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BENOIT MANDELBROT

ers of as it generated man who went deeply into the sub- Mandelbrot Makes Sense: pictures of ever increasing complexi- ject), it was said that “the French did A Book Review Essay ty using a deceptively minuscule quite useful before recursive rule, one that can be reap- Bourbaki” – as the secretive guild- A discussion of Benoit Mandelbrot’s The plied to itself repeatedly. You can like organization installed a truly (Mis)Behavior of Markets by Nassim look at the set at smaller and smaller top-down view of the subject matter, Nicholas Taleb resolutions without “ever” reaching insuring no corruption by earthly the limit; you will continue to see material. Indeed many physicists the recognizable shapes. have been horrified at the extent I closed this book feeling that it was classroom, may be beautiful and The introduction of was and side effects of such purism, with the first book in that pure notions; but they seem more not initially welcomed by the mathe- Murray Gell-Mann calling it the spoke directly to me. Not only that, present in the mind of mathemati- matical establishment. This method “Bourbaki Plague”, and attributing but the astonishing simplicity, real- cians and schoolteachers than in of pictorial presentation did not the divergence between pure mathe- ism, and relevance of the subject nature itself. Mountains are not tri- seem to correspond to what seemed matics and science to the obscure makes it the only general work in angles or pyramids; trees are not cir- “to be mathematics” in the self- language of the Bourbakists5. finance I’ve ever read that seemed to cles; straight lines are almost never defining discipline. It is thanks to its In a way, the separation between make sense. seen anywhere. To figure out how popularity with physicists and other and algebra can be seen as Benoit Mandelbrot makes sense. the world operates, we need a differ- applied scientists, themselves fol- the separation of images and words Just as he used us common readers ent geometry than the classical one lowing the lead of the general public in human expression and thought – outside the ivory tower to force his developed by Euclid of Alexandria (mostly computer “geeks”), that frac- just imagine a world in which ideas into science (where some 2400 years ago. Drawing on a tal geometry vindicated its way into images were barred. The Bourbaki- they became “part of the scientific list of then obscure (but subsequent- the now-broadened field of mathe- inspired purblindness does not just consciousness”1); he may just be the ly made famous) , matics. For The Fractal Geometryof limit the tools of analysis. Just like one to help turn economics into BM coined the word fractal geometry Nature made a splash when it came blindness, one of its effects is to something real. to describe these objects that are out a quarter century ago. It spread reduce contact with reality. Platonic This first essay is non-technical jagged yet self-similar in the sense across the artistic circles and led to top-down approaches are interesting and general2 (i.e. can be read by that small parts resemble, to some studies in aesthetics, architectural but they tend to choke under the someone without a mathematical degree, the whole (a more mathe- designs, even large industrial appli- occasional irrelevance of their pur- background) and focuses around matically appropriate designation cations. BM was even offered a posi- suits. It is telling that BM’s hero is the topics covered in this book. The would be the broader “self-affine” tion at a medical school! His talks Antaeus, son of Gaia the mother second one is more technical and it but, somehow, designations are were invaded by all manner of Earth, who needed periodic contact goes deeper into the epistemologi- sticky and, in this discussion, self- artists4, earning him the nickname with earth to replenish his strength. cal problems of “fat tails”, concen- similarity should be held to be “self- “the rock of mathematics”. The Owing to the vicissitudes of a tration, and extreme events. affine”). Leaves look like branches; computer age thus helped him clandestine life during the Nazi occu- What do leaves, commodity branches look like trees; rocks look become one of the most influential pation of France, the young Benoit prices, computer book sales, income like small mountains. If you look at mathematicians in history, in terms was spared some of the conventional distribution, the of Britain, the coast of Britain from an air- of the applications of his work, way Gallic education with the uninspir- cauliflowers, and the intricacies of plane, it resembles what you get before his acceptance by the ivory ing algebraic drills, becoming largely the vascular system have to do with using a magnifying glass. This char- tower. We will see that, in addition self-taught with some assistance one another? Mandelbrot’s work acter of self-affinity implies that one to its universality, his work possesses from his uncle Szolem, a prominent revolves around the simple practical deceivingly short and simple rule of an unusual attribute: it is remark- member of the French mathematical application of a concept called “frac- iteration can be used, either by a ably easy to understand. hierarchy and professor at the tal” in replacement for more com- computer, or more randomly, by A Polish-Lithuanian Jew who College de France. Instead, he devel- plicated mathematical tools that Mother Nature, to build shapes of found refuge in France as a child, BM oped an encyclopedic knowledge of are universally used without empiri- seemingly large . He is also a refugee from the French the history of mathematical cal justification. designed, or rather, according to Sir mathematical establishment protec- thought. He also gave free course to Triangles, squares, circles, and Roger Penrose3, discovered an object, tive of the “purity” of mathematics. his geometric bent. Untrained in the other geometric concepts that known as the “”, To borrow from the late probabilist usual equation solving techniques, caused many of us to yawn in the which became popular with follow- and probability thinker E. T. Jaynes (a he passed the entrance exam to the

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elite École Normale using purely geo- thinker who had the luxury to take than the theories of financial econo- strategy of going straight to practi- metric intuitions (this should be a his time to grow his ideas. mists, and, which is worrisome, tioners and the general public and hint for educators: consider how (Charmingly, BM, in his scientific more understandable by the common bypassing the academic establish- much more intuition you can devel- writings, when discussing a contri- man more than by the classically ment, a task that might appear easy op with images instead of words). But bution made by a mature mathe- trained economist – just as the com- with economics given that the he left after two days. Already stub- matician, mentions his age, such as puter graphic designer or a comput- public and professional standing of born, unruly and unmanageable, he “Cauchy, at the age of 64...”). It is erized teenager could get the point economists in general and finance moved to the more -ori- thanks to such maturation that he far more easily than a classically academics in particular is one of ented École Polytechnique. He then joins that category of the classical, trained . the lowest of any specialty. So the settled in the United States, working pre-academic specialization of It is not a well-known fact that mission of toppling these fake most of his life as an industrial scien- the wisdom-generating natural before his involvement with the and empirically invalid beliefs tist for IBM, with a few transitory and philosophers. roughness in the geometry in seems trivial. varied academic appointments. What does it all have to do with nature, BM started his career focus- Or is it? Finance academia, Indeed, thanks to the computer, he finance? Can we extend the concept ing on problems in unlike the physics establishment, could let the potent machine express of fractals and self-similarity to sta- and finance; it is certainly there that seems to work more like a religion his geometric hunches and lead tistical frequencies? It would make most of his ideas were refined. He than a science, with beliefs that have through the subject matter’s natural the concept of astonishing universal- initially wrote papers in the 1960s so far resisted any amount of empiri- course. Indeed, the computer played ity. This would make BM the true presenting his ideas on “infinite cal evidence (actually this statement two roles in the new science he Kepler of the social sciences. The helped conceive. First, these fractal analogy to Kepler is at two levels, objects, as we will see, could be gen- first in the building of insights This would make BM the true erated with a simple rule applied to rather than mere circuitry, second itself which is ideal for the automa- because you can step on his shoul- Kepler of the social sciences ... tion of a computer (or mother ders – the title of Kepler or “Newton nature). Second, in the generation of of the social sciences” is one so many visual intuitions lies a dialectic thinkers with grand ideas have tried first in the building of insights between the mathematician and the to grab (Marx for one aimed at being objects generated. A mathematical the Newton of the sciences of man). I rather than mere circuitry, scientist par excellence, in a subject am not in the business of defining matter that did not (then) exist insti- genius, but it seems to me that the second because you can step tutionally, he was held to be a mathe- mark of a genius is the ability to pick matician for scientists and a scientist up pieces that are fragmented in on his shoulders (particularly a physicist) by the math- people’s mind and binding them ematical establishment. And while together in one, a meta-connection variance”, getting some early accept- is quite mild; it works just like a reli- mathematicians burn out in their of the dots. ance, but rapidly causing anxiety in gion totally impervious to news twenties, he received his first aca- Do probabilities (more exactly, circles. He then from reality). The closest field to demic tenure at Yale when he was 75 cumulative frequencies) scale like moved to the less harmful fields of finance in the history of science years old. Indeed, after a stint at cauliflowers? If so, the implication is geometry and physics, returning to would be pre-Baconian as Harvard where computer and mathe- not trivial as we may be on to some- finance in 1995 when he started a practiced in the Middle Ages, either matics are subjected to a conceptual thing general, working across sci- very active production of scientific disdainful of observations or spin- separation, it is at Yale that BM6 got ences and fields. And if so, then the papers on financial . At eighty, ning them with theological argu- his dream job as a Professor of statistical attributes of financial he shows no sign of relenting, pro- ments. financial theory being a fad, Mathematical Sciences. And it took him markets can be made far more ducing, as I said, the deepest and not a science, it will take a fad, and half a century to fully realize what understandable than by the compli- most realistic finance book ever not necessarily a science, to unseat his work was united by an attribute: cated and middlebrow so-called printed. By writing The (Mis)Behavior its current set of beliefs. roughness, not just as a quality of “Gaussian” framework. Indeed there of Markets in collaboration with BM wrote his doctoral thesis on objects, but as a standalone field of is something about BM’s work that Richard Hudson, a long time jour- what seemed to be two subjects at study. It is impressive to see him as makes him and his ideas far more nalist at the Wall Street Journal, he once: mathematical linguistics and the embodiment of a scientific understandable to the common man seems to be employing the same statistical thermodynamics (de

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Broglie was the head of the thesis would the arrival of Bill committee). Before the advent of Gates to a town do to the average as a discipline, wealth there. such seemed quite strange. A It is worrisome because every quip goes to the effect that, of his student of statistics learns about two topics, the first did not exist yet mean and variance as the founda- and the second no longer existed. tions of their methods. But the unity between the two was The Gaussian, in contrast, is not the so-called “fat tails” and “power scalable. Most observations hover laws” that are now becoming around the mediocre, and devia- increasingly popular in physics and tions either way become increasing- social science, though not in eco- ly rare, to the point of there being nomics. The spark came from the so- events of an impossible occurrence. called Zipf’s “law” in linguistics, Take the number of adults heavier after the works of one George Zipf than 300lbs and those heavier than on the relative ranking in the fre- Figure 1 The Cauliflower theory of frequencies. This is the result of the application of 150lbs. The relation between the power laws dynamics to a wealth process. If you divide the area in smaller ordered sub- quency of words in a vocabulary. BM samples, you will see the same inequality prevailing. two numbers is not the same as the debunked Zipf’s belief in the separa- one prevailing between 600 and tion, thanks to these laws, between with wealth in excess of 20 million wars, and, of course, market move- 300lbs. The latter will be consider- social and the natural sciences: will be approximately the same in ments. The implication of these able smaller. It gets smaller as the these “fat tailed” phenomena also relation to those with more than 10 power laws is that, for most, there is number get larger – meaning that existed in physics. We are just blind million: about a quarter. This rela- generally no “standard” deviation there is no self-affinity. Deviations to them. tion (here the square of the ratio) is from the norm. In the previous from the norm decrease very rapid- BM later built on the works of called a scaling law, as it is retained example of wealth, if there are more ly, at an increasing rate, to the point the (then) unknown mathematician at all levels, no matter how large the than 1/4 the number of people with where some high number becomes Paul Lévy and, to a lesser degree, the number becomes (say two billion in a 2 times a given level of wealth than literally impossible. The increase of trader-economist Wilfredo Pareto to relation to one billion). What is criti- with a given level (more technically, the rate of the decrease is what pre- whom the original power law is cal here is that it does not vanish – when the tail exponent is higher vents scalability. BM calls this type attributed. The designation “L- frequencies get lower for higher than 2 since doubling the wealth of “mild”, as compared Stable” distributions, (for “Lévy- wealth levels, but the ratios between threshold here leads to an incidence to the “wild” one generated by Stable”), a.k.a. Pareto-Lévy distribu- two arbitrarily high numbers do of more than the square of the ratio), power laws. There is a beautiful sen- tions comes from Mandelbrot. I pre- not decrease! then we are dealing with undefined tence in the book differentiating fer to use the designation “PLM” Cauliflower? If you separate the variance. Now, worse, when the fre- between the two: “Markets often (Pareto-Lévy-Mandelbrot) for the frequencies you will find that the quency in the previous example leap, don’t glide”. more general case of a random series sub-samples resemble each other in drops by less than half, then we are To further see the link between with both independent and non- the degree of inequality in the differ- in a situation of extreme fat tails: finance and fractal geometry, pick a independent increments. ent ordered sub-sections, as can be there is no known average. Any arbi- financial chart. Just like the coast of Let us see how power laws, with shown in Figure 1. trary large number can take place Britain, self-similar at all resolu- their scalability, i.e., the asymptotic Note that the “tail” is the point that can disrupt the mean. The con- tions, monthly prices look “like” (i.e. settling of a series to a constant limit where the outcomes become scala- cept of average is meaningless, total- present an affinity with) hourly in the relationship between likeli- ble in cumulative probability; it does ly meaningless as a characterization charts. One has to shrink the hood of events, can be seen as an not have to be a transition point (it of the attributes of a very fat-tailed timescale more than the price scale application of fractal geometry. can be an asymptotic property as we process, such as computer firms. The in order to get the same effect. Consider wealth in America. tend towards it). This scalability notion of a “typical” computer com- Furthermore, if the stretching is Assuming we reached the “tail” , the seems to apply to a variety of phe- pany has nothing to do with any- done in a random manner, itself number of people with more than nomena like book sales, nodes on thing. Likewise characterizing a fractal, one ends up with what two million will be around a quarter , the relative size of cities, the “typical” writer provides no infor- Mandelbrot calls multifractal. of those with more than one mil- number of times an academic paper mation. Just consider how unstable In 1963 BM wrote a paper on the lion. Likewise the number of persons is cited, the number of casualties in these variables can be: imagine what properties of financial prices and

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found them to be scaling power laws ment, you may not be diversified as very sad history of modern finance. memory – in other words we are no of the anxiety-causing types – the much as conventional theory indi- It ends with the presentation of the longer dealing with serially inde- “infinite variance” variety. The paper cates. And conventional statistical evidence against these models. It pendent draws. The mathematics is was initially endorsed by the ortho- theory might make you jump to con- does not take a lot of empiricism to more intuitive and more realistic dox finance establishment, accept- sequences too quickly: your sample figure out that such risk measure is than what we are used to; indeed ing the implications that there is not size is smaller than you think. useless: the of there is no mathematics but graphs “standard” risk, no known risk. But I was trying to explain the differ- 1987 had, according to their models, and geometric intuitions. He pres- suddenly, these academics started ence between two modes of thinking, such a low probability, one in several ents the usual attacks on his model looking the other way as “modern the broad and the narrow, to an billion billion billion years , that it that consist in saying, “daily prices portfolio theory”, linking risk and investor. Remarkably, it corresponds should not have happened (probabil- might be nonGaussian but in the return, was born. There had to be a to the difference between power laws ities that low are no longer measura- long term things become Gaussian”. measure of risk, even if it presents the and Gaussians. As a method of risk ble; it is meaningless to argue Long term? After the bankruptcy? fatal contradiction of not working management, he follows the conven- whether to assign a 10-23 or a 10-12 Long Term Capital Management when you need it. The bell curve tional methodology of collecting probability to these). You do not was a “long term” idea as well. describes the equivalent of the odds past returns, building a database, need a lot of empirical work to real- Under leverage there is no such of an uncomfortable airplane ride, and simulating by drawing from the ize that a model is wrong: one single thing as long term. nothing about the risk of crash – but past, thanks to bootstrapping-style instance suffices to invalidate it. The third part wraps up with operators thought thanks to “sci- methods. Using such an approach Another piece of evidence more railing against finance theory ence” they were now in control. would make him select the largest among many is the hedge fund and some suggestions for further If you asked for the bridge possible deviation in the simulation Long Term Capital Management research. It includes a scene with between the arts and science, the as the worst scenario. A method of that went bust in 1998. It employed journalistic overtones of a visit to notion of fractal would come up. If say, fitting an “empirical probability 25 PhDs, and two “Nobel” medal- the laboratory of randomness spe- you ask about what bridges hard and distribution”, would do almost the lists in economics for their work in cialist Richard Olsen in Zurich. social sciences, the same scalable same. This is an interpolative finance. Aside from the fact that *** laws would come up. Doesn’t this method – of course the worst possi- their “Nobel” was mistakenly pre- This book has a crisp message make BM the universal scientist? ble move in the future is going to be sented for inventing a “formula” – about risk. The reviews were quite Most of the effects of similar to the one in the past, though the formula has been there for a favorable, but distressing for us NonGaussianism flow from the these moves did not take place in the while; what they did is make it fit empiricists as few commentators consequence that a small number of past’s past. After the stock market into the prevailing economic argu- got the point. People have difficulty observations might contribute dis- crash of 1987, they simulate using 22 ments. They used complicated dealing with the idea that one can proportionately to the total mean per cent as the worst daily deviation. mathematical models – they should write a general book on a financial and variance. Pending on the gravi- Don’t they realize that before the have had on their staff more street- topic without telling people about a ty, you either need a very large, possi- crash they would have used the pre- smart cabdrivers who do are privi- new foolproof (and secret) tech- bly infinite, sample to track the ceding worst case and missed on leged to not know economics. LTCM nique about how to double their properties. Indeed, if ten days in a such a big event? is a milestone as a catastrophe that money in 21 days. My book Fooled by decade represent 40 per cent of the Both the Gaussian and our con- was caused by the pseudo-science of Randomness generated hundreds of returns, which we tend to see rou- ventional wisdom are interpolative. economics, much like the side letters with the following class of tinely with financial securities, Power laws are extrapolative. You effects of those medieval medical complaints: “you tell us that it is much of conventional sampling the- look at the ratio of millionaires to remedies. mostly luck, which seems reason- ory goes out of the window. Consider bi-millionaires and can translate it The second part discusses the able, but you don’t tell us how to that under a Gaussian regime, since into the ratio of 10-millionaires to fractals theory and its relation to make money out of this luck”. these outliers represent a small 20-millionaires. Likewise the ratio the power laws. Those familiar with People are so conditioned by advice- share of total variations, you should between 5% and 10% moves allows BM’s ideas from ’s offering charlatans in business be able to obtain the properties of , you to infer the incidence of moves Chaos will see the usual themes pre- books that anything remotely away say, the stock market by being in it a in excess of 20%. sented. It ends with the multifractal from it seems, as I was told, quite small sub-segment of the time. I will rapidly go through the model where BM presents a memo- “odd”. BM’s, of course, does not give Diversification, too suffers from the details of the book . ry of prices similar to those of the you a recipe. It was therefore amus- consequences of scalability. Since fat The first part of the Mis-Behavior of floods by the Nile river; what hap- ing to see the book reviews com- tails create a winner-take-all environ- Markets, out of three, presents the pened a decade ago stays lurking in plaining about the “now what?” –

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how can we take these ideas home? fined variance”), implies that when the Swedes for his GARCH process, “mean-divergence” , and the other The answer is clear: get out of the you take a sample from a long series, made the following comment: more gullible “short volatility” who markets as we understand them every sub-sample yields a different whether or not you include the stock believe in models, “mean-reversion”, less, far less than we are led to measure of volatility. market crash of 1987 or not makes a “arbitrage”, the self-canceling activi- believe. That would be a significant Nor does it look like the fudging huge difference to the choice of ty called statistical arbitrage, and first step. What this book is about is of the finance models can produce model and its parameters. GARCH, similar things. In other words, there the variability of markets and their real results. I will omit discussing extremely fragile in its calibration, is the naive and the skeptic. , period. the repackaging of the Capital Asset is very sensitive to the inclusion of Scientists and academics tend to The central idea about risk man- Pricing Model under the newer such large observations from the squarely fall in the second category, agement that preoccupies me cur- “Arbitrage Pricing Theory”, except to deep past. Does it smell like unde- even when they trade, while veteran rently is as follows. If you save peo- bemoan that, seven years after fined variance to you? traders and real practitioners have ple in the process of drowning you LTCM, the most recent issue of the I am dedicating my next book, the first mindset. It was a surprise to are considered a hero. If you prevent Journal of Economic Perspectives7 cele- The Black Swan, to BM for his 80th encounter BM, a scientist of the people from drowning by averting a brates with some pomp the 40th birthday. I can now safely say, in “long-vol” category. flood you are considered to have anniversary of Modern Portfolio spite of my having had discussions It was also refreshing to find done nothing for them. Such asym- Theory. It is saddening to see that so with hundreds of hotshots, that he someone who shared the same aller- metry is apparent: you do not get few realize its epistemological dan- is the first person who ever taught gies. It was not just the notion of bonus points for telling agents to gers. One insightful and honest arti- me anything meaningful about my variance; small details can be reveal- avoid investing. They want “some- cle, by Fama and French, talks about subject matter of uncertainty. More ing. For instance, we both got inde- thing tangible”. the poor “empirical” results, accept- specifically, it was the first time in pendently offended by the same Likewise you do not go very far by telling people “we do not gain anything by talking about the vari- People readily mistake irreverence towards some ance”. They want a risk number, a correlation number and BM takes it class of accepted heroes for arrogance. A fair away from them (notice that unde- fined variance also means unde- fined correlation). approach would be to examine the targets of A simple implication of the con- fusion about risk measurement such irreverence applies to the research-papers-and- tenure-generating equity premium ing the notion that empirical my life that I had a conversation statement that “nature does not puzzle. It seems to have fooled econ- implies in practice, out of sample and with someone who can naturally make jumps”. omists on both sides of the fence realizing that, in the end, its appeal hold that the notion of “variance” is So time lost was made up and it (both neoclassical and behavioral lies far more in teaching MBA stu- meaningless in characterizing was refreshing to discover the per- finance researchers). They wonder dents than anything else. uncertainty – and we could move on sonal charm of the universal philoso- why stocks, adjusted for risks, yield Now there have been fixes to to a more meaningful discussion of pher and be privileged to his conver- so much more than bonds and come these equations to accommodate fat- the subject. I finally found someone sation partner. BM only lives five kilo- up with theories to “explain” such tails, to no avail. Every option trader I could talk to without feeling deep meters away from my house, which anomalies. Yet the risk-adjustment knows that volatility is variable – but strain and tension. means that we spent more time talk- can be faulty: take away the models such as GARCH, with close to There is more. He could commu- ing on the telephone than meeting Gaussian assumption and the puz- 10,000 academic publications, do nicate with the trader in me. I was in person (this is how these things zle disappears. Ironically, this sim- not seem to bring us closer to any- taken aback by how easily his ideas work). Conversations with him are ple idea makes a greater contribu- thing. Making volatility variable is spoke to me, down to the very practi- punctuated by opened-and-closed tion to your financial welfare than more complicated than we think: cal. We traders divide persons into parentheses, with tours of classical volumes of self-canceling trading there is the problem of the specifica- two categories: those with a “long literature, history, science, music, advice. tion of such variability. At the last volatility” frame of thought, who, in back to science, with digressions The possibility of “infinite vari- ICBI Madrid Derivatives Convention, general, never rule out blowups, rarely left hanging. Not surprisingly, ance” (or more appropriately “unde- Robert Engle, freshly medalled by change, trends, conspiracies, and he is an independent thinker in just

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about everything; he is a pack of honor. My reaction was to congratu- intuition; he is encyclopedic and is a late the Nobel committee: finally, Fat Tails, Asymmetric universal conversationalist. If you these Swedes seem to be serious manage to age well, you actually get about their prize. Not only have they Knowledge, and Decision better because you know so many helped to make economics more of a Making more things. And he has an astonish- science, but they also gave it the cre- ing memory (“une memoire dentials to help enrich other disci- ’s Essay in honor of Benoit d’éléphant”). plines. The abundance of data makes Mandelbrot’s 80th birthday Having read descriptions of his the field of economics an ideal labo- personality, I was taken aback by the ratory to develop insights and quanti- difference between the real man tative tools helpful to other sciences – and the reputation of “arrogance” – we can develop insights about which to me (as I am familiar with human nature from economic choic- such accusations) comes merely es (Kahneman and Tversky); we can from his targeted irreverence and also learn new mathematical meth- lack of willingness to put up with ods (Mandelbrot). I hereby ask the established truths and established Swedes to take some perspective and gods. People readily mistake irrever- think of those whom, a century from ence towards some class of accepted now, will be identified as having heroes for arrogance. A fair changed the way we view the world. approach would be to examine the The (Mis)Behavior of Markets: A Fractal targets of such irreverence. View of Risk, Ruin, and Reward by In a way BM is the exact opposite Benoit Mandelbrot & Richard of what I call the academic clerk: Hudson, Basic Books. Figure 1: A dataset of 2,500 prices. Infer the attributes. someone who is there to work on research like an obedient tax Introduction al increase by that one day. The gen- accountant. BM is a maverick, tena- FOOTNOTES Consider the following thought erating process for these draws is a cious, and idiosyncratic in his 1 Kenneth Falconer, Nature, 430/ 1 July experiment. You show an agent a set mere switching process, built approach; he seems to scorn formal- 2004. of data of 2,500 days worth of around a Gaussian, to which was 2 A shorter version of this book review was ities. It is all-natural that he would returns (the resulting asset price added the occasional drawing, once withdrawn from the Los Angeles Times, part- have had to counter resistance from W (t) being represented in Figure 1) in 2,500 days, from an infinite vari- ly because I got too close to Mandelbrot the clerks. I was in for a surprise: I after writing the review and did not want to and ask him to infer the attributes of ance kick. This implies that the total had the feeling of talking to a trader, bear the risk of personal conflict. what he saw. is of infinite variance. Those who capable of revising his views at a 3 , 2005, The Road to Reality, Odds are that he would tell you have not seen any such situation blip. And the man was simple, New York: Knopf. that the log-returns are Gaussian. should take a look at emerging mar- friendly, charming, the reverse of 4 John Brockman, 2005, Discussion with 2,500 days data set represents an ket currencies (those in a managed arrogant – except for his colorful Benoit Mandelbrot, www.edge.org ample sample size by any measure, regime). It can also apply to a hedge irreverence. Consider that one of his 5 See the posthumous Probability Theory: enough for the distribution to reveal fund returns: The properties of the colleagues, Michael Frame8 who was The Logic of Science by E.T. Jaynes, 2003, itself to us. Clearly all the attributes late hedge fund LTCM are not too dif- also told that BM was “arrogant”, Cambridge University Press. of a mild distributions are there: no ferent from what we just saw. The 6 See Mandelbrot's essay on www.edge.com accounts for his surprise upon hav- excess Kurtosis over that of a bigger the divergence between the 7 Journal of Economic Perspectives Vol. 18, ing to contradict BM on a critical Normal, no outliers, no jumps, no two regimes (the “normal” and the No. 3, Summer 2004 point. BM’s reply was “Marvelous. 8 See the personal testimony in Michel L. gaps; a histogram of the returns “unusual”), the worse the epistemo- The problem is more interesting Lapidus (editor): Fractal geometry and would reveal the Platonic Bell Shape. logical picture as more people will than I had expected”. applications: A Jubilee of Benoit Now we continue with the rest of tend to be fooled by what they saw. One final remark about recogni- Mandelbrot, Proceedings of Symposia in the story. We add one day, number tion. When Daniel Kahneman Pure Mathematics, 72, 1, American 2,501; one single day can show a The central problem of uncertainty received the Nobel medal many peo- Mathematical Society. quite different picture. What I call the central epistemologi- ple congratulated him on such an Picture 2 shows the information- cal problem of uncertainty1 is sum-

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marized as follows: we do not observe take into account the probability of the squared Gaussian variate) . Talmudic scholar), explains quite probability distributions, only ran- the candidate distribution being the This is exceedingly circular and eloquently that being an option dom draws from an unspecified gen- wrong one. You can use priors and reflects a severe lack of awareness of trader gives someone a philosophi- erator. So we need data to figure out probabilize with series of meta-prob- such circularity. cal approach along “long gamma” the probability distribution. How do abilities. Neither handy, nor con- lines, or, more formally in the deci- we gauge the sufficiency of the size of vincing, and it implies as Elie An easier solution sion theory literature: along a mind- the sample? Well, from the probabili- Ayache2 put it in this magazine “try- As an operator first and last, I believe set focused on the convexity of pay- ty distribution. If at the same time ing to find a random generator that there are, however, far more ele- offs. One comment I make here one needs data to figure out the prob- behind the random generator”. And mentary (and practical) ways to deal about Tony is that his definition of ability distribution, and the probabili- it does not escape the attacks by clas- with this problem, or at least to pro- philosopher is similar to mine (and ty distribution to figure out if we have sical Pyrrhonian skeptics: we seem to tect ourselves from its ill effects. Mandelbrot’s): a philosopher is enough data, then we have a severe be either 1) justifying belief with ref- How? I propose two approaches. someone who specializes in ideas, circular epistemological problem. erence of other belief, itself justified First, consider Pascal’s wager. We not in other people’s ideas – like Note here that fat tails are conta- by other belief, all the way up until can change our payoff structure to stamp collecting. Professional gious. If you combine two random some unargued dogma, which could accommodate what absence of philosophers can be like parasites. variables each following a power law be fragile (in this case some “known” knowledge we suffer from, and with To Tony, like for me, being long an distribution but with different expo- distribution or generator for the respect to which moments of the dis- option in the tail (or more generally “long nents, the result is a power law dis- time series) 2) justifying belief some- tribution. For instance, if the data convexity”) eliminates the need to try to tribution with, for tail exponent, the where in the loop with another pre- has “infinite” (or undefined) vari- figure out what we don’t know3. Only lower of the two. Here we have two viously derived belief and falling ance, one can avoid exposure to such an option trader could understand processes, one of finite, the other of back into severe circularity; finally 3) infinite tail by clipping the sensitivi- that – that’s what I am trying to gen- infinite variance; accordingly the the regress may never end and we ty to the offending part of the distri- eralize to all decision making infinite variance will prevail. stay at the beginning. bution. Purchasing a simple deriva- under uncertainty and convey to A traditional philosophical way Note that the quantitative-statis- tive(say, an extremely out-of-the- nontraders in my forthcoming The to deal with the regress argument, if tical literature is not thoughtful money call), if it such product is Black Swan. one follows the epistemological tra- enough or self-critical “to be even available, may provide a solution. It is key that we operators and ditions, would be to either 1) put wrong” on the subject. How? Our doubt can be targeted and reme- decision makers are capable of insu- your hands up and bemoan the Conventional tests of normality died by transactions. Tout simplement. lating ourselves from nasty parts of Problem of Induction, and find theo- study the square errors from a Second, what we call the mas- the distribution. It is a fact that a logical arguments to have some Gaussian and use a Gaussian- querade problem. The data cannot portfolio constituted of securities unquestioned belief or 2) proceed to inspired distribution (a special case tell us what is the probability distri- that have infinite variance does not a systematic layering: One can pose a of the Gamma distribution, the Chi- bution generating it; but it can easi- need to have infinite variance. How? meta-distribution, one that would Square, which is the distribution of ly tell us what such probability dis- If you are short a call spread with the tribution is not (or is not likely to be), position strike K, described as short and which moments of the distribu- a call struck at K, long another call tions we may not be able to compute. at K + y, you are “short volatility”, but you are not exposed to infinite Portfolios, infinite variance, and variance. Your payoff is capped. epistemic opacity Furthermore: the properties of your What many academic philosophers strategy are not fragile to parametric do not realize is that the limits of assumptions or choice of model. some knowledge may be of small Note here, in the earlier thought moment. I would rather use my ener- experiment, that the moments of gy in changing my payoff structure the distribution are very precarious; rather than getting into intractable the loss L (taken in Log returns) is so issues and playing philosophaster. large that the moments are insensi- My colleague, another option trader tive to the probability of the big loss and empirical philosopher Rabbi π. Indeed the pair π L (probability Figure 2: A dataset of 2,501 prices. What is the informational increase? Anthony (“Tony”) Glickman (also a times the payoff) is so large that we

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may never care about the size of the some questions put to me in this under aggregation. There have been Point 1: The slowness in the rate probability. It is so obvious that we magazine about skepticism and series of papers6 disagreeing with of convergence makes a cubic α should work to control L – or, if we asymmetric knowledge4, I will use Mandelbrot’s early work and its con- very seriously NonGaussian. can’t, to only enter transactions the argument that it is always easier clusions. Researchers tend to be If we accept that α is approximately where such L can be controlled. to figure out what the distribution is “skeptical” about the Lévy regime 3, “outside the Lévy regime”, we are Now the question: what if we not than what it is. Compare that to hypothesis producing, for more than still in trouble with respect to the can’t insulate ourselves from such the attributes of humans: a criminal a quarter century now, “evidence” to convergence to the Gaussian. Finite distributions? The answer is “do can masquerade as an honest citi- the effect that Mandelbrot’s early second moment implies conver- something else”, all the way to find- zen; an honest citizen cannot as easi- characterization of infinite variance gence under aggregation, but we ing another profession. Risk man- ly fake being a criminal. Many exten- is wrong – people seem to very badly need to remember that with α<4 agers frequently ask me what to do if sions of this point are accepted in need a Gaussian in order for them to have an undefined 4th moment. The the commonly accepted version of many fields: one single event consti- operate with the current academic implication is rather serious. Value-at-Risk does not work. They tutes a catastrophe; one needs many framework. Their methodology is Consider that the 4th moment is the still need to give their boss some days without an event to pronounce based on two arguments, first, the variance, corresponds to the error of number. My answer is: clip the tails an environment as catastrophe-free. “observation” of α>2 and, second, the measurement in the variance if you can; get another job if you This asymmetry is at the core of the examination of the behavior of (what we option traders call the can’t. “Otherwise you are defining skeptical empiricism: our body of the data when they lengthen the “Vvol”). It will be infinite! This yourself as a slave”. If your boss is knowledge is more readily increased time observation period. implies a quite nasty rate of conver- foolish enough to want you to guess by negative observations than by These studies are either inconse- gence. There will always be a a number (patently random), go confirming ones. Remarkably, we quential or wrong in their infer- NonGaussian jump in the extreme work for a shop that eliminates the can do something with this; it leads ences. First, it does not make much tail to make the tail scalable. exposure to its tails and does not get us to a ranking of the robustness of difference whether or not we are in a Another way to view it is that the into portfolios first then look for results. And remarkably, it is because Lévy regime since we don’t really observations that we are adding are measurement after. Indeed if like me I elect to behave operationally as if stay in the Gaussian regime in the likely to be biased towards the mid- you think that Modern Portfolio the market followed a Mandelbrot- parts of the distribution that matter. dle of the distribution, making it Theory is charlatanism (as con- stable process that I can build portfo- Second, we do not “have evidence” converge in the body but much more firmed by my trader’s observations lios that I am comfortable with. A that we are not in a Lévy regime. slowly in the tails. We can examine and empirical research, and Mandelbrot-stable variable is simply Third, we need to go beyond the this quantitatively. Take α = 3. It is Mandelbrot’s work), use portfolios here what is called a Levy stable, but “Lévy regime” and consider the easy to show7 that, in standard devia- that do not depend on their meas- with non-serially independent draws Mandelbrot regime by lifting the tion terms, outside (Log (n), with n urements. It is so easy to avoid traps. (what BM calls multifractal). We will too-restrictive assumption of inde- the number of observations, we stay return to the situation. pendent increments. I will get into in a scalable regime. Even if you add The asymmetric masquerade the details of the arguments next. up 1 million days, the Gaussian problem The α problem A power law (as we saw in the Take X a random variable, we have a ∼ −α thought experiment) can easily mas- power law P [X>x0] O(x0 ). querade as a Gaussian but not the Clearly we are told that if the first reverse (at least not easily). We can and second moments of the distribu- reject the Gaussian more easily than tion are defined, i.e., α>2, then, we can accept it. More generally, a under aggregation the series distribution with fat tails can show becomes Gaussian so we can use the milder tails than its “true” proper- conventional tools of analysis. Note ties, except, of course, when it is too here that this only holds if we have late. It will even tend to do so. The independent increments. small sample properties of these BM came up with papers in the processes are such that we are not 1960s5 showing cotton prices with likely to encounter large moves in tail α<2, in other words implying them. We can call that problem an Levy-stability; the distribution has fat “epistemic headwind”. To answers tails and does not become a Gaussian Figure 3: The regime densities.

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regime stops at 3.7 sigmas! Typical ate series of self-causing liquida- penny-wisdom since the conse- tions. What would that do to the FOOTNOTES REFERENCES quences of outside such moves are scalability? 1 See Taleb and Pilpel, 2004. See also Elie E. Ayache,2004a, The Back of Beyond, disproportionately large. Figure 3 Well, in such mechanism, the Ayache , 2004a Wilmott, 26-29 2 Ayache, 2004b. shows the two-regime densities. aggregation fattens the tails. Such is E. Ayache,2004b, A Beginning, in the end, 3 We can extend this convexity argument to The situation is reminiscent of the observation made by Sornette Wilmott, 6-11 the philosophy of probability in general. the value at risk problem. The tail of concerning the events leading up to M. Blyth, R. Abdelal and Cr. Parsons,2005, Take the subjectivist concept of probability Constructivist Political Economy, Preprint, the distribution is where our errors the crash of 1987,10 prompting him as degree of belief attributed to De Finetti. forthcoming, 2006: Oxford University Press. compound. That is where ironically to analyze the properties of “draw- Probability is held to be the price I am willing J.-P. Boucheaud and M. Potters, 2003, people like the precision. downs” independently. to fix in such a way as I would equivalently Theory of Financial Risks and Derivatives This brings us to the Mandelbrot buy or sell a state of the world, making sure I Pricing, From to Risk 11 Point 2: Absence of Evidence is multifractal generalization that remain consistent and avoid the “Dutch Management, 2 nd Ed., Cambridge Not Evidence of Absence: The shows that the process can have 1< α book” problem. Well, you do not have to put University Press. Small Sample Bias Problem < ∞. Indeed much of the work on sta- yourself in a situation which you have to X. Gabaix, P. Gopikrishnan, V.Plerou & Measuring an α>2 does not imply ble distributions is restrictive – trade. H.E. Stanley, 2003, A theory of power-law 4 see Ayache 2004. with any confidence that the “true” obsessively relying on the assump- distributions in financial market fluctuations, 5 Mandelbrot (1963) See also Mandelbrot α is not <2. I avoid the discrepancies tion of independence. Nature, 423, 267-270. (1997). here in the measurement results B. Mandelbrot, 1963, The variation of cer- 6 See Officer (1972), Stanley et al (2000), tain speculative prices. The Journal of from the various estimators, Final note and consequences for Gabaix et al (2003) Business, 36(4):394–419. whether Hill and Log-Log linear Financial Engineering and 7 See Sornette (2004) for the proof. See B. Mandelbrot, 1997, Fractals and Scaling regression. It just takes time (and Quantitative Finance. also Bouchaud and Potters (2003). in Finance, Springer-Verlag. data) for these distributions to reveal I conclude by saying that to many of 8 Mark Spitznagel brought this to my B. Mandelbrot, 2001a, Quantitative themselves. Simulate a series of sym- us the field of finance seem to be attention. Finance, 1, 113-123 metric random draws with α = 1.9 intricately linked to modern portfo- 9 Weron (2001). B. Mandelbrot, 2001b, Quantitative and you will recover an α close to 3 lio theory. I showed that it does not 10 See Sornette (2004) for the argument. Finance, 1, 124-130 with 106 samples. This is an argu- have to be so. And it does not take 11 Mandelbrot (2000a, 2000b). R. R. Officer 1972 J. Am. Stat. Assoc. 67, ment well known to many traders8 much to fix the problem. 807–12 and discussed in Weron (2003).9 D. Sornette, 2003, Why Stock Markets As we saw with the 2,500 day © Copyright 2005 by N. N. Taleb. Crash : Critical Events in Complex Financial Systems, Princeton University Press properties in the thought experi- D. Sornette ,2004, 2 nd Ed., Critical ment, matters can be even more Phenomena in Natural Sciences, Chaos, complex with a mixed process. In Fractals, Self-organization and Disorder: short, the fat tailed process tend to Concepts and Tools, (Springer Series in show the underestimation of the Synergetics, Heidelberg) observed volatility. H.E. Stanley, L.A.N. Amaral, P. Gopikrishnan, and V. Plerou, 2000, Scale Point 3: Where the Aggregation invariance and universality of economic fluc- Fattens the Tails. Many of these tuations, Physica A, 283,31-41 inferences and indeed much of N N Taleb and A Pilpel, 2004, I problemi the mathematics we are used to epistemologici del risk management in: Daniele Pace (a cura di) Economia del ris- assumes that we have independ- chio. Antologia di scritti su rischio e deci- ent draws sione economica, Giuffrè, Milano. Now consider the following intu- R. Weron, 2001,Levy-stable distributions ition: very bad moves generate very revisited: tail index > 2 does not exclude the large up or down moves. And also Levy-stable regime.International Journal of consider that this may only happen Modern Physics C (2001) 12(2), 209-223. in extreme circumstances, when the moves exceed a given threshold. Intuitively, a large loss might gener- W

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