garch models
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Lecture notes: Financial time series, ARCH and GARCH models Piotr Fryzlewicz Department of Mathematics University of Bristol Bristol BS8 1TW UK [email protected] http://www.maths.bris.ac.uk/~mapzf/ July 18, 2007 1 Fi nancial ti me series Let P k , k = 0,...,n, be a time series of prices of a fina ncial asset, e.g. daily quotes on a share, stock index, currency exchange rate or a commodity. Instead of analy sing P k , which often dis pla ys unit-root beha viour and thus cannot be mode lled as sta tionary, we often analyse log-returns on P k , i.e. the series y k = log P k − log P k−1 = log P k P k−1 = log 1 + P k − P k−1 P k−1 . By Tayl or-ex panding the abov e, we can see that y k is almost equivalent to the relative return (P k − P k−1 )/P k−1 . The reason we ty pical ly consider log-ret urns instead of relat ive 1
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