文档介绍：Estimating Volatilities and Correlations Chapter 17
1
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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)
2
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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
3
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Weighting Scheme
Instead of assigning equal weights to the observations we can set
4
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
ARCH(m) Model
In an ARCH(m) model we also assign some weight to the long-run variance rate, VL:
5
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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
6
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
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 l = for daily volatility forecasting
7
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
GARCH (1,1)
In GARCH (1,1) we assign some weight to the long-run average variance rate
Since weights must sum to 1
g + a + b =1
8
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
GARCH (1,1) continued
Setting w = gV the GARCH (1,1) model is
and
9
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Example
Suppose
The long-run variance rate is so that