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Among the vast assemblage of technical analysis tools, the ones based on Fibonacci recurrences in asset prices are relatively more scientific. In this paper, we review some of the popular technical analysis methodologies based on Fibonacci sequences and also advance a theoretical rationale as to why security prices may be seen to follow such sequences. We also analyze market data for an indicative empirical validation of the efficacy or otherwise of such sequences in predicting critical security price retracements that may be useful in constructing automated trading systems. © 2006 Peking University Press
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ePublications@bond
5-1-2006
A computational exploration of the efficacy of
Fibonacci Sequences in Technical analysis and
Sukanto Bhattacharya
Kuldeep Kumar
Bond University, Kuldeep_Kumar@bond.edu.au
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Recommended Citation
Sukanto Bhattacharya and Kuldeep Kumar. (2006) "A computational exploration of the efficacy of
Fibonacci Sequences in Technical analysis and trading" ,, .
ANNALS OF ECONOMICS AND FINANCE 1, 219–230 (2006)
A Computational Exploration of the Eﬃcacy of Fibonacci
Sequences in Technical Analysis and Trading
Sukanto Bhattacharya
and
Kuldeep Kumar
School of Information Technology Bond University, Australia
Among the vast assemblage of technical analysis tools, the ones based on
Fibonacci recurrences in asset prices are relatively more scientiﬁc. In this
paper, we review some of the popular technical analysis methodologies based
on Fibonacci sequences and also advance a theoretical rationale as to why
security prices may be seen to follow such sequences. We also analyse market
data for an indicative empirical validation of the eﬃcacy or otherwise of such
sequences in predicting critical security price retracements that may be useful
in constructing automated trading systems. c2006 Peking University Press
Key Words:Fibonacci geometry; Price patterns; Technical analysis; Trading
systems.
JEL Classiﬁcation Number :G1
1. INTRODUCTION
It is frequently observed in price charts that as signiﬁcant price moves
retrace themselves, support and resistance levels are more likely to occur at
certain speciﬁc retracement levels e.g. at 0.0%, 23.6%, 38.2%, 61.8%, 100%,
161.8%, 261.8% and 423.6%. Each of these numbers starting from 23.6% is
approximately 0.618 times the succeeding number and each number start-
ing from 38.2% is approximately 1.618 times the preceding number.
The number 1.618, sometimes called the golden mean, is of special math-
ematical signiﬁcance as it is the limiting value of the ratio Fn+1/Fnas n
219
1529-7373/2006
2006 by Peking University Press
All rights of reproduction in any form reserved.
220 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
tends to +. Here, the numbers Fnand Fn+1 are two successive numbers
in a Fibonacci series.
The general Fibonacci recurrence relation is given as follows:
Fn=Fn1+Fn2(1)
Dividing both sides of equation (1) by Fn1we obtain the following form:
Fn/Fn1= 1 + Fn2/Fn1(2)
As n , we have Fn/Fn1Fn1/Fn2. Putting Fn/Fn1as α, we
may therefore write as follows:
lim
n→∞
α= 1 + 1 (3)
Solving the above equation for α, we get α1.618, which is the limiting
value of the Fibonacci ratio for inﬁnitely large values of n. This ratio
has great historical signiﬁcance ancient Greek architects believed that
buildings constructed so as to make their perpendicular sides in the ratio
αwould render the most pleasing visual eﬀect. Therefore, many of the
ancient Greek and Egyptian works of architectural marvel are found to
reﬂect this golden mean property (Atanassov et. al. 2002).
In this paper, we seek to investigate whether any statistical evidence can
be found for security prices consistently showing such retracement patterns
in accordance with certain Fibonacci numbers. We must confess that our
work takes us beyond the peripheries of theoretical ﬁnance and into the
realms of pure technical analysis (TA) something which academicians
have always abhorred and have chosen to ignore despite its enormous pop-
ularity amongst the teeming millions practicing traders and investors all
over the world, both big and small. However, over the years a number of
papers have shown up which have dared to venture beyond classical ﬁnance
and take a second look at why security prices behave the way they actually
behave (e.g. Edwards and Magee, 1966; Brown and Jennings, 1989; Lo,
Mamaysky and Wang, 2000). With this paper, we join forces with them in
an attempt to unearth concrete statistical evidence (or lack thereof) that
will prove (or disprove) the eﬃcacy of TA.
2. FIBONACCI VECTOR GEOMETRY IMPLICATIONS
FOR TECHNICAL ANALYSIS
Fibonacci Vector Geometry (FVG) is a relatively modern branch of com-
putational geometry which studies geometric objects that can be sequen-
tially generated using Fibonacci-type recurrences.
A COMPUTATIONAL EXPLORATION OF THE EFFICACY 221
The n-th general Fibonacci vector is deﬁned as Gn= (Gn1, Gn, Gn+1);
{Gn}being a set of generalized Fibonacci vectors with G1=a, G2=band
the terms of the vector sequence satisfying the linear recurrence relation as
follows:
Gn+2 =Gn+1 +Gn(4)
Therefore, {Gn}=. . . , a, b, a+b, a +2b, a+ 3b, . . . , Fn2a+Fn1b, . . . where
aand bmay be interpreted as position vectors in Z3. Since each vector in
the sequence is of the form ma +nb, they individually lie on some plane
π(a, b) deﬁned by the point of origin θ(0,0,0) and two distinct points in
space Aand B, assuming that aand bare not collinear. The coeﬃcients
mand nare of course Fibonacci numbers. The vectors Gntend towards
an equilibrium limit ray originating from θ(0,0,0) as n (John Tee,
1994). In the context of security price movements, we will be concerned
only with positive values of the coordinates of the vectors aand bin case
of a signiﬁcant uptrend. A signiﬁcant downtrend will likewise mean that
we will be concerned only with negative coordinates of the vectors aand b.
2.1. Why would security prices seem to follow Fibonacci se-
quences?
This is a million-dollar question to which, unfortunately there is no sci-
entiﬁcally justiﬁed million-dollar answer as yet. Technical analysts would
often go a long way in putting their faith on what they visually inspect on
the price charts even in the absence of a thoroughly scientiﬁc reasoning.
And traders can and do make money based on the recurrent chart patterns.
In his addendum to the highly innovative and paper by Lo, Mamaysky
and Wang, (Lo, Mamaysky and Wang, 2000), Narasimhan Jagadeesh (2000)
has opined that serious academics and practitioners alike have long-held
reservations against technical analysis because most of the popular chart-
ing techniques are based on theoretically rather weak foundations. While
the chartists believe that some of the observed price patterns keep repeat-
ing over time, there is no plausible, scientiﬁcally justiﬁable explanation as
to why these patterns should indeed be expected to repeat.
While there is usually no debate regarding the “information content”
of price charts, Narasimhan (2000) argues that this information pertains
to past events and as such cannot be considered to have any practical
utility unless and until it helps market analysts to actually predict future
prices signiﬁcantly better than they can predict in the absence of such in-
formation. In our present paper, though we do not attempt to advance a
rigorously mathematical justiﬁcation as to why asset prices tend to some-
times follow predictable patterns like Fibonacci sequences, we do try and
provide a rational pointer to what might be a good enough explanation.
Of course, the topic is open to further exhaustive and incisive research, but
222 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
that we believe is largely what we originally wanted to achieve to make
and give it a fair chance to prove its eﬃcacy or otherwise in the light of rig-
orous theoretical investigation. For our part, in our present paper, we have
performed an indicative empirical investigation of the plausible predictive
usefulness of Fibonacci sequences as ﬁlters in automated trading systems.
2.2. Mathematical Representability of Certain Stochastic Processes
on a Sequence of Binary Trees with Fibonacci Nodes
One of the fundamental premises of many well-known asset-pricing mod-
els in theoretical ﬁnance is that of the temporal evolution of security prices
in accordance with some pre-speciﬁed stochastic process. For example,
the classical Black-Scholes option-pricing model assumes that stock prices
evolve over time in accordance with a speciﬁc stochastic diﬀusion process
known as the geometric Brownian motion.
Furthermore, all classical derivative valuation models assume that stock
prices evolve in a risk-neutral world, which implies that the expected return
from all traded securities is the risk-free rate and that future cash ﬂows can
be valued by discounting their future expected values at this expected rate.
This assumption of risk-neutrality enables a discretization of the continuous
geometric diﬀusion process in terms of a two-state price evolution and
forms the mathematical basis of the common numerical approach to option
pricing using multi-nodal binomial trees.
It has been shown (Turner, 1985) that certain stochastic processes can
be represented on a sequence of binary trees in which tree Tnhad Fn
nodes, Fnbeing the n-th element of a Fibonacci number sequence. It was
subsequently shown that generalized Fibonacci numbers can be used to
construct convolution trees whereby the sum of the weights assigned to the
nodes of Tnis equal to the n-th term of the convolution. That is, with
as the sum of the weights:
Ω(Tn) = XFjCnj+1 (5)
In equation (5), {Cn}is a general sequence used in weighting the nodes
with integers applying a speciﬁc sequential weighting scheme.
The idea we seek to convey here is that if a continuous evolution of asset
prices does follow a speciﬁc, time-dependent stochastic process, then there
could indeed be a discrete equivalent of that process whose convolutions
may be constructed out of generalized Fibonacci sequences.
Moreover, the Fibonacci retracements observed in security prices can
then be directly associated with the change in Gnand their oscillatory
convergence to the equilibrium limit ray in n steps. This is a geometric
analogue of the oscillatory changes in the Fibonacci ratio Fn/Fn1as it
A COMPUTATIONAL EXPLORATION OF THE EFFICACY 223
converges to α. The required conditions for this convergence are n+
and n > N; where Nis some critical value of nafter which the magnitudes
of |Gn|are increasing with n. That is, Nis some integer for which the
magnitude |G|=|Fn2a+Fn1b|is at its minimum. In price trends
of traded securities, quite obviously n denotes time-points e.g. close of
trading days; and hence can be said to satisfy these required convergence
conditions.
3. FIBONACCI SEQUENCES IN TECHNICAL ANALYSIS
A BRIEF REVIEW
Security prices are observed over time to climb up, slide down, pause
to consolidate and sometimes retrace, before continuing onward evolution.
A good number of technical analysts claim that these retracements often
reclaim ﬁxed percentages of the original price move and can be eﬀectively
predicted by the Fibonacci sequence.
We must however hasten to repeat that though there is a relatively strong
belief amongst technical analysts about the eﬃcacy of Fibonacci sequences
in security price prediction, yet to the best of knowledge of the authors no
scientiﬁc research has yet been directed towards establishing if at all there
is any grain of hard, mathematical truth to support this belief or even,
indeed, in the absence of any hard mathematical proof, is there in the very
least, any concrete empirical evidence to suggest likewise.
One of the more popular automated trading schemes based on the notion
of Fibonacci sequences is that of Harmonic Trading. This is a method-
ology that uses the recognition of speciﬁc Harmonic Price Patterns and
Fibonacci numbers to determine highly probable reversal points in stocks.
This methodology assumes that trading patterns or cycles, like many pat-
terns and cycles in life, repeat themselves. The key is to identify these pat-
terns, and to enter or exit a position based on a high degree of probability
that the same historic price action will occur. Essentially, these patterns
are price structures that contain combinations of consecutive Fibonacci re-
tracements and projections. By calculating the various Fibonacci aspects
of a speciﬁc price structure, this scheme attempts to indicate a speciﬁc area
to examine for potential turning points in price action.
J.M. Hurst (1973) outlined one of the most comprehensive references
to Harmonic Trading in his “cycles course”. He coined the well-known
principle of harmonicity that states: “The periods of neighbouring waves
in price action tend to be related by a small whole number.” It is believed
that Fibonacci numbers and price patterns manifest these relationships
and provide a means to determine where the turning points will occur in
signiﬁcant trends.
224 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
The analysis of Harmonic Price Patterns is based on the elements of plane
geometry and is related to the controversial Elliott Wave theory proposed
by R. N. Elliott (1935). However, the Fibonacci-based trading schemes
do try to account for the fact that speciﬁc price structures keep repeating
continually within the chaos of the markets. Hence, although conceptually
similar to the Elliott Wave approach in its examination of price movements,
trading schemes based on Fibonacci sequences require speciﬁc alignment
of the Fibonacci ratios to validate the price structures.
In this context, we should also perhaps point out that the Fibonacci
sequences do play a central role in imparting a semblance of scientiﬁc jus-
tiﬁcation to another common and albeit controversial charting tool that
of the Gann lines proposed by William Gann (1949).
During the early part of the last century, William Gann developed an
elaborate set of geometric rules that he proposed could predict market
price movements. Basically, Gann divided price action into “eighths” and
“thirds”. According to Gann, security prices should move in equal units of
price and time one unit of price increase occurring with the passage of
one unit of time. The division results in numbers such as 0.333, 0.375, 0.5,
0.625 and 0.667, which Gann used as crucial retracement values. Similarity
with the Fibonacci numbers is all too obvious.
proclaim, based on their interpretations of chart patterns that these re-
tracement levels show up repeatedly in the market because the stock mar-
ket is the ultimate mirror of mass psychology. It is a near-perfect recording
of social psychological states of human beings, reﬂecting the dynamic eval-
uation of their own productive enterprise, and thereby manifesting in its
very real patterns of progress and regress. Whether ﬁnancial economists
accept or reject their proposition makes no great diﬀerence to these tech-
nical analysts, as they happily rely on the self-compiled historical evidence
supporting their belief.
Obviously, if there is a large enough body of traders out there in the
market relying on the Fibonacci retracements to make their trading de-
cisions, then, in so doing they could end up self-validating the eﬃcacy of
the methodology in a circular reference! It is therefore a primary goal of
our current research to try and independently investigate the eﬃcacy or
otherwise of Fibonacci sequences in forecasting asset price structures.
3.1. Fibonacci maps
Besides the usual retracement analysis, other forms of Fibonacci studies
are also employed on historical price data to generate special graphs which
are collectively referred to by the chartists as Fibonacci maps. There are
several distinct types of such maps, each with their own interpretation
in terms of trading logic. However the basic idea is the same in all of
A COMPUTATIONAL EXPLORATION OF THE EFFICACY 225
them trying to identify if past price patterns bear some sort of a visual
resemblance to some form of Fibonacci representation and then reading
speciﬁc economic meanings in such resemblances. The two most common
ones are the Fibonacci arcs and the Fibonacci fans, both usually used with
Elliot Waves.
Fibonacci arcs:
Fibonacci arcs are constructed by ﬁrst ﬁtting a linear trend to the histor-
ical price data between two extreme points marking a crest and a trough.
Two arcs are then drawn, centred on the second extreme point, intersect-
ing the trendline at the Fibonacci levels of 38.2% and 61.8%, with a third
arc ﬁtted in between the two at the Gann level of 50%. Supports and
resistances are believed to localize in the proximity of the Fibonacci arcs.
The points where the arcs cross the price data will however depend on the
scaling of the chart but that does not aﬀect the way the trader interprets
them. The following Pound Sterling chart illustrates how the Fibonacci
arcs can indicate levels of support and resistance (points A”, B”, and
C”):
FIG. 1. Fibonacci arcs for Feb-Nov 1993 Pound-Sterling chart2
Fibonacci fans:
Like the arcs, Fibonacci fan lines are also constructed by ﬁtting a linear
trend between two extreme points marking a crest and a trough. Then
an imaginary vertical line is conceived through the second extreme point.
Three trendlines are then drawn from the ﬁrst extreme point so they pass
through this imaginary vertical line, again at the two Fibonacci levels of
38.2% and 61.8%, with the Gann level of 50% thrown in between. The
following chart of Texaco shows how prices found support and resistance
at the fan lines (points A”, B”, C”, D and E”):
226 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
FIG. 2. Fibonacci fans on Dec 1992-Nov 1993 TEXACO stock quotes4
4. COMPUTATIONAL INVESTIGATION OF THE
POTENTIAL UTILITY OF FIBONACCI SEQUENCES AS
Automated trading systems are usually collection of a set of rule-based
logic statements which generate a series of traﬃc lights regulating the trans-
actions. If the signals are favourable, the system gives an overall green sig-
nal implying that the trader should go ahead with the transaction or, if the
signals are not favourable, the system gives an overall red signal implying
that the trader should hold position and not go ahead with the transaction.
To what degree the underlying rule-based logic is transparent or not marks
a distinguishing feature between the so-called black box, grey box and tool
box systems.
The reliability of any rule-based system can be improved through in-
corporation of better ﬁlters for the trades. In this paper, we investigate
the utility of Fibonacci sequence as such a ﬁlter. Our objective is to try
and objectively identify any clear visual patterns resembling the Fibonacci
sequence in a set of historical market data.
What we have used is an input data set comprising of almost twelve-
hundred data points corresponding to the daily closing values of the S&P500
index from April 1, 1998 to March 31, 2003. A time-series plot of the data
is as follows:
We marked out seven critical retracements on the chart on August 31,
1998, December 20, 2000, April 4, 2001, September 21, 2001, August 23,
2002, October 9, 2002 and March 11, 2003. These seven points have been
evaluated based on the following formula which gives the maximum re-
tracement from t= 1 till t=j; that is:
Retracementj= [Indexjmax
t(Indexj)t]/Indexj(6)
A COMPUTATIONAL EXPLORATION OF THE EFFICACY 227
FIG. 3. Time series plot of Apr 1998-Mar 2003 S&P500 closing index values
If a row represents days of trading and a column represents the index val-
ues then the spreadsheet version of the above retracement formula could be
expressed in the form “= (E107 MAX (\$E\$2 : E107))/E107”, which in-
cidentally gives the retracement on the 106th day corresponding to August
31, 1998 of our S&P500 index historical data.
The spreadsheet formula yields the percentage retracements on these
seven days as 23.97%, 20.77%, 24.52%, 35.93%, 46.99%, 23.94% and 17.25%
respectively. Barring the ﬁfth and seventh retracements, the remaining
ones are indeed quite remarkably close to the Fibonacci levels of 23.6%
(the ﬁrst, second, third and sixth retracements) and 38.2% (the fourth
retracement)! These retracements have been marked out as critical simply
because an inspection of our spreadsheet output revealed that these were
the maximum retracements observable in our sample time-period.
A popular method employed by technical analysts to automatically gen-
erate buy-sell signals in computerized trading systems is by using the dif-
ference between a fast and a slow moving average and marking out the
cross-over points (Brock, Lakonishok and LeBaron, 1992). Often an expo-
nential moving average (EMA) is preferred over a simple moving average
because it is argued that an EMA is more sensitive and reﬂects changes
in price direction ahead of its simple counterpart. Fitting the EMA ba-
sically becomes the statistical equivalent of exponentially smoothing the
input data which calls for an optimal smoothing parameter to minimize
the sum-squared errors of the one-period lags. Mathematically, this boils
down to solving the following constrained, non-linear programming prob-
228 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
lem (NLPP):
Minimize SSEn+1 =X
t=1
[Pt+1 {αX
j=2
β(nj)Pj} β(n1)P1]2
Subject to α+β= 1; and 0 (α, β)1,
where Pjis the asset price on the jth day (7)
(Formal proof supplied in Appendix I)
Any approach towards an analytical solution to the above NLPP would
become exceedingly cumbersome, given the rather complex nature of the
polynomial objective function (Wilde, 1964). However, most spreadsheet
softwares do oﬀer useful numerical recipes to satisfactorily solve the prob-
lem and yield fair approximations of αand β.
We have calculated two exponential moving averages, a ﬁve-period one
and a ﬁfteen-period one as the fast and slow EMAs respectively. The
spreadsheet approximations of the optimal parameter values are 0.9916
and 0.9921 respectively.
We have thereafter calculated the diﬀerences between the fast and the
slow EMA and calculated its product-moment correlation with the diﬀer-
ence between the retracement percentages and their closest Fibonacci lev-
els. This yielded an rxy 0.3139. Now we perform the standard one-tailed
test of hypothesis to test the signiﬁcance of this correlation coeﬃcient:
H0:ρxy = 0
H1:ρxy >0
As t11.3559 > t0.05,1180 1.6461; we may reject H0and infer that a
signiﬁcant positive correlation exists between the two variables.
This does indicate that a fairly sophisticated automated trading system
based on a pattern learning algorithm like for example an adaptive neural
network could be trained on input Fibonacci sequences along with addi-
tional input vectors comprising of some common technical indicators like
the EMA cross-overs, to pick up recurring patterns in the historical price
data with plausible predictive utility. However more numerical tests are
obviously required to know the practical value of such predictions.
5. CONCLUSION AND GOAL OF FUTURE RESEARCH
In this paper we have taken a second look at one of the most popular
technical analysis methodology and have attempted to empirically examine
it under the light of statistical theory. We have taken the lead from Lo,
Mamaysky and Wang when they published their bold and innovative paper
on the foundations of technical analysis, for the ﬁrst time bringing what
was hitherto considered taboo within the coverage of traditional ﬁnancial
literature. We are inclined to opine that all of technical analysis is deﬁnitely
A COMPUTATIONAL EXPLORATION OF THE EFFICACY 229
not a voodoo science and there are in fact elements of true science lurking
in some of its apparently wishful formulations, just waiting to be uncovered
by the enterprising investigator. Our present paper is primarily intended to
inspire a few more academicians in Finance and allied ﬁelds, to take another
look at this widely criticized but scarcely researched area of knowledge.
The empirical result we obtained does appear to corroborate the claim
of technical analysts that there is some predictive utility associated with
Fibonacci sequences used as ﬁlters in automated trading systems. However
we must add that this is merely indicative and by no means conclusive em-
pirical evidence and more exhaustive studies are required before any deﬁnite
conclusion can be drawn. Nevertheless, our obtained evidence does warrant
a further incisive and potentially rewarding research into the topological
and statistical interrelationship of Fibonacci sequences with the prices of
securities being actively traded on the ﬂoors of the global ﬁnancial markets.
APPENDIX A
Statement:
SSEn+1 =Pt=1[Pt+1 {αPj=2 β(nj)Pj} β(n1)P1]2
Proof.
EM A1=P1
EM A2=αP2+βEM A1=αP2+βP1
EM A3=αP3+βEM A2=αP2+β(αP2+βEMA1)
=αP2+β(αP2+βP1); and
EM A4=αP4+βEM A3=αP4+β(αP3+αβP2+β2P2)
Generalizing up to kterms we therefore get:
EM At=αPt+αβPt1+αβ2Pt2+αβ3Pt3+· · · +αβt2P2+βt1P1
Therefore,
EM At+1 =αPt+1 +αβPt+αβ2Pt1+αβ3Pt2+· · · +αβt1P2+βtP1
(A.1)
But
EM At+1 =αPt+1 +βEM At
=αPt+1 +β(αPt+αβPt1+αβ2Pt2+· · · +βt1P1) (A.2)
It is easily seen that equation (A.1) is algebraically equivalent to equation
(A.2). Since we have already proved the case for k= 1,2,3 and 4, therefore,
230 SUKANTO BHATTACHARYA, AND KULDEEP KUMAR
by the principle of mathematical induction the general case is proved for
k=n.
Thus summing up and simplifying the expression, the one-period lag er-
ror in n+ 1 is evaluated as Pn+1 {αPj=2 β(nj)Pj} β(n1)P1. Now
squaring and summing over t= 1 to nterms, we have the required expres-
sion.
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