The document discusses the limitations of the Efficient Market Hypothesis (EMH) in explaining real financial market behavior, arguing for the Fractal Market Hypothesis (FMH) as a more accurate model. It highlights how actual market returns contradict the normal distribution predicted by EMH, exhibiting characteristics such as fat tails and leptokurtosis that suggest deeper underlying dynamics. The Q algorithm developed by TradersNow leverages the Levy index to forecast market trends by analyzing the behavior of financial time series data.

1. Overview of the
Fractal Market Hypothesis
& The Q Algorithm
Published by Professor Jonathan Blackledge DIT
and Kieran Murphy of TradersNow

2. Overview of the Fractal Market
Hypothesis & The Q Algorithm
Prof. Jonathan Blackledge Kieran Murphy
School of Electrical TradersNow Ireland
Engineering Systems, Dublin Docklands
Dublin Institute of Innovation Park,
Technology, 128-130 East Wall
Dublin 2, Ireland. Road, Dublin 3, Ireland
Email: 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
series
There 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

3. 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.

4. 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.

5. 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-Scholes
equation 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;

6. • 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

7. 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.

8. 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 based
analysis 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.

9. 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.

10. 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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