magnitude.In the context of equity returns this may be due to leverage effects⇒Negative shocks result in a fall in the value of the firm which increases the debt-equity ratio.As a result stockholders perceive the firm as being more risky ⇒volatility increases.2tσWarwick Business School 24
Asymmetric GARCH models : Threshold ARCH (TARCH) –Glosten, Jagannathanand Runkle(GJR) model
where: When (good news) the ARCH effect is .When (bad news) the ARCH effect is .If leverage effects are present then expect γ>0..0 if 00 if 1111>=<=−−−tttIεε12121121102−−−−+++=tttttIγεσβεαασ01>−tε1α01<−tεγα+1Dummy variableWarwick Business School 25
Asymmetric GARCH models: Exponential GARCH (EGARCH)
The log transformation ensures that the conditional variance is positive regardless of the parameter values⇒no need for non-negativity constraints.Also the effect of past shocks is exponential rather than quadratic(as in GARCH/TARCH)⎥⎦⎤⎢⎣⎡−+++=−−−−−πσεδσεγσβασ2loglog1111212tttttt0 0 and 00 0 and 0111111<⇒><>⇒<<−−−−−−ttttttσεγεγσεγεγBad newsGood newsIn the EGARCH modelleverage effects ⇒γ<0**Asymmetry coefficient hasopposite sign from TARCH**Warwick Business School 26
The News Impact Curve (NIC) .000.001.002.003.004.005.006.007-.15-.10-.05.00.05.10.15.20.25.30Value of lagged residualSIG2_GARCHSIG2_EGARCH.00.02.04.06.08.10.12.14-1.2-0.8-0.40.00.40.81.2Value of lagged residualSIG2_GARCHSIG2_TARCHGARCHTARCHGARCHEGARCHThe NIC describes the response of volatility to past shocks:•In the GARCH model the impact of shocks is symmetricabout the origin. •In the asymmetric models (TARCH/EGARCH) negative shocks have a biggerimpact on volatility than positiveshocks of the same magnitude.2tσThe asymmetry is more pronounced for EGARCH than TARCH due to the exponential relationship between volatility and past shocksWarwick Business School 27
GARCH-in-Mean (GARCH-M) (see Seminar 5)
CAPM applied to the market portfolio
gives:
The market return therefore varies
directlywith the conditional variance
(which is usually modelled as a GARCH(1,1) process as above).
This is an example of a GARCH-M model: the conditional variance enters the mean equation.
GARCH-M is a model of a time-varying risk premium.
Recall that mistakenly assuming a constantrisk premium may give rise to rejection of the EMH (see lecture 2)
ttftrrελσ++=22121102tttεβσαασ++=−Applying CAPM to the market portfolio λIs the MARKET PRICE OF RISK:()()[]()()[]()mmmfmmmfmfrrrrrrErrrrErrE≡==−=−=− ,,cov,cov22λσλσβ()()2mfmrrEσ−Warwick Business School 28
Conclusion
GARCH models have assumed a central role in empirical finance for modelling time varying volatility.
There is a proliferation of variations on the ‘plain vanilla’GARCH model (including models for absolute/power returns, multivariate models…)
Numerous applications in finance discussed well in Brooks Chp8 and Cuthbertson& NitzscheChp29).Warwick Business School 29
References
Bollerslev(1986), A generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327.
Brooks (2002), Introductory econometrics for finance, CUP: Cambridge. Chp8**
Cuthbertsonand Nitzsche(2004) Quantitative financial economics: stocks, bond
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