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In [[finance]], '''volatility clustering''' refers to the observation, first noted by [[Benoît Mandelbrot|Mandelbrot]] (1963), that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes."<ref>Mandelbrot, B. B., [https://www.jstor.org/stable/2351623 The Variation of Certain Speculative Prices], The Journal of Business 36, No. 4, (1963), 394-419</ref> A quantitative manifestation of this fact is that, while returns themselves are uncorrelated, absolute returns <math>|r_{t}|</math> or their squares display a positive, significant and slowly decaying autocorrelation function: corr(|r{{sub|t}}|, |r{{sub|t+τ}} |) > 0 for τ ranging from a few minutes to several weeks. This empirical property has been documented in the 90's by [[Clive Granger|Granger]] and Ding (1993)<ref>Granger, C.W. J., Ding, Z. [https://www.jstor.org/stable/20076016 Some Properties of Absolute Return: An Alternative Measure of Risk ], Annales d'Économie et de Statistique, No. 40 (Oct. - Dec., 1995), pp. 67-91</ref> and Ding and [[Clive Granger|Granger]] (1996)<ref>Ding, Z., Granger, C.W.J. [https://doi.org/10.1016/0304-4076(95)01737-2 Modeling volatility persistence of speculative returns: A new approach], Journal of Econometrics), 1996, vol. 73, issue 1, 185-215</ref> among others; see also.<ref>{{cite conference|last1=Cont|first1=Rama|date=2007|editor1-last=Teyssière|editor1-first=Gilles|editor2-last=Kirman|editor2-first=Alan|title=Volatility Clustering in Financial Markets: Empirical Facts and Agent-Based Models|publisher=Springer|pages=289–309|doi=10.1007/978-3-540-34625-8_10|book-title=Long Memory in Economics}}</ref> Some studies point further to long-range dependence in volatility time series, see Ding, Granger and [[Robert F. Engle|Engle]] (1993)<ref>Zhuanxin Ding, Clive W.J. Granger, Robert F. Engle (1993)
[https://doi.org/10.1016/0927-5398(93)90006-D A long memory property of stock market returns and a new model], Journal of Empirical Finance,
Volume 1, Issue 1, 1993, Pages 83-106</ref> and Barndorff-Nielsen and Shephard.<ref>{{cite encyclopedia |author= Ole E. Barndorff-Nielsen, Neil Shephard
</ref>
Observations of this type in financial time series go against simple random walk models and have led to the use of [[GARCH]] models and mean-reverting [[stochastic volatility]] models in financial forecasting and [[Derivative (finance)|derivatives]] pricing. The [[ARCH]] ([[Robert F. Engle|Engle]], 1982) and
==See also==
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