math 5076 project 2
TRANSCRIPT
Time Series Volatility Models-Different volatility estimation by
Equal Weight Model & EWMA & GARCH(1,1)
-By Di Wu & Zheng Rong
Agenda:• Equal Weight model
• EWMA
• GARCH(1,1)
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 model Importance of Volatility
EWMA(exponentially weight)
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.
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:
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
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 days covered by the data.
Covariance matrix of equal weight model
Covariance matrix of EWMA
GARCH(1,1)
Mean Reverting
Estimating GARCH(1,1) parameters
Solver!
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 considered
For K =15zero autocorrelationCan be rejected>=25
S&P 500 3/31/11—4/29/16
Long Run Volatility Per Year : 0.1528Ljung-Box: 26.25
Compare GARCH to VIX
VIX: Implied Volatility of S&P 500 index options
GRACH: sqrt(252)* GRACH(1,1) vol per day
Forecasting Future Volatility May-2-2016
10.67 GRACH vol vs 15.05 implied volLong run volatility per year 15.28
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?
Thank you