Overview of theFractal Market Hypothesis & The Q Algorithm Published by Professor Jonathan Blackledge DIT and Kieran Murphy of TradersNow
Overview of the Fractal MarketHypothesis & The Q AlgorithmProf. Jonathan Blackledge Kieran MurphySchool of Electrical TradersNow IrelandEngineering Systems, Dublin DocklandsDublin Institute of Innovation Park,Technology, 128-130 East WallDublin 2, Ireland. Road, Dublin 3, IrelandEmail: Email:email@example.com kieran@TradersNow.comhttp://eleceng.dit.ie/blackledge/ www.tradersnow.comWhy Efficient Market Hypothesis(EMH) is flawed for financial timeseriesThere are severe limitations that the standardEfficient Market Hypothesis (EMH) operates under.More specifically, until investors see the rather widegap between what EMH predicts and what theactual market does, investors have a difficult timeaccepting complexity as a viable model for whyand how markets behave as they do.People have spent lots of time writing about thefact that markets are not efficient, that investors arenot entirely rational and, as a result of these twothings, that stock prices or currency prices are notsome sort of random walk. By comparing how the
market should ‘act’ if it were truly efficient and howit actually does ‘act’, we might be able to becomecomfortable with the idea that another model forasset price behaviour may be operative.Below is a chart of what a normal frequencydistribution of returns would look like for an efficientstock market. While statisticians andmathematicians uniformly use the term “normaldistribution”, physicists sometimes call it a Gaussiandistribution and, because of its curved flaring shape,social scientists refer to it as the “bell curve.” Such a‘normal’ distribution of returns for the S&P 500 wouldlook like the below, where the X axis represents thestandard deviation of the expected returns and theY axis represents the probability of any given return’soccurrence:If we could illustrate that stock prices are not in factnormally distributed like the above theoretical chart,then we could then conclude both that investorsare not rationale (something that DanielKahneman’s and Amos Tversky’s research hasalready concluded) and that markets are not ef-ficient. By exhibiting inefficiency, theory would holdthat there are, in the least, variables, characteristics,and potential models (based on the FractalMarket Hypothesis) that exist that could help usbetter understand the market’s dynamics andpotentially predict future changes in asset pricesmore accurately.
So how does the actual distribution of S&P 500returns look relative to the ‘normal’ distribution of re-turns predicted by the Efficient Market Hypothesis?The chart below overlays a representation of theactual 5 day returns data for the SPX from 1928 to1989 (dashed line) on the ‘normal’ distribution chartfrom above.Note the significant discrepancy between the twoplots. The actual SPX return data is (1) skewed to theright, (2) shows a much larger frequency of returnsaround the mean (where X = 0 in this chart) but acorrespondingly smaller frequency of returns be-tween 1 and 2 standard deviations from mean, and(3) more frequent very large positive or negativereturns than predicted. The term ‘fat tails’ refers tothe higher-than-expected large positive or nega-tive returns while the term leptokurtosis refers to thehigher-than-expected peaks around the mean.The theoretical probability of seeing a 3 day returnlike that witnessed during the 1987 crash, the 1929bear market and a few times during the 2000-2002bear market is 1 occurrence in 7000 years. That ithas happened more than 4 times speaks directlythe discrepancy between the theoretical SPX re-turns and the actual returns. Again, the actual datadoesn’t fit the predicted data.
Importantly, the chart above is not novel at all;researchers (Turner & Weigel; “An analysis of stockmarket volatility”, Russell Research Commentaries,Frank Russell Companies, 1990) have known aboutthis ‘problem’ with EMH for decades. And whole ca-reers have been made in an attempt to explain thisphenomenon away. Too, further study of other assetmarkets – treasury bonds, currencies, other coun-tries’ stock markets, commodities - show the samecharacteristics of fat tails and leptokurtosis.The EMH is the basis for the Black-Scholes modeldeveloped for the Pricing of Options and CorporateLiabilities for which Scholes won the Nobel Prize foreconomics in 1997. However, there is a fundamen-tal flaw with this model which is that it is based ona hypothesis (the EMH) that assumes price move-ments, in particular, the log-derivate of a price, isnormally distributed and this is simply not the case.Indeed, all economic time series are character-ized by long tail distributions which do not conformto Gaussian statistics thereby making financial riskmanagement models such as the Black-Scholesequation redundant.What is the Fractal Market Hypothesis?The Fractal Market Hypothesis (FMH) is compound-ed in a fractional dynamic model that is non-sta-tionary and describes diffusive processes that havea directional bias leading to long tail distributions.The economic basis for the FMH is as follows:• The market is stable when it consists of investors covering a large number of investment horizons which ensures that there is ample liquidity for traders;• Information is more related to market sentiment and technical factors in the short term than in the long term - as investment horizons increase and longer term fundamental information dominates;
• If an event occurs that puts the validity of fundamental information in question, long-term investors either withdraw completely or invest on shorter terms (i.e. when the overall investment horizon of the market shrinks to a uniform level, the market becomes unstable);• Prices reflect a combination of short-term technical and long-term fundamental valuation and thus, short-term price movements are likely to be more volatile than long-term trades - they are more likely to be the result of crowd behavior;• If a security has no tie to the economic cycle, then there will be no long-term trend and short-term technical information will dominate.Unlike the EMH, the FMH states that information isvalued according to the investment horizon of theinvestor. Because the different investment horizonsvalue information differently, the diffusion of infor-mation is uneven.Unlike most complex physical systems, the agentsof an economy, and perhaps to some extent theeconomy itself, have an extra ingredient, an extradegree of complexity. This ingredient is conscious-ness which is at the heart of all financial risk man-agement strategies and is, indirectly, a governingissue with regard to the fractional dynamic modelnow being used by TradersNow Limited.By computing an index called the L’evy index, thedirectional bias associated with a future trend canbe forecast. In principle, this can be achieved forany financial time series, providing the algorithm hasbeen finely tuned with regard to theinterpretation of a particular data stream and theparameter settings upon which the algorithm relies.The L’evy index > 0 is the principal parameterassociated with a L’evy distribution whoseCharacteristic Function is given by (for thesymmetric case)with Probability Density Function (PDF) given by
where P(k) is the Fourier transform of p(x), a is aconstant and k is the spatial frequency.If we compare this PDF with a Gaussian distributiongiven by (ignoring scaling normalization constants)which is the case when g = 2 then it is clear that aL´evy distribution has a longer tail.This is illustrated in Figure 1.Fig. 1. Comparison between a Guassian distribution (blue) forB = 0.0001 and a L´evy distribution (red) for Y = 0.5 and p(0) = 1The long tail L´evy distribution represents astochastic process in which extreme events aremore likely when compared to a Gaussian process.This includes fast moving trends that occur in eco-nomic time series analysis. Moreover, the length of the tails of a L´evydistribution is determined by the value of the L´evyindex such that the larger the value of the index theshorter the tail becomes. Unlike the Gaussiandistribution which has finite statistical moments, theL´evy distribution has infinite moments and ‘longtails’.The statistics of (conventional) physical systems areusually concerned with stochastic fields that havePDFs where (at least) the first two moments (themean and variance) are well defined and finite.
L’evy statistics is concerned with statistical systemswhere all the moments (starting with the mean) areinfinite. Thus unlike a stochastic signal that isGaussian distributed and can be characterised bythe mean and variance (first two statisticalmoments), a L’evy distributed signal cannot becharacterised in the same way.The way to quantify such a stochastic signal isthrough the L´evy index itself. If this is done on amoving window basis for a given financial timeseries, a L´evy index function can be generatedthat, in effect, is a measure of the variations in thelength of the tail associated with the times series asa function of time. In turn, this function provides anindication of the likelihood of a trend taking placewhen decreases and the tail increases.It has long been known that financial time seriesdo not adhere to Gaussian statistics. This is the mostimportant of the shortcomings relating to the EMHmodel (i.e. the failure of the independence andGaussian distribution of increments assumption) andis fundamental to the inability for EMH basedanalysis such as the Black-Scholes equation toexplain characteristics of a financial signal such asclustering, flights and failure to explain events suchas ‘crashes leading to recession’.The limitations associated with the EMH areillustrated in Figure 2 which shows a (discrete)financial signal u(t), the first derivative of this signaldu(t)=dt (or “d prime” as it is called sometimes) anda synthesised (zeromean) Gaussian distributedrandom signal.
Fig. 2. Financial time series for the Dow-Jones value (close-of-day) from 02-04-1928 to 12-12-2007 (top), derivative of thesame time series (centre) and a zero-mean Guassian ditributedrandom signal (bottom).Clearly, there is a marked difference in thecharacteristics of a real financial signal and arandom Gaussian signal. This simple comparisonindicates a failure of the statistical independenceassumption which underpins the EMH and thesuperior nature of the L´evy based model thatunderpins the Fractal Market Hypothesis.The Q Algorithm and its benefitsThe L´evy index function (t) is simply related to the‘Fourier Dimension’ q(t) via the equationFor most financial data q(t) varies between 1 and 2as does for q in this range. Clearly as the L´evyindex decreases and the tail of the data getslonger, the value of q approaches 2. Thus as q(t)increases, so does the likelihood of a trendoccurring.
In this sense, q(t) provides a measure on thebehaviour of an economic time series in terms of atrend (up or down) or otherwise. By applying amoving average filter to q(t), a signal denoted byis obtained which provides an indication of whethera trend is occurring in the data over a user definedwindow (the period).This observation reflects a fundamental result,namely, that a change in the L´evy index precedesa change in the financial signal from which theindex is computed (from past data).In order to observe this effect more clearly, thegradientis taken.This provides the user with a clear indication of afuture trend based on the following observation:if Q’ > 0, the trend is positive; if Q’ < 0, the trend isnegative; if Q’ passes through zero a change in thetrend may occur.By establishing a tolerance zone associated with apolarity change in Q’, the importance of anyindication of a change of trend can be regulated inorder to optimise a buy or sell order.This is the principle basis and rationale for theq-algorithm
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