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留学生金融学课程作业需求:Models of time varying v

Empirical Finance
 Models of time varying volatility in empirical finance:
ARCH and GARCH models.
Module Leader: Dr Stuart Fraser
stuart.fraser@wbs.ac.uk
留学生论文网Room D1.18 (Social Studies)Warwick Business School 2
Introduction
Generalized-Autoregressive-Conditional-Heteroscedastic(GARCH)processes:
•Motivation for GARCH models.
•Examples of different types of GARCH model.
•Some applications of GARCH in finance.
•Identifying, estimating and testing GARCH models.
⇒Seminar 5: Modelling time-varying volatility in the FTSE All-Share Index excess returns.Warwick Business School 3
Motivation for GARCH processes
Two empirical features of financial return series are relevant in this context.
Firstly, there is the observation that over short horizons/holding periods of up to one month:
–Return volatility is time varying(there are periods of tranquilityand turbulence).
–Volatility clustering: large (small) price changes tend to be followed by further large (small) changes.
⇒volatility (risk) is positively correlated.
⇒Non-linear dependence in returns.
Secondly, the unconditional distributions of short-horizon returns have fat tails (leptokurtosis).Warwick Business School 4
1. Volatility clustering (see handout for Seminars 1 and 2)-.08-.06-.04-.02.00.02.04.06949596979899000102030405SP500_RETURNSWarwick Business School 5
⇒Non –linear dependence in returns
Correlogramof squaredSP500 returns
AutocorrelaPartial CorrelationAC PAC Q-Stat Prob |** |** 10.2070.207130.550 |** |* 20.20.164252.340 |** |* 30.2280.171410.190 |* |* 40.180.093508.640 |** |* 50.2050.115636.520 |* | 60.1550.044709.210 |* |* 70.1740.072801.50 |* | 80.1520.039871.820 |* | 90.1560.051946.10 |* | 100.160.051024.40Warwick Business School 6
2. Leptokurtic unconditional return distributions0100200300400500600700800-0.075-0.050-0.0250.0000.0250.050Series: SP500_RETURNSSample 1/03/1994 1/11/2006Observations 3033Mean 0.000412Median 0.000645Maximum 0.054248Minimum -0.070376Std. Dev. 0.010419Skewness -0.138163Kurtosis 6.818984Jarque-Bera 1852.783Probability 0.000000Warwick Business School 7
Volatility clusteringVolatility clustering implies volatility, h(r), is predictable from past information:A common measure of ex-antevolatility is the conditional variance: The ex-post (realized) volatility is: ()1−Ω=ttfhΩincludes past volatility and other relevant information.h is the conditional volatility.This measures ex-antevolatility.()()12−Ω−=tttrEhμThis is a one-step forecast of the varianceconditional on 1−Ωt()22ttrr≅−μμ≅0 for short horizon returnsestimator Range lnlnreturnsPower ,0 ,returns Absolute lowtthighttPPrr−>θθAlternative volatility/risk measures include:Warwick Business School 8



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