Math 5076 Project 2

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The document compares different models for estimating volatility over time: the equal weight model, EWMA, and GARCH(1,1). The equal weight model treats all past observations equally, while EWMA gives more recent observations greater weight. GARCH(1,1) incorporates mean reversion, making it theoretically more appealing. These models are applied to a portfolio of stocks to estimate volatility on a future date. EWMA produces higher volatility estimates as it weights recent high volatility more. While GARCH(1,1) fits well, the document concludes by questioning which model provides the best volatility estimates.

$Time Series Volatility Models -Different volatility estimation by Equal Weight Model & EWMA & GARCH(1,1) -By Di Wu & Zheng Rong$

$Agenda: • Equal Weight model • EWMA • GARCH(1,1)$

$Different volatility estimation by EWMA and Equal weight model We estimate the current volatility by is the daily return on day i. We estimate the current volatility by • Constant Volatility is far from perfect • Volatility, like asset’s price, is a stochastic process • Attempted to keep track the changing of volatility Equal weight modelImportance of Volatility EWMA(exponentially weight)$

$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 $471,025 Process: --path of volatility^2 from day 1 to day 501$

$Results • Sheets show the estimated daily standard deviations are much higher when EWMA is used than data are equally weighted. • Recall: • This is because volatilities were much higher during the period immediately preceding September 25, 2008, than during the rest of the 500 Covariance matrix of equal weight model Covariance matrix of EWMA$

$GARCH(1,1) Mean Reverting$

$Estimating GARCH(1,1) parameters Solver!$

$How Good is the Model? • Remove Autocorrelation • Ljung–Box statistic • where is the autocorrelation for a lag of k, K is the number of lags For K =15 zero autocorrelation Can be rejected >=25$

$S&P 500 3/31/11—4/29/16 sn 2 =0.0000043556+0.7993un-1 2 +0.153719sn-1 2 Long Run Volatility Per Year : 0.1528 Ljung-Box: 26.25$

$Compare GARCH to VIX VIX: Implied Volatility of S&P 500 index options GRACH: sqrt(252)* GRACH(1,1) vol per day$

$Forecasting Future Volatility May-2-2016 se =0.0000927+(0.7993+0.1537)*(0.00004286-0.0000927) 10.67 GRACH vol vs 15.05 implied vol Long run volatility per year 15.28$

$Summary • The key feature of the EWMA is that it does not give equal weight to the observations on the ui^2 . • The more recent an observation, the greater the weight assigned to it. • GARCH(1,1) incorporates mean reversion ------theoretically more appealing Which model is better?$

$Thank you$

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Math 5076 Project 2

1. Time Series Volatility Models -Different volatility estimation by Equal Weight Model & EWMA & GARCH(1,1) -By Di Wu & Zheng Rong

2. Agenda: • Equal Weight model • EWMA • GARCH(1,1)

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

8. GARCH(1,1) Mean Reverting

9. Estimating GARCH(1,1) parameters Solver!

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

11. S&P 500 3/31/11—4/29/16 sn 2 =0.0000043556+0.7993un-1 2 +0.153719sn-1 2 Long Run Volatility Per Year : 0.1528 Ljung-Box: 26.25

12. Compare GARCH to VIX VIX: Implied Volatility of S&P 500 index options GRACH: sqrt(252)* GRACH(1,1) vol per day

13. Forecasting Future Volatility May-2-2016 se =0.0000927+(0.7993+0.1537)*(0.00004286-0.0000927) 10.67 GRACH vol vs 15.05 implied vol Long run volatility per year 15.28

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?

15. Thank you

Editor's Notes

#9: 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.
#10: 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.
#11: 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

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Math 5076 Project 2

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Editor's Notes