Black Swans and fractals: a better way to understand financial markets?

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INTRODUCTION 
The development of fractal geometry, with the expression fractal appearing for the first 
time in 1975 in the book entitled Les Objects Fractals: Forme, Hasard et Dimension and 
written by the great mathematician Benoit Mandelbrot, has represented on the one hand the 
seeding of a new and innovative perspective from which the entire world can be observed 
and has contributed on the other hand to the creation of a modern and original system 
which manages to describe complex shapes, natural forms and real objects in terms of a 
few simple rules. 
More specifically, whereas the traditional figures contemplated by Euclidean geometry, 
namely dimensionless points, one-dimensional lines, two-dimensional planes and three-
dimensional solids, despite their being pure, symmetric and smooth, do not succeed in 
precisely describing the universe around people, rough, asymmetric and subject-to-decay 
fractals manage instead to deal effectively with all the irregularities observable in nature: 
therefore, in the same way as Euclidean geometry can be considered as the suitable set of 
rules to struggle with perfect and ordered frameworks, so similarly fractal geometry can be 
imagined as the system capable of handling with imperfect and chaotic situations, thus 
putting itself for being the most appropriate tool in order to successfully cope with the 
apparently flawed and faulty real world. 
As a matter of fact, fractals have been always viewed as a powerful new frontier of 
mathematics which could have turned out to have remarkable implications also in a large 
number of further sectors and therefore fractal geometry has long since succeeded in 
capturing the attentions of numerous researchers, from pure mathematicians to natural 
scientists, with economists and financers included: in particular, these latter ones have tried 
to extend the main properties and the principal insights behind fractal objects to the 
financial universe, moving from the belief that since fractal geometry manages to explain 
and describe some natural, intricate and chaotic phenomena in a much more precise and 
accurate manner than Euclidean geometry, at the same time it may have put itself for being 
a much more suitable system to figure out the hidden mechanisms dominating financial 
assets and the whole world of finance.  
Based on this latter assumption, the goal of my thesis is therefore to highlight the peculiar 
connection between fractals and financial assets and to propose a new way to observe,
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analyse and possibly better understand financial markets – namely, my purpose is to 
introduce the so called hypothesis of fractal markets and its main characteristics.   
Anyway, since such an hypothesis of fractal markets happens to be an alternative to 
classical capital market theory taught in almost every business school around the world, the 
first chapter of my dissertation focuses on what I have defined as the old way and 
highlights in particular some empirical observations and findings which seem to contradict 
the main assumptions on which the entire house of modern finance is built, namely the 
insights of rational investors, efficient markets and random walk originally proposed by 
Louis Bachelier at the beginning of the twentieth century and afterwards largely developed 
and improved by other eminent scientists, such as Eugène Fama, Harry Markowitz, 
William Sharpe, Fischer Black and Myron Scholes.  
However, since pointing out some weaknesses apparently characterising the system of 
classical finance without identifying some possible instruments to be used for the creation 
of an alternative theory would be nothing but a meaningless operation, the second chapter 
of my dissertation concentrates on fractals and fractal geometry, by providing a definition 
of fractal objects, distinguishing between deterministic and random fractals and 
introducing the concept of fractal dimension.  
Eventually, having first highlighted the need of a revision of the entire house of modern 
finance and having afterwards proposed fractals as the new tools to be used for an 
innovative approach, the third and the fourth chapters introduce what I have defined as the 
new way, namely the hypothesis of fractal markets: more precisely, the apparently non-
existent similarities between fractals and financial assets are made evident and some 
remarkable concepts belonging to the universe of fractal geometry are extended to the 
context of financial markets. Moreover, the core elements of the hypothesis of fractal 
markets, such as the insights of stable Paretian distribution and of long-range dependence, 
the technique corresponding to the rescaled-range analysis, the peculiar Noah effect and 
Joseph effect and the complex idea of multifractal trading time, are all deeply examined 
and explained from a theoretical point of view and translated into practical terms: in 
particular, whereas the third chapter consists in an extensive description of such core 
elements in a general and abstract way, the fourth chapter includes instead some practical 
examples showing how the hypothesis of fractals markets and its devices can be helpful to 
understand the financial framework in concrete, not just through the use of some statistical 
relationships and tools developed by famous and successful researchers, but also recurring
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to a relatively simple macro, which automatically executes the rescaled-range analysis of 
financial series, that I have personally developed using the program Visual Basic for 
Application in Excel.  
In conclusion, some further considerations in the light of both the hypothesis of fractal 
markets from a general viewpoint and the results obtained in some practical examples are 
spelled out, particularly focusing on the role that such an hypothesis can play in the 
financial framework and trying to find the best and correct answer to the question that I 
have proposed as title of my thesis, that is Black Swans and fractals: a better way to 
understand financial markets?
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Chapter 1 – THE OLD WAY 
1.1 Louis Bachelier and the ThØorie de la SpØculation 
The foundations of modern financial theory can be traced back to Bachelier’s doctoral 
thesis ThØorie de la SpØculation. Bachelier’s dissertation was defended in front of a jury of 
mathematicians including Henri PoincarØ, one of the most celebrated mathematicians of all 
time: it did not deal with a conventional topic in mathematical vogue, such as differential 
equations, complex numbers or function theory, and the jury observed in a critical way that 
“the subject chosen by M. Bachelier is a bit distant from those usually treated by our 
candidates (Mandelbrot and Hudson 2008:45)”, thus not assigning to the thesis the highest 
grade, which was instead reserved for those memoirs able to solve challenging problems in 
one of the conventional mathematical disciplines. Although Bachelier’s work was not 
properly awarded, his creative and innovative ideas were not completely lost: in fact, the 
dissertation appeared on a major journal and other research workers, mathematicians and 
economists began reading and analysing it; however, the crucial rediscovery of Bachelier 
and the definitive recognition of the worthiness of his thesis started to take place only after 
the crisis of 1929, to such an extent that the economist Paul Cootner eventually admitted 
that Bachelier was “so outstanding in his work that we can say that the study of speculative 
prices has its moment of glory at its moment of conception (Mandelbrot and Hudson 
2008:46)”.  
In fact, Bachelier was the first not only to formulate some questions regarding how asset 
prices move, but also to propose some possible answers; he developed concepts and 
formulae on which economists have afterwards built the entire house of modern finance.  
 
The factors that determine activity on the Exchange are innumerable, 
with events, current or expected, often bearing no apparent relation to 
price variation. Beside somewhat natural causes for variation come 
artificial causes: The Exchange reacts to itself, and the current trading 
is a function, not only of prior trading, but also of its relationship to the 
rest of the market. The determination of this activity depends on an 
infinite number of factors: It is thus impossible to hope for 
mathematical forecasting. Contradictory opinions about these variations
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are so evenly divided that at the same instant buyers expect a rise and 
sellers a fall.  
The calculus of probability can doubtless never be applied to market 
activity, and the dynamics of the Exchange will never be an exact 
science. But it is possible to study mathematically the state of the 
market at a given instant – that is to say, to establish the laws of 
probability for price variation that the market at that instant dictates. If 
the market, in effect, does not predict its fluctuations, it does assess 
them as being more or less likely, and this likelihood can be evaluated 
mathematically.  
Louis Bachelier, opening lines of the ThØorie de la SpØculation 
(Mandelbrot and Hudson 2008:50) 
 
In the first section of his dissertation, Bachelier describes in an accurate and precise 
manner the different products available in the French financial market of the epoch: his 
attention is focused on stocks, bonds, forward contracts and options traded on the Paris 
exchange, which is at the beginning of the twentieth century one of the world capitals of 
financial trading.  
After this preliminary part, in the second section of the thesis he used mathematical 
modelling to explain the movements of stock and bond prices. Bachelier starts his work 
pointing out two possible perspectives from which a particular event may be analysed: on 
the one hand, the ex-post facto perspective; on the other hand, the ex-ante facto 
perspective. The former approach is nothing but the cause-and-effect story: something 
happens and prices react. Although this way turns out to be very easy to follow once the 
event has happened, making forecasts beforehand is usually a challenging and daunting 
task; in fact, in the real world causes are often obscure, information may be unknown, not 
available, misunderstood or misrepresenting reality; rarely the connection between causes 
and effects can be correctly worked out until the event considered actually happens: it is 
already hard to predict new events and it is even harder to forecast how the market will 
react. Moreover, some news may generate ambiguous situations: for example, in the past 
the threat of war has determined both a Dollar rise and a Dollar fall, clearly showing how
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understanding hidden mechanisms of the markets is not that easy. The most innovative 
element in Bachelier’s thesis is however the latter approach, commonly defined as the ex-
ante facto perspective: instead of searching for cause-and-effect relationships, he tries to 
estimate something different, namely the odds that a price will move.  
Following this new and unconventional way, Bachelier notices an unexpected and strange 
connection between how prices move up and down and the process of diffusion of heat 
through a substance: although both phenomena cannot be precisely forecasted, since details 
at the level of individual investors in the market or of single particles in matter are too 
complicated, in both cases it is possible to back away from the details and focus instead on 
the general pattern of probability which characterises the whole system. According to this 
perspective, financial markets can be viewed as fair games similar to the tossing of a coin. 
If a person has a coin and the coin is unweighted, then on each toss it is as likely to come 
up head as tail; if an investor wins ten dollars for every head and loses ten dollars for every 
tail, after a large number of tosses the probability theories predict that he should expect a 
profit of zero; moreover, no matter what happened on the prior tosses, each time a person 
tosses a coin, the odds relative to head and tail do not change but remain the same. 
Assuming that financial markets in every moment take account of all relevant information 
and that prices are in equilibrium, with supply and sellers paired respectively with demand 
and buyers, the coin-reasoning also works for financial markets: unless new information 
outcrops, changes in prices are not even expected and every other eventual move is as 
likely to be up as down, according to the idea that every price change is completely 
unrelated to the others.  
In other words, Bachelier sustains that prices tend to follow a random walk: they are 
examples of independent and identically distributed variables, the best expectation of 
future prices is today’s price and each price variation is not related to the last, meaning that 
the market has no memory. Moreover, Bachelier observes that plotting price changes over 
a long period of time onto a graph, the result is nothing but the famous Gaussian bell curve 
where the many small changes are concentrated in the centre of the picture and the few 
large variations are located at the extreme edges (Courtault et al 2000; Mandelbrot and 
Hudson 2008:43-57). 
These assumptions have to be considered as the milestones of the whole study realised by 
Bachelier and based on them, the entire set of mathematical and statistical tools developed
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relating to the Gaussian distribution may start to be applied and used for the analysis of 
financial markets.  
1.2 Eugène Fama and the Efficient Market Hypothesis 
The innovative insights introduced by Bachelier through his dissertation ThØorie de la 
SpØculation gave a strong impulse to several following studies conducted both by 
economists and by mathematicians. His work was rediscovered in particular during the 
years following Wall Street dramatic crash of 1929 and in this regard a remarkable role 
was played by the American mathematician and statistician Jimmie Savage: in fact, after 
having deeply studied Bachelier’s work, he sent to a dozen of eminent economists a 
postcard, whose message was a simple and direct question, “do any of you guys know 
about a 1914 French book on the theory of speculation by some French professor named 
Bachelier? (Samuelson et al 2006:vii)”. He managed to draw attentions to Bachelier’s 
thesis and accordingly numerous books and writings began following one another; among 
them, of particular interest and value was undoubtedly the article produced by the 
American economist Eugène Fama and entitled Efficient Capital Markets: A Review of 
Theory and Empirical Work: its worthiness is due to the fact that Fama revisited and 
improved Bachelier’s innovative assumptions and ideas, eventually managing to develop 
one of the main and most influential tool of modern finance, namely the Efficient Market 
Hypothesis (Samuelson et al 2006:1-4). 
The Efficient Market Hypothesis developed by Fama describes the market as an ideal 
framework or as a fair game, in which on the one hand prices fully reflect and discount all 
relevant information and on the other hand price variations can be due only to unexpected 
news: in other words, similarly to the insights proposed by Bachelier’s ThØorie de la 
SpØculation, this definition of the market means both that it is not reasonable to assume 
that prices will change unless new information is received and that the price of a specific 
day is not at all related to the previous day’s price, meaning in turn that the market has no 
memory and prices are independent. Moreover, the Efficient Market Hypothesis assumes 
that such a market is characterised by the fact that buyers and sellers continuously balance 
one another: although it is acceptable that they may have different opinions, since one may 
be a bull whereas the other one is a bear or vice-versa, Fama assumes that they always 
agree on prices, which therefore can be considered correct and fair. Another central 
assumption in the context of the Efficient Market Hypothesis regards investors and their 
behaviours: people are in fact supposed to be rational and to act accordingly, thus being
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able to understand and evaluate what information is important and what is not; the main 
implication of this latter assumption, if it is extended to an extreme situation, suggests that 
the market is made up of too many people to be wrong.  
Based on the three principal concepts involved in the Efficient Market Hypothesis, namely 
the market is priced so that information is already discounted, price changes are due only 
to new information and follow a random walk, investors are a large group of rational 
people, one of the most relevant implications linked to the system refined by Fama claims 
that the market cannot be beaten. Nevertheless, unlike what predicted by the theoretical 
framework, in the real world this latter statement does not always hold and to justify the 
fact that some investors have managed and still manage to beat the market, Fame himself 
has introduced a distinction among three forms of the Efficient Market Hypothesis: the 
weak form, the semi-strong form and the strong form (Fama 1970).  
The weak form of the Efficient Market Hypothesis claims that prices on traded assets 
already reflect all relevant information which is available, so that the market at any time 
discounts in one day’s price of an asset all its past prices. Such a definition of the Efficient 
Market Hypothesis implies that technical analysis is nothing but flawed, because there is 
no way to predict future prices of an asset based on the prices it has had during the past. 
Although on the one hand an excess return cannot be achieved in the long run following 
investment strategies based on the tools and techniques of technical analysis, on the other 
hand the weak form of the Efficient Market Hypothesis asserts instead that fundamental 
analysis can be very helpful to identify assets which are undervalued or overvalued and 
therefore can guarantee some excess return (Fama 1970). 
In the context of the weak form of the Efficient Market Hypothesis and in particular 
concerning the uselessness of technical analysis, a nice example is that of a clever chart 
reader, who claims to have spotted a particular trend in the old price records, according to 
which every January stock prices tend to rise rapidly: following a simple investment 
strategy, namely buying in December and selling in January, he is inclined to believe that 
he may get rich on that information. However, the assumption of a big and efficient 
market, on the one hand made up of several investors at least as smart and careful as the 
chart reader and on the other hand priced so that all relevant and available information is 
already discounted, implies that other traders are able to spot the trend or at least the 
trading strategy on it quite soon, thus determining a situation in which more and more 
people try to anticipate the January rally by buying in December. But such a phenomenon
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in turn creates the opportunity of a December rally, therefore leading a large number of 
traders to buy in November and to hope for selling in the following month at a higher 
price: eventually, since people are rational and continue to behave this way, the entire 
process is spread out over so many months that eventually it ceases to be even noticeable 
(Mandelbrot and Hudson 2008:55-56).  
The semi-strong form of the Efficient Market Hypothesis claims both that prices at any 
time reflect all relevant and available information and that prices instantly change to adapt 
and to reflect possible new information. Such a definition of the Efficient Market 
Hypothesis implies on the one side that neither technical analysis nor fundamental analysis 
can be used to achieve some excess return and on the other side that adjustments to new 
information have to be instantaneous and of a reasonable size (Fama 1970).  
In the context of the semi-strong form of the Efficient Market Hypothesis, a proper 
example is that of a financial analyst, who focuses his attention on a particular listed 
society and by chatting with its bankers and competitors concludes that the company’s debt 
is getting too large, thus meaning that it will be soon forced either to cut its dividends, to 
borrow more or to sell an important asset to keep affording it. However, if the market is 
efficient in such a way as predicted by the Efficient Market Hypothesis in its semi-strong 
form, then the financial analyst has no reason to believe that he may get rich on that 
information: in fact, either some other experts can immediately spot the same problem and 
advise their own clients to sell that company’s share short or some bankers, scared by the 
possibility of a loan default, can decide to charge the society extra for its routine credit 
lines. In other words, the market soon becomes aware of the situation of danger and 
according to the assumption that information is quickly priced into the stock, the price of 
the stock immediately drops (Mandelbrot and Hudson 2008:56).  
The strong form of the Efficient Market Hypothesis claims that prices instantly reflect all 
information in a market, even hidden or inside information. Such a definition of the 
Efficient Market Hypothesis implies that no one can have excess return, since even private 
and restricted information cannot be helpful to obtain some advantages (Fama 1970).  
In the context of the strong form of the Efficient Market Hypothesis, a good example is 
that of a CEO of a company, who starts cashing in his stock options because he knows that 
the debt of his company is nothing but a time bomb. Although he should be the unique or 
one of the few owners of such an inside information, considering the environment as
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predicted by the Efficient Market Hypothesis in its strong form, the chances that he may 
get rich on that information are more or less zero: in fact, other traders soon spot the 
unusual and suspicious CEO’s behaviour and notice that the captain is abandoning ship. As 
consequence, they predict something bad is about to happen and start selling the stock, thus 
determining a dramatic fall of the stock price (Mandelbrot and Hudson 2008:56).  
 
In short, the evidence in support of the efficient market model is 
extensive, and (somewhat uniquely in economics) contradictory 
evidence is sparse. Nevertheless, we certainly do not want to leave the 
impression that all issues are closed. The old saw, “much remains to be 
done”, is relevant here as elsewhere. Indeed, as is often the case in 
successful scientific research, now that we know we’ve been in the 
past, we are able to pose and (hopefully) to answer an even more 
interesting set of questions for the future. In this case the most pressing 
field of future endeavor is the development and testing of models of 
market equilibrium under uncertainty. When the process generating 
equilibrium expected returns is better understood (and assuming that 
some expected return model turns out to be relevant), we will have a 
more substantial framework for more sophisticated intersecurity tests of 
market efficiency.    
Eugène Fame, closing lines of the article Efficient Capital Markets: A 
Review of Theory and Empirical Work (Fama 1970) 
 
In general, the entire financial community recognised the worthiness of the Efficient 
Market Hypothesis and the role played in its context by the ThØorie de la SpØculation; 
however, as a matter of fact, it was commonly believed that some more tools were needed 
to eventually succeed in the construction of the entire house of modern finance. And in this 
regard, whereas the ThØorie de la SpØculation and the Efficient Market Hypothesis may be 
imagined as the foundations of such a house of modern finance, several other theories and 
models may be considered to be its improvements and decorations – and among others, a 
relevant role is played in particular by Modern Portfolio Theory, Capital Asset Pricing