Advertisement

ARCH Processes

  • Gilles Zumbach
Part of the Springer Finance book series (FINANCE)

Abstract

The ARCH processes are very important as they can capture correctly the empirical stylized facts regarding the heteroscedasticity. The GARCH(1,1) process is introduced first. Two equivalent formulations are presented: the original formulation of Engle and another form that is easier to understand intuitively and that allows for natural multiscale generalizations. This process is investigated in detail, in particular volatility forecasts and lagged correlations as they are the step stones for more complex analytical computations. Some variations around GARCH(1,1) are introduced, like I-GARCH(1), I-GARCH(2), and EGARCH(1,1). This set of simple processes allows one to understand the generic differences between linear and affine processes, and the implications for mean reversion, integrated processes, and the asymptotic properties. These simple processes pave the way for the rich family of multicomponent ARCH processes. Several variations of the multicomponent processes are studied, with a trade-off between simplicity and analytical computations versus more accurate stylized facts. In particular, these processes can reproduce the long memory observed in the empirical data. The mug shots are used to display the long-memory heteroscedasticity and the non-time reversal invariance. For a subclass of the multicomponent ARCH processes, a volatility forecast can be derived analytically. Then, the FIGARCH process is presented, and the difficulties related to the fractional difference operator with a finite cut-off are discussed. Advanced sections are devoted to the trend effect, the sensitivity with respect to the estimated parameters, the long-term dynamics of the mean volatility, and the empirical finding that the ARCH processes are always close to an instability limit. The induced dynamic for the volatility is derived from the ARCH equations, showing that it is a new type of process, clearly different from the stochastic volatility equation. The chapter concludes with a suggestion for a good and robust overall ARCH volatility process, with fixed parameters, that can be used in most practical situations.

Keywords

Stochastic Volatility Implied Volatility Financial Time Series Volatility Process Volatility Forecast 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 14.
    Baillie, R.T., Bollerslev, T., Mikkelsen, H.-O.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econom. 74(1), 3–30 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 25.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 26.
    Borland, L., Bouchaud, J.-P.: On a multi-timescale statistical feedback model for volatility fluctuations. J. Investm. Strat. 1(1), 65–104 (2011) Google Scholar
  4. 29.
    Bouchaud, J.-P.: Lessons from the 2008 crisis. In: The Endogenous Dynamics of Markets: Price Impact, Feedback Loops and Instabilities. Risk Publication (2011) Google Scholar
  5. 36.
    Carr, P., Geman, H., Madan, D., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002) Google Scholar
  6. 38.
    Chicheportiche, R., Bouchaud, J.-P.: The fine-structure of volatility feedback. Technical report (2012). Available at http://ssrn.com/abstract=2081675
  7. 41.
    Chung, C.F.: Estimating the fractionally integrated GARCH model. Unpublished working paper, National Taiwan University, Taipei, TW (1999) Google Scholar
  8. 45.
    Corradi, V.: Reconsidering the continuous time limit of the GARCH(1,1) process. J. Econom. 96, 145–153 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 52.
    Ding, Z., Granger, C.: Modeling volatility persistence of speculative returns: a new approach. J. Econom. 73, 185–215 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 55.
    Duan, J.-C.: The GARCH option pricing model. Math. Finance 5, 13–32 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 59.
    Engle, R.F.: Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 61.
    Engle, R.F., Bollerslev, T.: Modelling the persistence of conditional variances. Econom. Rev. 5, 1–50 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 77.
    Granger, C., Ding, Z.: Varieties of long memory models. J. Econom. 73, 61–77 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 84.
    Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: Financial market models with levy processes and time-varying volatility. J. Bank. Finance 32, 1363–1378 (2008) CrossRefGoogle Scholar
  15. 105.
    Menn, C., Rachev, S.T.: A GARCH option pricing model with α-stable innovations. Eur. J. Oper. Res. 163, 201–209 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 109.
    Mina, J., Xiao, J.: Return to riskmetrics: the evolution of a standard. Technical report, Risk Metrics Group (2001). Available at http://www.riskmetrics.com/pdf/rrmfinal.pdf
  17. 114.
    Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 115.
    Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econom. Theory 6, 318–334 (1990) CrossRefGoogle Scholar
  19. 136.
    Sentana, E.: Quadratic ARCH models. Rev. Econ. Stud. 62(4), 639 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 141.
    Teyssière, G.: Double long-memory financial time series. QMW Working paper 348, University of London, UK (1996) Google Scholar
  21. 150.
    Zumbach, G.: The Pitfalls in Fitting GARCH Processes. Advances in Quantitative Asset Management. Kluwer Academic, Dordrecht (2000) Google Scholar
  22. 152.
    Zumbach, G.: Volatility processes and volatility forecast with long memory. Quant. Finance 4, 70–86 (2004) Google Scholar
  23. 154.
    Zumbach, G.: The riskmetrics 2006 methodology. Technical report, RiskMetrics Group (2006). Available at: www.riskmetrics.com and www.ssrn.com
  24. 156.
    Zumbach, G.: Time reversal invariance in finance. Quant. Finance 9, 505–515 (2009) MathSciNetCrossRefGoogle Scholar
  25. 157.
    Zumbach, G.: Volatility conditional on price trends. Quant. Finance 10, 431–442 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 158.
    Zumbach, G.: Volatility forecasts and the at-the-money implied volatility: a multi-components ARCH approach and its relation with market models. Quant. Finance 11, 101–113 (2010) MathSciNetCrossRefGoogle Scholar
  27. 166.
    Zumbach, G., Fernández, L.: Option pricing with realistic ARCH processes. Technical report, Swissquote Bank (2011, submitted). Available at www.ssrn.com
  28. 167.
    Zumbach, G., Fernández, L., Weber, C.: Realistic processes for stocks from one day to one year. Technical report, Swissquote Bank (2010, submitted). Available at www.ssrn.com

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gilles Zumbach
    • 1
  1. 1.Consulting in Financial ResearchSaconnex d’ArveSwitzerland

Personalised recommendations