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Business papers School of Business

5-1-2006

A computational exploration of the efficacy of

Fibonacci Sequences in Technical analysis and

trading

Sukanto Bhattacharya

Kuldeep Kumar

Bond University, Kuldeep_Kumar@bond.edu.au

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Fibonacci Sequences in Technical analysis and trading" ,, .

http://epublications.bond.edu.au/business_pubs/32

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

Department of Business Administration Alaska Paciﬁc University, USA

E-mail: sbhattacharya@alaskapaciﬁc.edu

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

Copyright c

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=Fn−1+Fn−2(1)

Dividing both sides of equation (1) by Fn−1we obtain the following form:

Fn/Fn−1= 1 + Fn−2/Fn−1(2)

As n→ ∞, we have Fn/Fn−1≈Fn−1/Fn−2. Putting Fn/Fn−1as α, we

may therefore write as follows:

lim

n→∞

α= 1 + 1/α (3)

Solving the above equation for α, we get α≈1.618, 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= (Gn−1, 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, . . . , Fn−2a+Fn−1b, . . . 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

die-hard academicians shake oﬀ their inhibitions about technical analysis

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 Ω

as the sum of the weights:

Ω(Tn) = XFjCn−j+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/Fn−1as 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|=|Fn−2a+Fn−1b|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.

Practicing chartists and traders who advocate Fibonacci retracements

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

FILTERS IN AUTOMATED TRADING SYSTEMS

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= [Indexj−max

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(n−j)Pj} − β(n−1)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 t≈11.3559 > 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(n−j)Pj} − β(n−1)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+αβPt−1+αβ2Pt−2+αβ3Pt−3+· · · +αβt−2P2+βt−1P1

Therefore,

EM At+1 =αPt+1 +αβPt+αβ2Pt−1+αβ3Pt−2+· · · +αβt−1P2+βtP1

(A.1)

But

EM At+1 =αPt+1 +βEM At

=αPt+1 +β(αPt+αβPt−1+αβ2Pt−2+· · · +βt−1P1) (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(n−j)Pj} − β(n−1)P1. Now

squaring and summing over t= 1 to nterms, we have the required expres-

sion.

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