3. Different volatility estimation by EWMA and Equal weight
model
We estimate the current
volatility by
is the daily return on
day i.
We estimate the current
volatility by
• Constant Volatility is
far from perfect
• Volatility, like asset’s
price, is a stochastic
process
• Attempted to keep
track the changing of
volatility
Equal weight modelImportance of Volatility EWMA(exponentially
weight)
4. Four-Index Example
Portfolio
• Dow Jones $ 4
million
• FTSE 100 $ 3
million
• CAC 40 $ 1
million
• Nikkei 225 $ 2
million
Initial Data
Data is from 07/08/2006
to 25/09/2008, totally
501 days with 500 daily
returns.
Find Volatility
We are going to
estimate the volatility
on tomorrow,
26/09/2008.
Comparing the results
from two models.
5. Equal weight
model • Calculate daily returns
• Find variance-corvariance
matrix by
• From the matrix, we could find
portfolio Std. Thus, we find one
day 99% VaR is $217,757
Process:
6. EWMA(exponentially
weight) • Calculate daily returns
• Find variance-corvariance
matrix by
• From the matrix, we could find
portfolio Std. Thus, we find one
day 99% VaR is $471,025
Process:
--path of volatility^2 from day 1 to day 501
7. Results
• Sheets show the estimated
daily standard deviations are
much higher when EWMA is
used than data are equally
weighted.
• Recall:
• This is because volatilities
were much higher during the
period immediately preceding
September 25, 2008, than
during the rest of the 500
Covariance matrix of equal weight model
Covariance matrix of EWMA
10. How Good is the
Model?
• Remove
Autocorrelation
• Ljung–Box statistic
• where is the
autocorrelation for a
lag of k, K is the
number of lags
For K =15
zero
autocorrelation
Can be
rejected
>=25
14. Summary
• The key feature of the EWMA is
that it does not give equal weight
to the observations on the ui^2 .
• The more recent an observation,
the greater the weight assigned
to it.
• GARCH(1,1) incorporates mean
reversion ------theoretically more
appealing
Which model is better?
The parameter belta can be interpreted as a ‘‘decay rate’’. It is similar to in the EWMA model.
But the GARCH(1,1) assigs weights that decline exponentially to past u^2, it also assigns some weight to the long-run average volatility.
The GARCH (1,1) model recognizes that over time the variance tends to get pulled back to a long-run average level of VL.
In practice, variance rates do tend to be mean reverting. The GARCH(1,1) model incorporates mean reversion, whereas the EWMA model does not. GARCH (1,1) is therefore theoretically more appealing than the EWMA model.
We are interested in choosing omega, alpha, and belta to maximize the sum of the numbers in the sixth
column. This involves an iterative search procedure.
Occasionally Solver gives a local maximum, so testing a number of different starting values for parameters is a good idea.
Informally, it is the similarity between observations as a function of the time lag.
considering the autocorrelation structure for the variables u_i^2 / sigma_i^2
Autocorrelation of the errors violates the ordinary least squares (OLS) assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (BLUE).
Durbin–Watson statistic
