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