math 5076 project 2

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