Training course for PhD/Master students

Title:  An Introduction to Volatility Modeling (ARCH models)

(Slides)

Abstract: Volatility modeling plays an important role in many financial applications, and it has held the attention of academics and practitioners over the last four decades. Mandelbrot (1963) and Fama (1965) showed that stock market volatility exhibits the volatility clustering property, a phenomenon which has been modeled by Engle’s (1982) Autoregressive Conditional heteroscedasticity (ARCH) model and its extension, Bollerslev’s (1986) GARCH model. The construction of the two models ARCH and GARCH was in contradiction with the asymmetric properties of financial series (e.g., leverage effect). To address this problem, many nonlinear extensions ARCH model have been proposed such as EGARCH (Nelson 1991), PGARCH (Ding et al. 1993), TGARCH (Zakoian 1994), MS-ARCH (Hamilton, 1989; Francq, Roussignol, Zakoian 2001), and Self-Exciting Threshold ARCH (Saidi, 2003; Saidi and Zakoian, 2006). In the existing literature, there are more than 300 GARCH types (Hansen and Lunde, 2005).

In this course, we will present a quick introduction to the volatility modeling using univariate and multivariate Autoregressive Conditional Heteroskedasticity (ARCH) models and discuss their applications. The course will present in the fist part the statistical properties of financial time series and then review the probabilistic and statistical frameworks of some selected ARCH models.

References:

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2. Ding, Z., Granger, C.W.J. and Engle, R.F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance (1) pp. 83 - 106.

3. Engle, R.-F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1007.

4. Fama, E.-F. (1965). The behavior of stock-market prices. J Bus 38(1):34–105.

5. Francq, C., Roussignol, M. and Zakoïan, J.- M. (2001). Conditional heteroskedasticity driven by hidden Markov chains. Journal of Time Series Analysis 22, 197-220.

6. Francq, C. and Zakoïan, J.-M. (2019). GARCH Models: Structure, Statistical Inference and Financial Applications, 2nd Edition. Wiley.

7. Hamilton J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time series and the Business Cycle, Econometrica, Vol. 57, No. 2.

8. Hansen, P. R. and Lunde, A. (2005). A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH (1,1)? Journal of Applied Econometrics, 20, 873-889.

9. Mandelbrot, B. (1963). New methods in statistical economics. J Polit Econ 71(5):421–440.

10. Nelson DB (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59:347–370

11. Saïdi, Y. (2003). Étude probabiliste et statistique de modèles conditionnellement hétéroscédastiques non linéaires. PhD Thesis, University of Lille.

12. Saïdi, Y. and Zakoian, J.-M. (2006). Stationarity and geometric ergodicity of a class of nonlinear ARCH models. The Annals of Applied Probability, 16(4): 2256–2271.

13. Zakoïan, J.-M. (1994). Threshold heteroskedastic models. J. Econom. Dynam. Control 18, 931–955.