Volatility bands with predictive validity∗Dimitris N. PolitisProfessor of Mathematics and EconomicsUniversity of California, San DiegoLa Jolla, CA 92093-0112, USAwww.math.ucsd.edu/∼politisAbstractThe issue of volatility bands is re-visited. It is shown how therolling geometric mean of a price series can serve as the centerline of anovel set of bands that enjoy a number of favorable properties includingpredictive validity.Acknowledgment. Many thanks are due to A. Venetoulias of QuadrantManagement for introducing me to Bollinger bands, to K. Thompson ofGranite Portfolios for the incitement to re-visit this issue, and to two re-viewers for their helpful comments.∗PapertoappearintheJournal of Technical Analysis in Jan./Feb. 2007.1Volatility bands with predictive validityAbstract: The issue of volatility bands is re-visited. It is shown how the rollinggeometric mean of a price series can serve as the centerline of a novel set of bandsthat enjoy a number of favorable properties including predictive validity.IntroductionConsider financial time series data P1,... ,Pncorresponding to recordingsof a stock index, stock price, foreign exchange rate, etc.; the recordings maybe daily, weekly, or calculated at different (discrete) intervals. Also considerthe associated percentage returns X1,... ,Xn.Asiswellknown,wehave:Xt=Pt− Pt−1Pt−1 log Pt− log Pt−1, (1)the approximation being due to a Taylor expansion under the assumptionthat Xtis small; here log denotes the natural logarithm.Eq. (1) shows how/why the logarithm of a price series, i.e., the seriesLt:= log Pt, enters as a quantity of interest. Bachelier’s (1900) originalimplication was that the series Ltis a Gaussian random walk, i.e., Brown-ian motion. Under his simplified setting, the returns {Xt} are (approxi-mately) independent and identically distributed (i.i.d.) random variableswith Gaussian N(0,σ2) distribution.Of course, a lot of water has flowed under the bridge since Bachelier’spioneering work. The independence of returns was challenged first by Man-delbrot (1963) who pointed out the phenomenon of ‘volatility clustering’.Then, the Gaussian hypothesis was challenged in the 1960s by Fama (1963)who noticed that the distribution of returns seemed to have fatter tails than2the normal. Engle’s (1982) ARCH models attempt to capture both of theabove two phenomena; see Bollerslev et al. (1992) and Shephard (1996)for a review of ARCH/GARCH models and their application. More re-cently, Politis (2003, 2004, 2006) has developed a model-free alternative toARCH/GARCH models based on the notion of normalizing transformationsthat we will find useful in this paper.On top of an ARCH-like structure, the working assumption for manyfinancial analysts at the moment is that the returns {Xt} are locally sta-tionary, i.e., approximately stationary when only a small time-window isconsidered, and approximately uncorrelated. As far as the first two mo-ments are concerned, this local stationarity can be summarized as:1EXt 0and Var(Xt) σ2. (2)Here E and Vardenote expectation and variance respectively. Letting Covdenote covariance, the approximate uncorrelatedness can be described by:Cov(Xt,Xt−k) 0 for all k>0. (3)Note, that since Xt= Lt− Lt−1, the condition EXt 0ofeq.(2)isequivalent toELt= µtwith µt µt−1. (4)Thus, the mean of Ltand the variance of Xtcan be thought to be approxi-mately constant within the extent of a small time-window.1The approximate constancy of the (unconditional) variance in eq. (2) does not con-tradict the possible presence of the conditional heteroscedasticity (ARCH) phenomenonand volatility clustering.3A simple way to deal with the slowly changing mean µtis the popularMoving Average employed in financial analysis to capture trends. Further-more, the notion of volatility bands (Bollinger bands) has been found usefulin applied work. It is important to note, though, that the usual volatilitybands do not have predictive validity; see e.g. Bollinger (2001). Similarly,no claim can be made that a desired percentage of points fall within theBollinger bands.Nevertheless, it is not difficult to construct volatility bands that dohave predictive validity, i.e., prescribe a range of values that—with highprobability—will ‘cover’ the future price value Pn+1. We construct suchpredictive bands in the paper at hand; see eq. (22) in what follows. Todo this, the notion of a geometric mean moving average will turn out veryuseful.Before proceeding, however, let us briefly discuss the notion of predictiveintervals. The issue is to predict the future price Pn+1on the basis of theobserved data P1,... ,Pn.DenotebyˆPn+1our predictor; this is a ‘point’predictor in the sense that it is a point on the real line. Nevertheless, withdata on a continuous scale, it is a mathematical certainty that this pointpredictor—however constructed—will result in some error.So we may define the prediction error wn+1= Pn+1−ˆPn+1and studyits statistical properties. For example, a good predictor will result intoEwn+1=0andVar(wn+1) that is small. The statistical quantification ofthe prediction error may allow the practitioner to put a ‘margin-of-error’around the point predictor, i.e., to construct a ‘predictive interval’ with adesired coverage level.The notion of ‘predictive interval’ is analogous to the notion of ‘confi-dence interval’ in parameter estimation. The definition goes as follows: a4predictive interval for Pn+1is an interval of the type [A, B]whereA, B arefunctions of the data P1,... ,Pn. The probability that the future price Pn+1is actually found to lie in the predictive interval [A, B] is called the interval’scoverage level.The coverage level is usually denoted by (1 − α)100% where α is chosenby the practitioner; choosing α =0.05 results into the popular 95% cover-age level. The limits A, B must be carefully selected so that a prescribedcoverage level, e.g. 95%, is indeed achieved (at least approximately) in prac-tice; see e.g. Geisser (1993) for more details on predictive distributions andintervals.I. Smoothing and predictionUnder eq. (2)–(4) and given our data L1,... ,Ln, a simple nonparametricestimator of µn+1is given by a general Moving Average2in the log-pricedomain, i.e., byMALn,θ,q=q−1k=0θkLn−k, (5)where q is the length of the Moving Average window, and θksome weightsthat sum to one, i.e.,q−1k=0θk=1. The simplest choice is letting θk=1/q forall