Abstract
Figure 12.1 illustrates how volatility can vary dramatically over time in financial markets. This figure is a semilog plot of the absolute values of weekly changes in AAA bond interest rates. Larger absolute changes occur in periods of higher volatility. In fact, the expected absolute change is proportional to the standard deviation. Because many changes were zero, 0.005% was added so that all data could plot on the log scale. A spline was added to show changes in volatility more clearly. The volatility varies by an order of magnitude over time; e.g., the spline (without the 0.005% added) varies between 0.017% and 0.20%. Accurate modeling of time-varying volatility is of utmost importance in financial engineering. The ARMA time series models studied in Chapter 4 are unsatisfactory for modeling volatility changes and other models are needed when volatility is not constant.
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References
Alexander, C. (2001) Market Models: A Guide to Financial Data Analysis, Wiley, Chichester.
Bera, A. K., and Higgins, M. L. (1993) A survey of Arch models, Journal of Economic Surveys, 7, 305–366. (Reprinted in Jarrow (1998).)
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedastic-ity, Journal of Econometrics, 31, 307–327.
Bollerslev, T. and Engle, R. F. (1993) Common persistence in conditional variances, Econometrica, 61, 167–186.
Bollerslev, T., Chou, R. Y., and Kroner, K. F. (1992) ARCH modelling in finance, Journal of Econometrics, 52, 5–59. (Reprinted in Jarrow (1998))
Bollerslev, T., Engle, R. F., and Nelson, D. B. (1994) ARCH models, In Handbook of Econometrics, Vol IV, Engle, R.F., and McFadden, D.L., Elsevier, Amsterdam.
Bollerslev, T., Engle, R. F., and Wooldridge, J. M. (1988) A capital asset pricing model with time-varying covariances, Journal of Political Economy, 96, 116–131.
Carroll, R. J. and Ruppert, D. (1988) Transformation and Weighting in Regression, Chapman & Hall, New York.
Duan, J.-C., (1995) The GARCH option pricing model, Mathematical Finance, 5, 13–32. (Reprinted in Jarrow (1998).)
Duan, J-C. and Simonato, J. G. (2001) American option pricing under GARCH by a Markov chain approximation, Journal of Economic Dynamics and Control, 25, 1689–1718.
Enders, W. (1995) Applied Econometric Time Series, Wiley, New York.
Engle, R. F. (1982) Autoregressive conditional heteroskedasticity with esti-mates of variance of U.K. inflation, Econometrica, 50, 987–1008.
Engle, R. F. and Ng, V. (1993) Measuring and testing the impact of news on volatility, Journal of Finance, 4, 47–59.
Gourieroux, C. and Jasiak, J. (2001) Financial Econometrics, Princeton University Press, Princeton, NJ.
Heston, S. and Nandi, S. (2000) A closed form GARCH option pricing model, The Review of Financial Studies, 13, 585–625.
Hsieh, K. C. and Ritchken, P. (2000) An empirical comparison of GARCH option pricing models, working paper.
Jarrow, R. (1998) Volatility: New Estimation Techniques for Pricing Derivatives, Risk Books, London. ( This is a collection of articles, many on GARCH models or on stochastic volatility models, which are related to GARCH models. )
Pindyck, R. S. and Rubinfeld, D.L. (1998) Econometric Models and Economic Forecasts, Irwin/McGraw Hill, Boston.
Ritchken, P. and Trevor, R. (1999) Pricing options under generalized GARCH and stochastic volatility processes, Journal of Finance, 54, 377–402.
Rossi, P. E. (1996) Modelling Stock Market Volatility, Academic, San Diego.
SAS Institute (1993) SAS/ETS User’s Guide, Version 6, 2nd Ed., SAS In-stitute, Cary, NC.
Tsay, R. S. (2002) Analysis of Financial Time Series, Wiley, New York.
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Ruppert, D. (2004). GARCH Models. In: Statistics and Finance. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6876-0_12
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DOI: https://doi.org/10.1007/978-1-4419-6876-0_12
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