.10 Example • Suppose • The long-run variance rate is 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 − . .