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