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Volatility Bands with Predictive Validity



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Volatility bands with predictive validity Dimitris N Politis Professor of Mathematics and Economics University of California San Diego La Jolla CA 92093 0112 USA www math ucsd edu politis Abstract The issue of volatility bands is re visited It is shown how the rolling geometric mean of a price series can serve as the centerline of a novel set of bands that enjoy a number of favorable properties including predictive validity Acknowledgment Many thanks are due to A Venetoulias of Quadrant Management for introducing me to Bollinger bands to K Thompson of Granite Portfolios for the incitement to re visit this issue and to two reviewers for their helpful comments Paper to appear in the Journal of Technical Analysis in Jan Feb 2007 1 Volatility bands with predictive validity Abstract The issue of volatility bands is re visited It is shown how the rolling geometric mean of a price series can serve as the centerline of a novel set of bands that enjoy a number of favorable properties including predictive validity Introduction Consider nancial time series data P1 Pn corresponding to recordings of a stock index stock price foreign exchange rate etc the recordings may be daily weekly or calculated at di erent discrete intervals Also consider the associated percentage returns X1 Xn As is well known we have Xt Pt Pt 1 log Pt log Pt 1 Pt 1 1 the approximation being due to a Taylor expansion under the assumption that Xt is small here log denotes the natural logarithm Eq 1 shows how why the logarithm of a price series i e the series Lt log Pt enters as a quantity of interest Bachelier s 1900 original implication was that the series Lt is a Gaussian random walk i e Brownian motion Under his simpli ed setting the returns Xt are approximately independent and identically distributed i i d random variables with Gaussian N 0 2 distribution Of course a lot of water has owed under the bridge since Bachelier s pioneering work The independence of returns was challenged rst by Mandelbrot 1963



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