Overview of the Fractal Market Hypothesis

& The Q Algorithm

Published by Professor Jonathan Blackledge DIT and Kieran Murphy of TradersNow

Overview of the Fractal Market Hypothesis & The Q Algorithm

Prof. Jonathan Blackledge Kieran MurphySchool of Electrical TradersNow IrelandEngineering Systems, Dublin Docklands Dublin Institute of Innovation Park,Technology, 128-130 East Wall Dublin 2, Ireland. Road, Dublin 3, IrelandEmail: Email:jonathan.blackledge@dit.ie kieran@TradersNow.com http://eleceng.dit.ie/blackledge/ www.tradersnow.com

Why Efficient Market Hypothesis (EMH) is flawed for financial time seriesThere are severe limitations that the standard Efficient Market Hypothesis (EMH) operates under. More specifically, until investors see the rather wide gap between what EMH predicts and what the actual market does, investors have a difficult time accepting complexity as a viable model for why and how markets behave as they do. People have spent lots of time writing about the fact that markets are not efficient, that investors are not entirely rational and, as a result of these two things, that stock prices or currency prices are not some sort of random walk. By comparing how the

market should ‘act’ if it were truly efficient and how it actually does ‘act’, we might be able to become comfortable with the idea that another model for asset price behaviour may be operative. Below is a chart of what a normal frequency distribution of returns would look like for an efficient stock market. While statisticians and mathematicians uniformly use the term “normal distribution”, physicists sometimes call it a Gaussian distribution 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 would look like the below, where the X axis represents the standard deviation of the expected returns and the Y axis represents the probability of any given return’s occurrence:

If we could illustrate that stock prices are not in fact normally distributed like the above theoretical chart, then we could then conclude both that investors are not rationale (something that Daniel Kahneman’s and Amos Tversky’s research has already concluded) and that markets are not ef-ficient. By exhibiting inefficiency, theory would hold that there are, in the least, variables, characteristics, and potential models (based on the Fractal Market Hypothesis) that exist that could help us better understand the market’s dynamics and potentially predict future changes in asset prices more accurately.

So how does the actual distribution of S&P 500 returns look relative to the ‘normal’ distribution of re-turns predicted by the Efficient Market Hypothesis? The chart below overlays a representation of the actual 5 day returns data for the SPX from 1928 to 1989 (dashed line) on the ‘normal’ distribution chart from above.

Note the significant discrepancy between the two plots. The actual SPX return data is (1) skewed to the right, (2) shows a much larger frequency of returns around the mean (where X = 0 in this chart) but a correspondingly smaller frequency of returns be-tween 1 and 2 standard deviations from mean, and (3) more frequent very large positive or negative returns than predicted. The term ‘fat tails’ refers to the higher-than-expected large positive or nega-tive returns while the term leptokurtosis refers to the higher-than-expected peaks around the mean. The theoretical probability of seeing a 3 day return like that witnessed during the 1987 crash, the 1929 bear market and a few times during the 2000-2002 bear market is 1 occurrence in 7000 years. That it has happened more than 4 times speaks directly the discrepancy between the theoretical SPX re-turns and the actual returns. Again, the actual data doesn’t fit the predicted data.

Importantly, the chart above is not novel at all; researchers (Turner & Weigel; “An analysis of stock market volatility”, Russell Research Commentaries, Frank Russell Companies, 1990) have known about this ‘problem’ with EMH for decades. And whole ca-reers have been made in an attempt to explain this phenomenon away. Too, further study of other asset markets – treasury bonds, currencies, other coun-tries’ stock markets, commodities - show the same characteristics of fat tails and leptokurtosis.

The EMH is the basis for the Black-Scholes model developed for the Pricing of Options and Corporate Liabilities for which Scholes won the Nobel Prize for economics in 1997. However, there is a fundamen-tal flaw with this model which is that it is based on a hypothesis (the EMH) that assumes price move-ments, in particular, the log-derivate of a price, is normally distributed and this is simply not the case.

Indeed, all economic time series are character-ized by long tail distributions which do not conform to Gaussian statistics thereby making financial risk management 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 have a 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 is valued according to the investment horizon of the investor. Because the different investment horizons value information differently, the diffusion of infor-mation is uneven.

Unlike most complex physical systems, the agents of an economy, and perhaps to some extent the economy itself, have an extra ingredient, an extra degree of complexity. This ingredient is conscious-ness which is at the heart of all financial risk man-agement strategies and is, indirectly, a governing issue with regard to the fractional dynamic model now being used by TradersNow Limited.

By computing an index called the L’evy index, the directional bias associated with a future trend can be forecast. In principle, this can be achieved for any financial time series, providing the algorithm has been finely tuned with regard to the interpretation of a particular data stream and the parameter settings upon which the algorithm relies.

The L’evy index > 0 is the principal parameter associated with a L’evy distribution whose Characteristic Function is given by (for the symmetric case)

with Probability Density Function (PDF) given by

where P(k) is the Fourier transform of p(x), a is a constant and k is the spatial frequency.

If we compare this PDF with a Gaussian distribution given by (ignoring scaling normalization constants)

which is the case when g = 2 then it is clear that a L´evy distribution has a longer tail.

This is illustrated in Figure 1.

Fig. 1. Comparison between a Guassian distribution (blue) for B = 0.0001 and a L´evy distribution (red) for Y = 0.5 and p(0) = 1

The long tail L´evy distribution represents a stochastic process in which extreme events are more 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´evy distribution is determined by the value of the L´evy index such that the larger the value of the index the shorter the tail becomes. Unlike the Gaussian distribution which has finite statistical moments, the L´evy distribution has infinite moments and ‘long tails’.

The statistics of (conventional) physical systems are usually concerned with stochastic fields that have PDFs where (at least) the first two moments (the mean and variance) are well defined and finite.

L’evy statistics is concerned with statistical systems where all the moments (starting with the mean) are infinite. Thus unlike a stochastic signal that is Gaussian distributed and can be characterised by the mean and variance (first two statistical moments), a L’evy distributed signal cannot be characterised in the same way.

The way to quantify such a stochastic signal is through the L´evy index itself. If this is done on a moving window basis for a given financial time series, a L´evy index function can be generated that, in effect, is a measure of the variations in the length of the tail associated with the times series as a function of time. In turn, this function provides an indication of the likelihood of a trend taking place when decreases and the tail increases.

It has long been known that financial time series do not adhere to Gaussian statistics. This is the most important of the shortcomings relating to the EMH model (i.e. the failure of the independence and Gaussian distribution of increments assumption) and is fundamental to the inability for EMH basedanalysis such as the Black-Scholes equation to explain characteristics of a financial signal such as clustering, flights and failure to explain events such as ‘crashes leading to recession’.

The limitations associated with the EMH are illustrated in Figure 2 which shows a (discrete) financial signal u(t), the first derivative of this signal du(t)=dt (or “d prime” as it is called sometimes) and a synthesised (zeromean) Gaussian distributed random 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 the same time series (centre) and a zero-mean Guassian ditributed random signal (bottom).

Clearly, there is a marked difference in the characteristics of a real financial signal and a random Gaussian signal. This simple comparison indicates a failure of the statistical independence assumption which underpins the EMH and the superior nature of the L´evy based model that underpins the Fractal Market Hypothesis.

The Q Algorithm and its benefits

The L´evy index function (t) is simply related to the ‘Fourier Dimension’ q(t) via the equation

For most financial data q(t) varies between 1 and 2 as does for q in this range. Clearly as the L´evy index decreases and the tail of the data gets longer, the value of q approaches 2. Thus as q(t) increases, so does the likelihood of a trend occurring.

In this sense, q(t) provides a measure on the behaviour of an economic time series in terms of a trend (up or down) or otherwise. By applying a moving average filter to q(t), a signal denoted by

is obtained which provides an indication of whether a trend is occurring in the data over a user defined window (the period).

This observation reflects a fundamental result, namely, that a change in the L´evy index precedes a change in the financial signal from which the index is computed (from past data).

In order to observe this effect more clearly, the gradient

is taken.

This provides the user with a clear indication of a future trend based on the following observation: if Q’ > 0, the trend is positive; if Q’ < 0, the trend is negative; if Q’ passes through zero a change in the trend may occur.

By establishing a tolerance zone associated with a polarity change in Q’, the importance of any indication of a change of trend can be regulated in order to optimise a buy or sell order.

This is the principle basis and rationale for the q-algorithm

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