40

Wilmott magazine 41

sense of the financial world-view that Mandelbrot has

always been on the brink of over-turning. A beggar, ragged and

starving was traveling swiftlythrough the forest when he

chanced upon the cottage of a meanold woman. He knocked on her doorand was unfazed when she refuseda request for a little bread, insteadhe asked for a pot, suitable for cook-ing soup – that he might preparehis favorite meal, and then he with-drew a stone from the sack he car-ried on his back. Intrigued thewoman asked what dish might heprepare with that? To which he

ical, is the seedof actual knowl-edge. When dis-cussing the Black-Scholes paradigm whichhas led to far from svelte pro-prietary models, Mandelbrotdraws an analogy to aspirin. Thereare many forms of aspirin availablein the market today, but the processof evolution over a considerableexpanse of time means that each isconstituted of a multitude of ingre-dients, and to identify the activeone is close to impossible.

There is an old fairy tale, whichmight be apt here in capturing a

What describes themarket? What

underpins the struc-ture of financial trad-

ing? Do we acceptwholesale the intu-

itions of Brownianmotion, the normal dis-

tribution, the continuousnature of prices, the Black-

Scholes world, the post-rationalizations of GARCH?

When talking to BenoitMandelbrot we have an immediatesense that what we see in the behav-ior of the markets, with unfetteredvision and a pure trust in the empir-

42 Wilmott magazine

answered, as he drew water from thewell – Stone Soup. Soon enough thewater in the pot was boiling over thefire he made by the well, he deposit-ed the stone in the water and shortlyafter took a sip of the boiling liquid –with relish he smacked his lips andsaid it was good but would be somuch better with a pinch of salt; ofcourse he would share the soup with

the miserly hag but not until it methis exacting standards. Eager to tastethis delicacy the woman soon pro-vided the salt and so the story con-tinues in similar fashion until pep-per, carrots, onions, potatoes, somefowl, and various herbs had beenadded. After the hearty meal thewoman asked if the beggar wouldpart with the stone for some consid-eration. After some wrangling themagical rock found pride of place inthe woman’s kitchen whilst the beg-gar walked away, a little less swiftlywhat with a full stomach and a sackon his back laden with gold.

The misbehavior of marketsOver the last forty years BenoitMandelbrot has revolutionizedthinking and practice in areas asdiverse as statistical physics, meteor-ology, geomorphology, linguistics,computer graphics and naturallymathematics. The fractal view of the

world, of which he is the father, isone of the most important scientificdiscoveries of the last 300 years. Butdespite great triumphs and plauditsacross the world, he is an outlier, ananomaly in whatever field he lightsupon. This is the nature of a geniuswho the usually irreverent NassimNicholas Taleb has dubbed a mod-ern-day Kepler.

Now Mandelbrot has returnedpublicly to the world of finance, notthat his interest was ever reallyturned from it – as we will see.Finance was the perennial challengefor Mandelbrot from the early daysof research into cotton prices whilstserving as a one-man Google at IBMin the 1960s. In his latest book, The(Mis) Behavior of Markets, written withRichard Hudson, Mandelbrot pres-ents a review of all the strands whichhave led to the accepted view of thefinancial world. His multifractalbased approach, has, despite initialenthusiasm from the economicscommunity, been the stalking horseto the hybridized legacy of Bachelierunder which everyone officiallylabors. Now, in this special issue welook at the ideas that Mandelbrotoffers up as the foundation of anentirely new approach to the mar-kets. Nassim Taleb presents his verypersonal review of Mandelbrot’sbook, and an essay discussing some

key ideas in Mandelbrot’s work. Butfirst we begin with an extendedinterview with Benoit Mandelbrot,looking at the formative factors thathave defined his career.

Nassim has called The (Mis)Behavior of

Markets “the deepest and most realis-tic finance book ever printed.” Couldyou outline some of your motivations

behind your approach in the bookwith a view to explaining why it cangarner such high praise?Well let me begin with the secondpart of the question. I think thepraise very much has to be sharedwith Richard. The work itself waslargely done by me quite a while

ago, between 1962 and 1972,it had remained in a statethat it could not be under-stood by most of the peopleto whom it was addressed,but it was very fortunatethat Richard Hudson wasthinking of leaving hisposition at The WallStreet Journal to start his

own periodical and he wanted towrite a book in between. His skills asa journalist are very important tothe fact that the book has been sowell received so widely.

The approach in the book is onethat I would describe as being aboutideas in science. I became very inter-

BENOIT MANDELBROT

The conventional model became accepted without question and the fact that it didn't fitthe data was very widely understood but utterly disregarded, and it developed into asubstantial body of research and development

Mandelbrot inLondon, 2004

Wilmott magazine 43

ested ineconomicsand science asa student in Parisworking with somevery important people,none of them professionaleconomists, but one of them –an engineer – working for one ofthe very large French nationalizedindustries was talking to studentsabout economics. I was not regis-tered as an economics student but Idid attend his lectures and I had avery strong impression that the fieldwas not as advanced as they hadbeen, because this man who was avery high-up official in the Frenchelectricity board was a planningadvisor for Charles de Gaulle, wasinvolved in the problem of optimiza-tion in the industry. There was nosense that economics was a veryaccomplished field and that it need-ed to start from the basics. One ofthe changes that made me enter thefield very strongly in 1961, untilthen I dabbled a bit in things like thedistribution of income which was astandard topic in Paris at the time. Iviewed that as being the mostimportant part of my work. But in1961 things changed when I realizedthat I had developed techniques andways of representing reality thatcould work equally well in describ-ing the variation in financial prices.The change was actually amazing;it’s actually chastening to think howthin a thread everything was hang-ing; because it depended on me see-

ing afigure ona blackboardin the office of acolleague who I hadgone to visit, I was givinga lecture at his school. Thepicture reminded me of some-thing, and I thought I could dosomething about it – so I becamehooked very rapidly.

In the early sixties the avail-ability of computers, very simple

and clumsy ones by modern stan-dards, was becoming more wide-spread. People could begin to workon things that before were simplytalked about, for example the col-league whom I visited that dayHendrik Houthakker commentedthat until then we were talkingabout analyzing data but the datawas unavailable and the computerswere also unavailable. But suddenlythe computers became available andeveryone was able to analyze data.That was the key point for me,because Houthakker was saying thatthe analysis was not as it was sup-posed to be, it was very unsuccessful– the picture was very murky. So Idecided to clarify things. At thesame time many others entered thefield. I didn't know them yet butthey acted very differently. Theytried to apply to prices an approachwhich was basically physics, namelyBrownian motion, and the randomwalk.

So The Random Walk Down WallStreet was a very successful book, theprocess became a very popularapproach and it was also rapidly dis-covered that the whole pricing ques-

tion had been addressed byBachelier in the early 1900s butnobody had been paying attentionuntil 1961 or so. It was a remarkablecase where Bachelier’s discovery wasalmost invisible. He had been knownfor pure mathematics, but even thatwas very limited. So in 1960 the twoapproaches were very much on theminds of people. On the one handthe Bachelier approach and on theother hand mine. Both came fromthe same source, which is the desireamongst quantitatively minded peo-ple to create the simplest possiblepicture of how prices vary, not thatthey could go in to much detail, butat least they could identify the maincharacteristics of price variation.

Then the paths diverge, the con-ventional model became acceptedwithout question and the fact that itdidn’t fit the data was very widelyunderstood but utterly disregarded,and it developed in to a substantialbody of research and development.My work was more critical and tooka very much more demandingstance more demanding stance onthe question of fit to the data. In par-ticular, one aspect of prices whichthe other model denied which wasthe discontinuity of prices. I felt thatby emphasizing discontinuity I was

very much emphasizing a key fact ofprice variation, because many distri-butions one sees are physical or bio-logical in nature; number of peopleworking for a company, how manytons of coal per week are needed topower a country in a given year, theprice of cotton. All of these quanti-ties have a meaning, which is clearlyunderstood in physical terms out-side of economics.

The aspect of economics and thefree market that is unique is the roleof anticipation, that the price of aproduct is not only due to the cost ofmanufacture but also the fact thatone expects the product to becomescarcer. So the equity of the companyselling the product does not onlydepend on the quantity they canmake in a year but also the amountthat the market expects them to sell.Now in the case of coal the prices arediscontinuous, what happens whena new coalmine opens? But of coursewe know how much coal there isbeforehand. But price and anticipa-tion are completely free and theprice can jump from one moment tothe next, this is a fundamentalaspect of economics. Therefore forme I could see that the price changesfor cotton were very much largerthan usual and these price changes

BENOIT MANDELBROT

To go step by step is perfectlylegitimate. But there is a problemwhich is that with step by stepone gets very used to the stepwhich one has taken; one stays atit longer than is justified.

did not happen gradually but withina very short time. Within the daythen because the day was the onlyunit of time at that moment, butnow within the minute or the sec-ond, the price goes from 100 to

70 to 200 within seconds, thisis something which was

quite clear and to me theessential character. I

thought that this wasan insight that was

essential to a properunderstanding ofprice variation.

So, my moti-vation was toexplore that insuccessiveapproxima-tions. That’s one

aspect that dif-ferentiated my

approach to the con-ventional approach because

they started with Brownianmotion, the random walk, and

then assumed that the only differ-ence between one financial instru-ment and another was its volatility.Volatility is just one number.Otherwise the rules by which thedifferent financial instruments varyare the same. I understand that to bean extremely gross simplificationbecause the data we had in 1960 wasvery limited, but still there was sub-stantial market data collected in old

books by the National Bureau ofEconomic Research.

This Bureau began in New Yorkbut was then moved to Cambridge,Massachusetts, linking up in termsof personnel to Harvard and MIT. Sothe Bureau had published in the1930s a number of books withprices, in particular a man namedMacaulay had published them.Some of these books were aboutinterest rates, dollar–sterlingexchange, and it was quite clear thatthe price changes of dollars versussterling were not moving and vary-ing continuously in time, moreoften it was a very brutal change inthe price. It was also quite clear inthe case of interest rates that theydon’t vary like say equities whichcannot become negative – interestrates can go below zero, well maybenot on those data. But it was a differ-ent kind of instrument so wethought how can the same rulesapply to both? That seemed to me tobe quite difficult to perceive, soearly on I felt that we needed a rich-er and more complex basic struc-ture and I developed one in threestages. One in the article in 1963which is now very widely quoted,then a series of articles in the late1960s, and a series of articles whichwere started in 1972 (but were most-ly done in the 1990s) which wereabout the multifractal model.

Why do you think the adoption ofBrownian motion and the randomwalk, this smoothing process (despitethe fact that empirically the evidencedidn't exist that adequately describedthe way the markets work) was takenup in such a huge way?Well it’s an interesting topic which Idid not follow blow by blow but Imust say that part of it is perfectlylegitimate. I mean perfectly legiti-mate in terms of the model pro-posed by physics, because whetherwe like it or not physics is such anamazing proportion of what weknow about the world, at least theinanimate world, that most of theother sciences that lay claim to somekind of quantitative aspect do willy-nilly follow the example of physics.Well, to follow the example ofphysics one good idea is to begin bytrying the simplest theory, and it isquite legitimate that one tries it, onederives it as far as it can go, and thenone sees that it doesn’t work and onesays ‘well this theory has failed, wemust do better.’

In my work in physics there aremany examples of theories proposedby great physicists whom everybodypraises even though the theory isincorrect, but it was the right thingto do in order to get the first under-standing. And the second under-standing that came beyond it raisesthe first theory by one, which ismuch closer to reality and of coursemuch more delicate and more com-plicated. So to go step by step is per-fectly legitimate. But there is a prob-lem which is that with step by step isperfectly legitimate. But there is aproblem which is that with step bystep one gets very used to the stepwhich one has taken, one stays at itlonger than is justified. Perhaps with this situation much ofthe work that was done in applying

It is quite clear that a certain elementof expectation of stability which I

find in very many people that I met in my life was totally

absent in my case

BENOIT MANDELBROT

44 Wilmott magazine

Wilmott magazine 45

Brownian motion and the randomwalk was done by people who weremathematicians, therefore com-pletely unfamiliar with the ways oftheories in the natural sciences, orby physicists with comparatively lit-tle training, or by engineers – thepeople that first came into this fieldin the 1960s were not proficientmodel makers, they were muchyounger people who were not experi-enced in the processes of modelmaking in the delicate parts ofnature. But I have great difficulty injudging this question because Inever asked people why they movedfrom one theory to another.

Then, there was one elementwhich I did follow very closely, whichwas how the attitude towards mechanged and it was quite interestingto follow because early on in 1962when I came up with the theorythere was an enormous amount ofenthusiasm within the economicscommunity towards my work, andmany expressions of the fact that thiswas correct in terms of expressingwhat we knew and also complimentsabout my skill in tackling the prob-lem of difficult data. But then theyobserved that my first theory was notquite accurate, that some odd thingswere coming up. My reaction uponobserving these difficulties was todeepen my theory and to not make it

more complicated but just to make ita bit different to take account of thesephenomena, which is why I alwayssay that my theory went throughthree stages in which I first observedthe long stages of the changes, andthen observed the long stages of thedependence and then the two com-bined in the multifractal model.

But that requires a certainamount of participation and therethey felt that as the effects went thiswas an effective theory but whilechoosing between two effective theo-ries let’s choose the easier one,because there is no comparison interms of mathematical complexitybetween Brownian motion and mywork. The Brownian motion couldbe undertaken right on the spot by alarge number of people who hadlearned it for the purposes of puremathematics or physics and there-fore had what you would call a run-ning start, and my work representeda very big innovation.

So for quite a number of years Ihad to fend off questioning like ‘youcannot write a book about thismodel because no one cares aboutit!’ Now it has been done and therehave been many books written aboutmy innovations of the 1960s but sci-ence does not move unless there isan inheritance and Brownianmotion inherited a great deal from

electrical engineering and physicsand mathematics. But very little hasbeen inherited with my modelsbecause there is a very clear breakwith the past.

Your family were refugees fromPoland. How did this affect yourworldview? Well it is quite clear that a certainelement of expectation of stabilitywhich I find in very many peoplethat I met in my life was totallyabsent in my case. My parents, myfamily, both my mother and father’ssides had lived in Lithuania for verymany centuries but they moved toWarsaw because of economic oppor-tunities. I was born in Warsaw, buttheir life from when I was born hadbeen filled with sudden disruptionwhich were quite unlike my lateracquaintances and friends in the UShad experienced. My parents weremoved several times: first by WorldWar One and then by the Russianrevolution, and so they didn't havethe luxury of basic stability that oth-ers may have had.

Also Poland was not well servedby France in 1919. There was verywidespread opinion that Polanddeserved to be reconstituted afterover 100 years of being divided, butFrance went overboard and treatedGermany and Russia in a way that

nobody around me felt to be stable.Poland, when I was a child, hadalmost more than half of its citizenswho were not ethnic Poles - whichwas a very high proportion – andthose non-ethnic Poles were not ter-ribly happy with the Polish occupa-tion of everything. So it was not astable country and the prospect of astable future was very slim. Also ofcourse, the Soviet Union was tryingto influence Poland and there wasmuch Communist activity. NaziGermany became an increasinglyimportant factor too, because theywanted to separate from the Polishcorridor what once used to beGerman land; and the fact that thethreat of the Germans conqueringPoland was always felt when I was asmall child. I always felt in a threat-ened position.

And then we moved to Franceunder conditions which wererushed but quite rational. My par-ents felt that things were so bad thatmy mother at the age of 50 aban-doned her profession, she was aphysician, and became a housewife,a very lowly housewife in Paris.Shortly afterwards Word War II start-ed. Our life was by no means theworst, but it was by all means unpre-dictable, I can safely say that untilthe liberation of France in 1944 I hadnot experienced any kind of stable

BENOIT MANDELBROT

I found their attitude very oppressive; there was onlyone good way of doing mathematics, there were onlya small number of topics which were worthy, and mostwere either too much physics, too vague, not devel-oped enough or not difficult enough perhaps

46 Wilmott magazine

sense of remaining in one place for the length of my life, andwhether that makes me more con-scious of these possibilities in life is something that many people mayask but for which there is no possible answer.

Another aspect of my life: I foundschool extremely easy. Much of myintellectual development occurredoutside of school. It was the goodside of very widespread unemploy-ment, there were many people in myfamily, uncles and so on who couldnever find a steady job, becausethere were so few steady jobs. Thedepression in Poland was far worsethan in France or in Britain, evenGermany, therefore I have a recollec-tion of endless conversations playingchess with uncles who were very bril-liant people and they told me storieswhich were much more interestingthan those at school. Therefore Icould say that my schooling alto-gether including until I was twentywas extremely disorganized whichI’m sure made a very big contribu-tion to the kind of attitude I took tolife as an adult.

When I turned twenty I had achoice. Maybe I'll go back a step. Atone time during the war I had to stopgoing to school because it was toodangerous and too expensive also,but mainly too dangerous and I hadto use all my energy just to keep bodyand soul together for quite a while.

But after a particularly dangerousand complicated episode some peo-ple who were helping me to getthrough arranged for me to beadmitted to a class in Lyon. The pur-pose was primarily to be in a placethat was sheltered. There were peo-ple there who knew who I was butwould not say anything and I wasjust expected to sit there and recu-perate from some particularly shakyand horrible events.

But things turned out rather dif-ferently because I discovered at thatpoint that I had a rather freakishgift. Freakish amongst mathematicsstudents. The professor there wouldpresent a problem in terms of a for-mula but I did not see the problem, Isaw pictures, what occurred wastotally unconscious, I saw that whatwas being asked was the intersectionof this shape with that shape and ifyou look at them the properties arequite clear. What was special wasthat I was looking at pictures andseeing pictures that others did notsee and secondly I was creating pic-tures out of nowhere to fit the prob-lem that the professor was setting,and these pictures were not in hismind; that’s what he always told me.

That was more of a curiosity, butsuddenly my position changedbecause instead of keeping quiet andout of trouble I was becoming veryconspicuous and my professor wastelling my fellow students that afterthe war when exams could be taken I

would either be number one if I getthe right geometry or number nth if Idon’t! Because I was incapable ofwriting in the four hours those twen-ty pages of algebra which wererequired to solve that same problem.After the liberation those examswere held and I guessed the rightgeometries and I found myself at thefork deciding between the two mostdemanding schools in France. Bothof them were extremely demandingand one had very tiny classes, if youwere to compare it to Britain one waslike a very small Cambridge collegewhile the other was like the largestCambridge college. They were verydifferent schools and I went throughan agonizing choice: on the one handone of the schools presented a nearguarantee of a very simple life, whilethe second did not. But the firstschool was very narrow in its taste,very abstract and would not accom-modate my approach in geometry, soI opted for the second school whichcreated a great deal of surprise.

It is quite clear that I had beentempered by the war and I had gonethrough a number of experiencesand I must have been quite a hardperson to deal with at twenty, I hadlearned to be completely independ-ent and to trust my luck but also tothink quickly on my feet. I didn’t par-ticularly like to be in dangerous situ-ations but I had been in so many thatI thought it was just something thatyou do. So the desire for simplicitywas completely lost and the accept-ance of complexity won, I neverthought then how deeply this wouldchange my life, but it certainlyaffected it all the way.

A younger brother of my father,was a very successful academic, hewas a professor and probably had thebest chair in Paris. By then he hadreturned from the war which he

Therewas a feeling of

fear and revulsionfor what I liked in

my accidentaleducation, whichwas finding order

in this totalmessiness ofnature, and Ihoped that I

would be able tofind it in expressly

mathematicalterms

Wilmott magazine 47

BENOIT MANDELBROT

spent in the US and then the UK, sohad returned around the time that Iwas taking those exams and he wasvery much pushing towards thestraight and simple alternative. Myfather was very much pushing theother way. It was a very strong fightbetween the two of them and everyargument for and against each ofthe options was discussed by veryskilled people in great detail. I didnot act in ignorance at all and it wasvery clear that I was free and respon-sible and I never accused anybody ofhaving pushed me. As a matter offact, in a way, both of them wonbecause the career I had combinedwhat both of them were hoping for.

Can you tell us a little about theBourbaki movement?Well, this is a very interesting phe-nomenon. This was a direct conse-quence of two circumstances. One,the times and secondly, just oneman. The times were just after WorldWar One. Now, France in World WarOne had lost about two million peo-ple, many were killed, but even thosewho survived – if they had lived for ayear in those trenches – were verydeeply wounded for life. The kind ofextraordinary hard work, involve-ment, devotion and full-time com-mitment that you required for exam-ple in science was not somethingthey could envision. Schools werepopulated by older men who had notbeen to war and very young menwho had been too young to be in thewar, and that was it. The young peo-ple felt a very sharp discontinuity inFrench intellectual life.

At the same time the frontiersopened and France was not as isolat-ed as it had been before the war andParis was a very popular place forpeople to go and people started tocollaborate, so the world was no

longer divided. The younger peoplefelt that they did not have much sup-port amongst their teachers becausethey were not the normal number ofyears older, but much older. One ofthose younger people was namedAndré Weil. He was extremely bril-liant, very precocious, and verymuch a leader so he put together hisfriends who were in the first approx-imation all the best students ofmathematics in the 1920s, a dozenof them plus my uncle SzolemMandelbrojt, who had come fromPoland and who had joined themearly on and then left them becausehe was of a different attitude. Butearly on it was just a group of youngpeople who were friends of AndréWeil who was motivating them toget together, to work together toestablish a new school. But the main

thing was there was a very sharp dis-continuity within French mathe-matical tradition.

I think traditions must change,but there had been before a verygreat deal of continuity. France's tra-dition was substantially differentfrom the British and the German,and the Swedes managed to havetheir own tradition which was sepa-rate again from everybody else. TheRussians had their tradition, whichwas a pre-revolutionary traditionthat was modified, but there wascontinuity. The Russian civil war andrevolution were not as destructive ofan entire generation as the wartime

trenches of 1914, so in France it wasjust a new departure that dependedvery much on the personal tastes ofthis man André Weil. Now, there aremany sorts of superficial features;they never published a list of mem-bers, everybody knew it but it wasnever published. They never pub-lished any kind of proclamation; itwas perfectly well known what theywere about to do. Early on they werejust a group of young people whowere very good, the very best, veryambitious and very much in opposi-tion to the older men who were inthe mold of Poincaré and that tradi-tion. The older French tradition ofmathematics was very broad, goingfrom the very abstract to theextremely concrete.

Well, by the time I was twentythose people who were about twenty

years older were about to takepower. I found their attitude veryoppressive; there was only one goodway of doing mathematics, therewere only a small number of topicswhich were worthy, and most topicswere either too much physics, or toovague or not developed enough, ornot difficult enough perhaps.Whatever it is there was a list ofdesirable topics and a list of undesir-able topics and the mood again wasa mood of unity, the sense of peoplewho had found the comfort of a posi-tion of dominance was very strong.In particular they despised geome-try, you could not find any refer-

ences to the real world – mathe-matics was entirely separatefrom the real world.

I was faced with this verybrutally because while I wasstudying for those exams myuncle explained it to me andeven before the war I spentpart of the summer in myuncle’s house in centralFrance so I heard aboutthem when I was fifteenor sixteen. It didn’t mat-ter much but I knewabout them and myuncle was very pleasedto be part of this elitegroup who were goingto change mathemat-ics and were going tomake it different andmuch better.

Those people were unquestionablyextremely brilliant, they were thecream of French mathematics byevery standard and several of themmade very great contributionsthrough their activities, but in termsof giving the tone to quantitative sci-ences they were a complete, unmiti-gated disaster.

For example, the physicists of mygeneration, those who claimed to bephysicists, ended up being muchmore proud proving some mathe-matical result than having discov-ered something in physics. A man-whom I very much respected, a geolo-gist of all people, was very proud of

The first striking aspect of mytime at IBM was that 35 yearslater those 50 oddballs had allbecome well-known scientists

48 Wilmott magazine

the fact that his PhD dissertation wasin the style of Bourbaki. Anotherman, an economist who I also verymuch respected, also wrote his dis-sertation in the style of Bourbaki.There was this feeling of fear andrevulsion for what I liked in my acci-dental education, which was findingorder in this total messiness ofnature, and I hoped that I would beable to find it in expressly mathe-matical terms. I hoped to be a cre-ative mathematician, but the mainthing I liked was that miraculousmoment when a mess becomesorganized by a few simple formulaeand that thing was universallydespised as being a very strange atti-tude. There was also this element ofbeing together in a group, and per-haps I had not developed the abilityto be part of a group. The secondgeneration who took over were not assharp in their organization, and thatfirst generation were really domi-nant in the 1950s and 1960s, but itwas evident to me in the 1940s that Iwould live under their shadow.

So this was a motivation in leavingFrance?Well, this was the motivating factorin two large decisions in my life; thefirst was in not joining ÉcoleNormale. I in fact did join ÉcoleNormale briefly which was thesmaller of the two schools, it had fif-teen students in both mathematicsand science it was obvious that I wasanomalous within this communitywhich either followed Bourbaki orSchopenhauer (?) and anything I didwould push me even further fromthem. When the opportunity aroseagain by chance to take a differentpath, I did leave France.You moved to America.I didn’t move to America, I moved toIBM. It sounds like a silly distinction

because IBM was in America butwhen I decided to move to America Iwould never have thought of IBM asa natural place to have moved to. Anatural place would have been aleading American university but atthat time the leading American uni-versities were very much in moodthe way the French had been underBourbaki, I can add that in Englandthere was a separate developmentunder the leadership of Hardy atCambridge and he was differentfrom Bourbaki in a way because hewas more individual as opposed tobeing in an organized group, buthis ideas were very similar and I sus-pect that had I been at Cambridgeor Oxford I would have also beenfaced with the big trend of the daywhich was Hardy. In America theschools were very much influencedby two elements, one was thatAndré Weil came to America andtherefore had a personal influenceafter having influenced France,first in Chicago and then Princetonand several other members ofBourbaki also spent the war inAmerica due to personal condi-tions. And then naturally there isan attraction to the simple solutionsomebody to complicate problemsis good and that was the element offashion. So I went to IBM which atthe time was completely removedfrom this world.

Did you feel initially that you hadfound a home at IBM?Not immediately, it took time. Firstof all I had a few things that I want-ed to do which in the French envi-ronment I couldn’t do because Iwould have possibilities that werenot necessarily linked to my taste.And again being in the other line ofwork I didn’t have any support orhelp from the community. It’s very

important to understand one aspectof scientific life; that happy medi-um between an organized institu-tion which has no room for any-thing outside of its purpose and theother where everyone works at cross-purposes and nothing is everachieved. IBM was in between, it wasa very interesting time very muchruled by the succession of Thomas JWatson Sr. to his son Thomas JWatson Jr. and everyone knows thisas a very heavy event in US industrialhistory, and in each case is very rep-resentative of the time.

It was a very big company, verysuccessful, but what he was manu-facturing was getting obsolete, theywere very well-built tabulatingmachines that lasted forever butthey were no longer necessary.Watson Jr. had to rebuild the compa-ny from scratch, he had to take thecompany into electronics, whichwas an extraordinary achievement,the change was not that differentfrom RCA but RCA did not have agood succession even though theywere very big and for IBM they start-ed out behind and it took severalyears to sort out. I was hired as a con-sultant because I had this qualifica-tion of having an enormous memo-ry of trivia, so I knew about this andthat, it was easier to ask me than tospend a long time in the library.That was a strange way of making aliving, but that gave me time to domy grounding which means to startmy work in finance and also mywork in physics which I cared verystrongly about it was being reorgan-ized every second, it was growing,moving into a new building, shak-ing itself up, and something mostremarkable happened, which Ithink is a lesson forever.

Sputnik happened and suddenlythe resources that were available for

sciences grew massively, andthe places that took theseresources were laboratories likeMIT and they are now major centersof research and development. Theycould suddenly hire whomever theywanted and the laboratories grewwithout changing very much andretaining the same structure, organi-zation and purpose. I go now and it isthe same place as it was fifty yearsago, but so much bigger. It’s amazing.

IBM was much the same, sud-denly we could grow and hire, andthose who applied were a mixture ofall kinds of people including a sub-stantial number of, well, oddballs.Young guys who for one reason oranother did not fit the pattern, have a clean record, etc. etc. Themajor decision IBM took was to hirea number of those people. Many left shortly afterward, but manystayed; and the first striking aspectof my time at IBM was that 35 yearslater when I retired those 50 odd-balls had all become well-known scientists. Everyone had becomemembers of the National Academyof Sciences, everyone had receivedmajor awards, because they were better placed than well-trainedpeople for the new era that wascoming up.

The kind of freewheeling struc-ture that was then in place did notprevent good people from achievingtheir potential. A few madeextremely important practical dis-coveries which led to new devices, afew made discoveries which led topatents which continue to be verylucrative for IBM. In fact this experi-ment, which was never repeated, isan excellent advertisement for theway in which science and engineer-ing are now over-organized becauseit was better to have the freewheel-ing sort of structure.

BENOIT MANDELBROT

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BENOIT MANDELBROT

ers of chaos theory as it generatedpictures of ever increasing complexi-ty using a deceptively minusculerecursive rule, one that can be reap-plied to itself repeatedly. You canlook at the set at smaller and smallerresolutions without “ever” reachingthe limit; you will continue to seethe recognizable shapes.

The introduction of fractals wasnot initially welcomed by the mathe-matical establishment. This methodof pictorial presentation did notseem to correspond to what seemed“to be mathematics” in the self-defining discipline. It is thanks to itspopularity with physicists and otherapplied scientists, themselves fol-lowing the lead of the general public(mostly computer “geeks”), that frac-tal geometry vindicated its way intothe now-broadened field of mathe-matics. For The Fractal Geometry ofNature made a splash when it cameout a quarter century ago. It spreadacross the artistic circles and led tostudies in aesthetics, architecturaldesigns, even large industrial appli-cations. BM was even offered a posi-tion at a medical school! His talkswere invaded by all manner ofartists4, earning him the nickname“the rock star of mathematics”. Thecomputer age thus helped himbecome one of the most influentialmathematicians in history, in termsof the applications of his work, waybefore his acceptance by the ivorytower. We will see that, in additionto its universality, his work possessesan unusual attribute: it is remark-ably easy to understand.

A Polish-Lithuanian Jew whofound refuge in France as a child, BMis also a refugee from the Frenchmathematical establishment protec-tive of the “purity” of mathematics.To borrow from the late probabilistand probability thinker E. T. Jaynes (a

man who went deeply into the sub-ject), it was said that “the French didquite useful mathematics beforeBourbaki” – as the secretive guild-like organization installed a trulytop-down view of the subject matter,insuring no corruption by earthlymaterial. Indeed many physicistshave been horrified at the extentand side effects of such purism, withMurray Gell-Mann calling it the“Bourbaki Plague”, and attributingthe divergence between pure mathe-matics and science to the obscurelanguage of the Bourbakists5.

In a way, the separation betweengeometry and algebra can be seen asthe separation of images and wordsin human expression and thought –just imagine a world in whichimages were barred. The Bourbaki-inspired purblindness does not justlimit the tools of analysis. Just likeblindness, one of its effects is toreduce contact with reality. Platonictop-down approaches are interestingbut they tend to choke under theoccasional irrelevance of their pur-suits. It is telling that BM’s hero isAntaeus, son of Gaia the motherEarth, who needed periodic contactwith earth to replenish his strength.

Owing to the vicissitudes of aclandestine life during the Nazi occu-pation of France, the young Benoitwas spared some of the conventionalGallic education with the uninspir-ing algebraic drills, becoming largelyself-taught with some assistancefrom his uncle Szolem, a prominentmember of the French mathematicalhierarchy and professor at theCollege de France. Instead, he devel-oped an encyclopedic knowledge ofthe history of mathematicalthought. He also gave free course tohis geometric bent. Untrained in theusual equation solving techniques,he passed the entrance exam to the

I closed this book feeling that it wasthe first book in economics thatspoke directly to me. Not only that,but the astonishing simplicity, real-ism, and relevance of the subjectmakes it the only general work infinance I’ve ever read that seemed tomake sense.

Benoit Mandelbrot makes sense.Just as he used us common readersoutside the ivory tower to force hisfractal ideas into science (wherethey became “part of the scientificconsciousness”1); he may just be theone to help turn economics intosomething real.

This first essay is non-technicaland general2 (i.e. can be read bysomeone without a mathematicalbackground) and focuses aroundthe topics covered in this book. Thesecond one is more technical and itgoes deeper into the epistemologi-cal problems of “fat tails”, concen-tration, and extreme events.

What do fern leaves, commodityprices, computer book sales, incomedistribution, the coast of Britain,cauliflowers, and the intricacies ofthe vascular system have to do withone another? Mandelbrot’s workrevolves around the simple practicalapplication of a concept called “frac-tal” in replacement for more com-plicated mathematical tools thatare universally used without empiri-cal justification.

Triangles, squares, circles, andother geometric concepts thatcaused many of us to yawn in the

Mandelbrot Makes Sense: A Book Review EssayA discussion of Benoit Mandelbrot’s The(Mis)Behavior of Markets by NassimNicholas Taleb

classroom, may be beautiful andpure notions; but they seem morepresent in the mind of mathemati-cians and schoolteachers than innature itself. Mountains are not tri-angles or pyramids; trees are not cir-cles; straight lines are almost neverseen anywhere. To figure out howthe world operates, we need a differ-ent geometry than the classical onedeveloped by Euclid of Alexandriasome 2400 years ago. Drawing on alist of then obscure (but subsequent-ly made famous) mathematicians,BM coined the word fractal geometryto describe these objects that arejagged yet self-similar in the sensethat small parts resemble, to somedegree, the whole (a more mathe-matically appropriate designationwould be the broader “self-affine”but, somehow, designations aresticky and, in this discussion, self-similarity should be held to be “self-affine”). Leaves look like branches;branches look like trees; rocks looklike small mountains. If you look atthe coast of Britain from an air-plane, it resembles what you getusing a magnifying glass. This char-acter of self-affinity implies that onedeceivingly short and simple rule ofiteration can be used, either by acomputer, or more randomly, byMother Nature, to build shapes ofseemingly large complexity. Hedesigned, or rather, according to SirRoger Penrose3, discovered an object,known as the “Mandelbrot set”,which became popular with follow-

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thinker who had the luxury to takehis time to grow his ideas.(Charmingly, BM, in his scientificwritings, when discussing a contri-bution made by a mature mathe-matician, mentions his age, such as“Cauchy, at the age of 64...”). It isthanks to such maturation that hejoins that category of the classical,pre-academic specialization of the wisdom-generating naturalphilosophers.

What does it all have to do withfinance? Can we extend the conceptof fractals and self-similarity to sta-tistical frequencies? It would makethe concept of astonishing universal-ity. This would make BM the trueKepler of the social sciences. Theanalogy to Kepler is at two levels,first in the building of insightsrather than mere circuitry, secondbecause you can step on his shoul-ders – the title of Kepler or “Newtonof the social sciences” is one so manythinkers with grand ideas have triedto grab (Marx for one aimed at beingthe Newton of the sciences of man). Iam not in the business of defininggenius, but it seems to me that themark of a genius is the ability to pickup pieces that are fragmented inpeople’s mind and binding themtogether in one, a meta-connectionof the dots.

Do probabilities (more exactly,cumulative frequencies) scale likecauliflowers? If so, the implication isnot trivial as we may be on to some-thing general, working across sci-ences and fields. And if so, then thestatistical attributes of financialmarkets can be made far moreunderstandable than by the compli-cated and middlebrow so-called“Gaussian” framework. Indeed thereis something about BM’s work thatmakes him and his ideas far moreunderstandable to the common man

than the theories of financial econo-mists, and, which is worrisome,more understandable by the commonman more than by the classicallytrained economist – just as the com-puter graphic designer or a comput-erized teenager could get the pointfar more easily than a classicallytrained mathematician.

It is not a well-known fact thatbefore his involvement with theroughness in the geometry innature, BM started his career focus-ing on problems in social scienceand finance; it is certainly there thatmost of his ideas were refined. Heinitially wrote papers in the 1960spresenting his ideas on “infinite

variance”, getting some early accept-ance, but rapidly causing anxiety infinancial economics circles. He thenmoved to the less harmful fields ofgeometry and physics, returning tofinance in 1995 when he started avery active production of scientificpapers on financial risk. At eighty,he shows no sign of relenting, pro-ducing, as I said, the deepest andmost realistic finance book everprinted. By writing The (Mis)Behaviorof Markets in collaboration withRichard Hudson, a long time jour-nalist at the Wall Street Journal, heseems to be employing the same

strategy of going straight to practi-tioners and the general public andbypassing the academic establish-ment, a task that might appear easywith economics given that the public and professional standing ofeconomists in general and financeacademics in particular is one of the lowest of any specialty. So themission of toppling these fake and empirically invalid beliefsseems trivial.

Or is it? Finance academia,unlike the physics establishment,seems to work more like a religionthan a science, with beliefs that haveso far resisted any amount of empiri-cal evidence (actually this statement

is quite mild; it works just like a reli-gion totally impervious to newsfrom reality). The closest field tofinance in the history of sciencewould be pre-Baconian medicine aspracticed in the Middle Ages, eitherdisdainful of observations or spin-ning them with theological argu-ments. financial theory being a fad,not a science, it will take a fad, andnot necessarily a science, to unseatits current set of beliefs.

BM wrote his doctoral thesis onwhat seemed to be two subjects atonce: mathematical linguistics andstatistical thermodynamics (de

elite École Normale using purely geo-metric intuitions (this should be ahint for educators: consider howmuch more intuition you can devel-op with images instead of words). Buthe left after two days. Already stub-born, unruly and unmanageable, hemoved to the more engineering-ori-ented École Polytechnique. He thensettled in the United States, workingmost of his life as an industrial scien-tist for IBM, with a few transitory andvaried academic appointments.Indeed, thanks to the computer, hecould let the potent machine expresshis geometric hunches and leadthrough the subject matter’s naturalcourse. Indeed, the computer playedtwo roles in the new science hehelped conceive. First, these fractalobjects, as we will see, could be gen-erated with a simple rule applied toitself which is ideal for the automa-tion of a computer (or mothernature). Second, in the generation ofvisual intuitions lies a dialecticbetween the mathematician and theobjects generated. A mathematicalscientist par excellence, in a subjectmatter that did not (then) exist insti-tutionally, he was held to be a mathe-matician for scientists and a scientist(particularly a physicist) by the math-ematical establishment. And whilemathematicians burn out in theirtwenties, he received his first aca-demic tenure at Yale when he was 75years old. Indeed, after a stint atHarvard where computer and mathe-matics are subjected to a conceptualseparation, it is at Yale that BM6 gothis dream job as a Professor ofMathematical Sciences. And it took himhalf a century to fully realize whathis work was united by an attribute:roughness, not just as a quality ofobjects, but as a standalone field ofstudy. It is impressive to see him asthe embodiment of a scientific

This would make BM the trueKepler of the social sciences ...first in the building of insightsrather than mere circuitry, second because you can step on his shoulders

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would the arrival of Bill Gates to a town do to the averagewealth there.

It is worrisome because everystudent of statistics learns aboutmean and variance as the founda-tions of their methods.

The Gaussian, in contrast, is notscalable. Most observations hoveraround the mediocre, and devia-tions either way become increasing-ly rare, to the point of there beingevents of an impossible occurrence.Take the number of adults heavierthan 300lbs and those heavier than150lbs. The relation between thetwo numbers is not the same as theone prevailing between 600 and300lbs. The latter will be consider-able smaller. It gets smaller as thenumber get larger – meaning thatthere is no self-affinity. Deviationsfrom the norm decrease very rapid-ly, at an increasing rate, to the pointwhere some high number becomesliterally impossible. The increase ofthe rate of the decrease is what pre-vents scalability. BM calls this typeof randomness “mild”, as comparedto the “wild” one generated bypower laws. There is a beautiful sen-tence in the book differentiatingbetween the two: “Markets oftenleap, don’t glide”.

To further see the link betweenfinance and fractal geometry, pick afinancial chart. Just like the coast ofBritain, self-similar at all resolu-tions, monthly prices look “like” (i.e.present an affinity with) hourlycharts. One has to shrink thetimescale more than the price scalein order to get the same effect.Furthermore, if the stretching isdone in a random manner, itselffractal, one ends up with whatMandelbrot calls multifractal.

In 1963 BM wrote a paper on theproperties of financial prices and

wars, and, of course, market move-ments. The implication of thesepower laws is that, for most, there isgenerally no “standard” deviationfrom the norm. In the previousexample of wealth, if there are morethan 1/4 the number of people witha 2 times a given level of wealth thanwith a given level (more technically,when the tail exponent is higherthan 2 since doubling the wealththreshold here leads to an incidenceof more than the square of the ratio),then we are dealing with undefinedvariance. Now, worse, when the fre-quency in the previous exampledrops by less than half, then we arein a situation of extreme fat tails:there is no known average. Any arbi-trary large number can take placethat can disrupt the mean. The con-cept of average is meaningless, total-ly meaningless as a characterizationof the attributes of a very fat-tailedprocess, such as computer firms. Thenotion of a “typical” computer com-pany has nothing to do with any-thing. Likewise characterizing a“typical” writer provides no infor-mation. Just consider how unstablethese variables can be: imagine what

with wealth in excess of 20 millionwill be approximately the same inrelation to those with more than 10million: about a quarter. This rela-tion (here the square of the ratio) iscalled a scaling law, as it is retainedat all levels, no matter how large thenumber becomes (say two billion inrelation to one billion). What is criti-cal here is that it does not vanish –frequencies get lower for higherwealth levels, but the ratios betweentwo arbitrarily high numbers do not decrease!

Cauliflower? If you separate thefrequencies you will find that thesub-samples resemble each other inthe degree of inequality in the differ-ent ordered sub-sections, as can beshown in Figure 1.

Note that the “tail” is the pointwhere the outcomes become scala-ble in cumulative probability; it doesnot have to be a transition point (itcan be an asymptotic property as wetend towards it). This scalabilityseems to apply to a variety of phe-nomena like book sales, nodes onGoogle, the relative size of cities, thenumber of times an academic paperis cited, the number of casualties in

Figure 1 The Cauliflower theory of frequencies. This is the result of the application ofpower laws dynamics to a wealth process. If you divide the area in smaller ordered sub-samples, you will see the same inequality prevailing.

Broglie was the head of the thesiscommittee). Before the advent ofInformation Theory as a discipline,such mixing seemed quite strange. Aquip goes to the effect that, of histwo topics, the first did not exist yetand the second no longer existed.But the unity between the two wasthe so-called “fat tails” and “powerlaws” that are now becomingincreasingly popular in physics andsocial science, though not in eco-nomics. The spark came from the so-called Zipf’s “law” in linguistics,after the works of one George Zipfon the relative ranking in the fre-quency of words in a vocabulary. BMdebunked Zipf’s belief in the separa-tion, thanks to these laws, betweensocial and the natural sciences:these “fat tailed” phenomena alsoexisted in physics. We are just blindto them.

BM later built on the works ofthe (then) unknown mathematicianPaul Lévy and, to a lesser degree, thetrader-economist Wilfredo Pareto towhom the original power law isattributed. The designation “L-Stable” distributions, (for “Lévy-Stable”), a.k.a. Pareto-Lévy distribu-tions comes from Mandelbrot. I pre-fer to use the designation “PLM”(Pareto-Lévy-Mandelbrot) for themore general case of a random serieswith both independent and non-independent increments.

Let us see how power laws, withtheir scalability, i.e., the asymptoticsettling of a series to a constant limitin the relationship between likeli-hood of events, can be seen as anapplication of fractal geometry.Consider wealth in America.Assuming we reached the “tail” , thenumber of people with more thantwo million will be around a quarterof those with more than one mil-lion. Likewise the number of persons

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memory – in other words we are nolonger dealing with serially inde-pendent draws. The mathematics ismore intuitive and more realisticthan what we are used to; indeedthere is no mathematics but graphsand geometric intuitions. He pres-ents the usual attacks on his modelthat consist in saying, “daily pricesmight be nonGaussian but in thelong term things become Gaussian”.Long term? After the bankruptcy?Long Term Capital Managementwas a “long term” idea as well.Under leverage there is no suchthing as long term.

The third part wraps up withmore railing against finance theoryand some suggestions for furtherresearch. It includes a scene withjournalistic overtones of a visit tothe laboratory of randomness spe-cialist Richard Olsen in Zurich.

***This book has a crisp message

about risk. The reviews were quitefavorable, but distressing for usempiricists as few commentatorsgot the point. People have difficultydealing with the idea that one canwrite a general book on a financialtopic without telling people about anew foolproof (and secret) tech-nique about how to double theirmoney in 21 days. My book Fooled byRandomness generated hundreds ofletters with the following class ofcomplaints: “you tell us that it ismostly luck, which seems reason-able, but you don’t tell us how tomake money out of this luck”.People are so conditioned by advice-offering charlatans in businessbooks that anything remotely awayfrom it seems, as I was told, quite“odd”. BM’s, of course, does not giveyou a recipe. It was therefore amus-ing to see the book reviews com-plaining about the “now what?” –

ment, you may not be diversified asmuch as conventional theory indi-cates. And conventional statisticaltheory might make you jump to con-sequences too quickly: your samplesize is smaller than you think.

I was trying to explain the differ-ence between two modes of thinking,the broad and the narrow, to aninvestor. Remarkably, it correspondsto the difference between power lawsand Gaussians. As a method of riskmanagement, he follows the conven-tional methodology of collectingpast returns, building a database,and simulating by drawing from thepast, thanks to bootstrapping-stylemethods. Using such an approachwould make him select the largestpossible deviation in the simulationas the worst scenario. A method ofsay, fitting an “empirical probabilitydistribution”, would do almost thesame. This is an interpolativemethod – of course the worst possi-ble move in the future is going to besimilar to the one in the past, thoughthese moves did not take place in thepast’s past. After the stock marketcrash of 1987, they simulate using 22per cent as the worst daily deviation.Don’t they realize that before thecrash they would have used the pre-ceding worst case and missed onsuch a big event?

Both the Gaussian and our con-ventional wisdom are interpolative.Power laws are extrapolative. Youlook at the ratio of millionaires tobi-millionaires and can translate itinto the ratio of 10-millionaires to20-millionaires. Likewise the ratiobetween 5% and 10% moves allowsyou to infer the incidence of movesin excess of 20%.

I will rapidly go through thedetails of the book .

The first part of the Mis-Behavior ofMarkets, out of three, presents the

very sad history of modern finance.It ends with the presentation of theevidence against these models. Itdoes not take a lot of empiricism tofigure out that such risk measure isuseless: the stock market crash of1987 had, according to their models,such a low probability, one in severalbillion billion billion years , that itshould not have happened (probabil-ities that low are no longer measura-ble; it is meaningless to arguewhether to assign a 10-23 or a 10-12probability to these). You do notneed a lot of empirical work to real-ize that a model is wrong: one singleinstance suffices to invalidate it.

Another piece of evidenceamong many is the hedge fundLong Term Capital Managementthat went bust in 1998. It employed25 PhDs, and two “Nobel” medal-lists in economics for their work infinance. Aside from the fact thattheir “Nobel” was mistakenly pre-sented for inventing a “formula” –the formula has been there for awhile; what they did is make it fitinto the prevailing economic argu-ments. They used complicatedmathematical models – they shouldhave had on their staff more street-smart cabdrivers who do are privi-leged to not know economics. LTCMis a milestone as a catastrophe thatwas caused by the pseudo-science ofeconomics, much like the sideeffects of those medieval medicalremedies.

The second part discusses thefractals theory and its relation tothe power laws. Those familiar withBM’s ideas from James Gleick’sChaos will see the usual themes pre-sented. It ends with the multifractalmodel where BM presents a memo-ry of prices similar to those of thefloods by the Nile river; what hap-pened a decade ago stays lurking in

found them to be scaling power lawsof the anxiety-causing types – the“infinite variance” variety. The paperwas initially endorsed by the ortho-dox finance establishment, accept-ing the implications that there is not“standard” risk, no known risk. Butsuddenly, these academics startedlooking the other way as “modernportfolio theory”, linking risk andreturn, was born. There had to be ameasure of risk, even if it presents thefatal contradiction of not workingwhen you need it. The bell curvedescribes the equivalent of the oddsof an uncomfortable airplane ride,nothing about the risk of crash – butoperators thought thanks to “sci-ence” they were now in control.

If you asked for the bridgebetween the arts and science, thenotion of fractal would come up. Ifyou ask about what bridges hard andsocial sciences, the same scalablelaws would come up. Doesn’t thismake BM the universal scientist?

Most of the effects ofNonGaussianism flow from the consequence that a small number ofobservations might contribute dis-proportionately to the total meanand variance. Pending on the gravi-ty, you either need a very large, possi-bly infinite, sample to track theproperties. Indeed, if ten days in adecade represent 40 per cent of thereturns, which we tend to see rou-tinely with financial securities,much of conventional sampling the-ory goes out of the window. Considerthat under a Gaussian regime, sincethese outliers represent a smallshare of total variations, you shouldbe able to obtain the properties of ,say, the stock market by being in it asmall sub-segment of the time.Diversification, too suffers from theconsequences of scalability. Since fattails create a winner-take-all environ-

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fined variance”), implies that whenyou take a sample from a long series,every sub-sample yields a differentmeasure of volatility.

Nor does it look like the fudgingof the finance models can producereal results. I will omit discussingthe repackaging of the Capital AssetPricing Model under the newer“Arbitrage Pricing Theory”, except tobemoan that, seven years afterLTCM, the most recent issue of theJournal of Economic Perspectives7 cele-brates with some pomp the 40thanniversary of Modern PortfolioTheory. It is saddening to see that sofew realize its epistemological dan-gers. One insightful and honest arti-cle, by Fama and French, talks aboutthe poor “empirical” results, accept-

ing the notion that empiricalimplies in practice, out of sample andrealizing that, in the end, its appeallies far more in teaching MBA stu-dents than anything else.

Now there have been fixes tothese equations to accommodate fat-tails, to no avail. Every option traderknows that volatility is variable – butmodels such as GARCH, with close to10,000 academic publications, donot seem to bring us closer to any-thing. Making volatility variable ismore complicated than we think:there is the problem of the specifica-tion of such variability. At the lastICBI Madrid Derivatives Convention,Robert Engle, freshly medalled by

the Swedes for his GARCH process,made the following comment:whether or not you include the stockmarket crash of 1987 or not makes ahuge difference to the choice ofmodel and its parameters. GARCH,extremely fragile in its calibration,is very sensitive to the inclusion ofsuch large observations from thedeep past. Does it smell like unde-fined variance to you?

I am dedicating my next book,The Black Swan, to BM for his 80thbirthday. I can now safely say, inspite of my having had discussionswith hundreds of hotshots, that heis the first person who ever taughtme anything meaningful about mysubject matter of uncertainty. Morespecifically, it was the first time in

my life that I had a conversationwith someone who can naturallyhold that the notion of “variance” ismeaningless in characterizinguncertainty – and we could move onto a more meaningful discussion ofthe subject. I finally found someoneI could talk to without feeling deepstrain and tension.

There is more. He could commu-nicate with the trader in me. I wastaken aback by how easily his ideasspoke to me, down to the very practi-cal. We traders divide persons intotwo categories: those with a “longvolatility” frame of thought, who, ingeneral, never rule out blowups,change, trends, conspiracies, and

“mean-divergence” , and the othermore gullible “short volatility” whobelieve in models, “mean-reversion”,“arbitrage”, the self-canceling activi-ty called statistical arbitrage, andsimilar things. In other words, thereis the naive and the skeptic.Scientists and academics tend tosquarely fall in the second category,even when they trade, while veterantraders and real practitioners havethe first mindset. It was a surprise toencounter BM, a scientist of the“long-vol” category.

It was also refreshing to findsomeone who shared the same aller-gies. It was not just the notion ofvariance; small details can be reveal-ing. For instance, we both got inde-pendently offended by the same

statement that “nature does notmake jumps”.

So time lost was made up and itwas refreshing to discover the per-sonal charm of the universal philoso-pher and be privileged to his conver-sation partner. BM only lives five kilo-meters away from my house, whichmeans that we spent more time talk-ing on the telephone than meetingin person (this is how these thingswork). Conversations with him arepunctuated by opened-and-closedparentheses, with tours of classicalliterature, history, science, music,back to science, with digressionsrarely left hanging. Not surprisingly,he is an independent thinker in just

how can we take these ideas home?The answer is clear: get out of themarkets as we understand themless, far less than we are led tobelieve. That would be a significantfirst step. What this book is about isthe variability of markets and theirrisks, period.

The central idea about risk man-agement that preoccupies me cur-rently is as follows. If you save peo-ple in the process of drowning youare considered a hero. If you preventpeople from drowning by averting aflood you are considered to havedone nothing for them. Such asym-metry is apparent: you do not getbonus points for telling agents toavoid investing. They want “some-thing tangible”.

Likewise you do not go very farby telling people “we do not gainanything by talking about the vari-ance”. They want a risk number, acorrelation number and BM takes itaway from them (notice that unde-fined variance also means unde-fined correlation).

A simple implication of the con-fusion about risk measurementapplies to the research-papers-and-tenure-generating equity premiumpuzzle. It seems to have fooled econ-omists on both sides of the fence(both neoclassical and behavioralfinance researchers). They wonderwhy stocks, adjusted for risks, yieldso much more than bonds and comeup with theories to “explain” suchanomalies. Yet the risk-adjustmentcan be faulty: take away theGaussian assumption and the puz-zle disappears. Ironically, this sim-ple idea makes a greater contribu-tion to your financial welfare thanvolumes of self-canceling tradingadvice.

The possibility of “infinite vari-ance” (or more appropriately “unde-

People readily mistake irreverence towards someclass of accepted heroes for arrogance. A fairapproach would be to examine the targets ofsuch irreverence

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IntroductionConsider the following thoughtexperiment. You show an agent a setof data of 2,500 days worth ofreturns (the resulting asset priceW (t) being represented in Figure 1)and ask him to infer the attributes ofwhat he saw.

Odds are that he would tell youthat the log-returns are Gaussian.2,500 days data set represents anample sample size by any measure,enough for the distribution to revealitself to us. Clearly all the attributesof a mild distributions are there: noexcess Kurtosis over that of aNormal, no outliers, no jumps, nogaps; a histogram of the returnswould reveal the Platonic Bell Shape.

Now we continue with the rest ofthe story. We add one day, number2,501; one single day can show aquite different picture.

Picture 2 shows the information-

al increase by that one day. The gen-erating process for these draws is amere switching process, builtaround a Gaussian, to which wasadded the occasional drawing, oncein 2,500 days, from an infinite vari-ance kick. This implies that the totalis of infinite variance. Those whohave not seen any such situationshould take a look at emerging mar-ket currencies (those in a managedregime). It can also apply to a hedgefund returns: The properties of thelate hedge fund LTCM are not too dif-ferent from what we just saw. Thebigger the divergence between thetwo regimes (the “normal” and the“unusual”), the worse the epistemo-logical picture as more people willtend to be fooled by what they saw.

The central problem of uncertainty What I call the central epistemologi-cal problem of uncertainty1 is sum-

Fat Tails, AsymmetricKnowledge, and DecisionMakingNassim Nicholas Taleb’s Essay in honor of BenoitMandelbrot’s 80th birthday

Figure 1: A dataset of 2,500 prices. Infer the attributes.

1 Kenneth Falconer, Nature, 430/ 1 July

2004.

2 A shorter version of this book review was

withdrawn from the Los Angeles Times, part-

ly because I got too close to Mandelbrot

after writing the review and did not want to

bear the risk of personal conflict.

3 Roger Penrose, 2005, The Road to Reality,

New York: Knopf.

4 John Brockman, 2005, Discussion with

Benoit Mandelbrot, www.edge.org

5 See the posthumous Probability Theory:

The Logic of Science by E.T. Jaynes, 2003,

Cambridge University Press.

6 See Mandelbrot's essay on www.edge.com

7 Journal of Economic PerspectivesVol. 18,

No. 3, Summer 2004

8 See the personal testimony in Michel L.

Lapidus (editor): Fractal geometry and

applications: A Jubilee of Benoit

Mandelbrot, Proceedings of Symposia in

Pure Mathematics, 72, 1, American

Mathematical Society.

FOOTNOTES

honor. My reaction was to congratu-late the Nobel committee: finally,these Swedes seem to be seriousabout their prize. Not only have theyhelped to make economics more of ascience, but they also gave it the cre-dentials to help enrich other disci-plines. The abundance of data makesthe field of economics an ideal labo-ratory to develop insights and quanti-tative tools helpful to other sciences –we can develop insights abouthuman nature from economic choic-es (Kahneman and Tversky); we canalso learn new mathematical meth-ods (Mandelbrot). I hereby ask theSwedes to take some perspective andthink of those whom, a century fromnow, will be identified as havingchanged the way we view the world. ■ The (Mis)Behavior of Markets: A FractalView of Risk, Ruin, and Reward byBenoit Mandelbrot & RichardHudson, Basic Books.

about everything; he is a pack ofintuition; he is encyclopedic and is auniversal conversationalist. If youmanage to age well, you actually getbetter because you know so manymore things. And he has an astonish-ing memory (“une memoired’éléphant”).

Having read descriptions of hispersonality, I was taken aback by thedifference between the real manand the reputation of “arrogance” –which to me (as I am familiar withsuch accusations) comes merelyfrom his targeted irreverence andlack of willingness to put up withestablished truths and establishedgods. People readily mistake irrever-ence towards some class of acceptedheroes for arrogance. A fairapproach would be to examine thetargets of such irreverence.

In a way BM is the exact oppositeof what I call the academic clerk:someone who is there to work onresearch like an obedient taxaccountant. BM is a maverick, tena-cious, and idiosyncratic in hisapproach; he seems to scorn formal-ities. It is all-natural that he wouldhave had to counter resistance fromthe clerks. I was in for a surprise: Ihad the feeling of talking to a trader,capable of revising his views at ablip. And the man was simple,friendly, charming, the reverse ofarrogant – except for his colorfulirreverence. Consider that one of hiscolleagues, Michael Frame8 who wasalso told that BM was “arrogant”,accounts for his surprise upon hav-ing to contradict BM on a criticalpoint. BM’s reply was “Marvelous.The problem is more interestingthan I had expected”.

One final remark about recogni-tion. When Daniel Kahnemanreceived the Nobel medal many peo-ple congratulated him on such an

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the squared Gaussian variate) . This is exceedingly circular andreflects a severe lack of awareness ofsuch circularity.

An easier solutionAs an operator first and last, I believethat there are, however, far more ele-mentary (and practical) ways to dealwith this problem, or at least to pro-tect ourselves from its ill effects.How? I propose two approaches.

First, consider Pascal’s wager. Wecan change our payoff structure toaccommodate what absence ofknowledge we suffer from, and withrespect to which moments of the dis-tribution. For instance, if the datahas “infinite” (or undefined) vari-ance, one can avoid exposure to suchinfinite tail by clipping the sensitivi-ty to the offending part of the distri-bution. Purchasing a simple deriva-tive(say, an extremely out-of-the-money call), if it such product isavailable, may provide a solution.Our doubt can be targeted and reme-died by transactions. Tout simplement.

Second, what we call the mas-querade problem. The data cannottell us what is the probability distri-bution generating it; but it can easi-ly tell us what such probability dis-tribution is not (or is not likely to be),and which moments of the distribu-tions we may not be able to compute.

Portfolios, infinite variance, andepistemic opacityWhat many academic philosophersdo not realize is that the limits ofsome knowledge may be of smallmoment. I would rather use my ener-gy in changing my payoff structurerather than getting into intractableissues and playing philosophaster.My colleague, another option traderand empirical philosopher RabbiAnthony (“Tony”) Glickman (also a

Talmudic scholar), explains quiteeloquently that being an optiontrader gives someone a philosophi-cal approach along “long gamma”lines, or, more formally in the deci-sion theory literature: along a mind-set focused on the convexity of pay-offs. One comment I make hereabout Tony is that his definition ofphilosopher is similar to mine (andMandelbrot’s): a philosopher issomeone who specializes in ideas,not in other people’s ideas – likestamp collecting. Professionalphilosophers can be like parasites.To Tony, like for me, being long anoption in the tail (or more generally “longconvexity”) eliminates the need to try tofigure out what we don’t know3. Only an option trader could understandthat – that’s what I am trying to gen-eralize to all decision making under uncertainty and convey tonontraders in my forthcoming TheBlack Swan.

It is key that we operators anddecision makers are capable of insu-lating ourselves from nasty parts ofthe distribution. It is a fact that aportfolio constituted of securitiesthat have infinite variance does notneed to have infinite variance. How?If you are short a call spread with theposition strike K, described as shorta call struck at K, long another callat K + y, you are “short volatility”,but you are not exposed to infinitevariance. Your payoff is capped.Furthermore: the properties of yourstrategy are not fragile to parametricassumptions or choice of model.

Note here, in the earlier thoughtexperiment, that the moments ofthe distribution are very precarious;the loss L (taken in Log returns) is solarge that the moments are insensi-tive to the probability of the big lossπ. Indeed the pair π L (probabilitytimes the payoff) is so large that we

marized as follows: we do not observeprobability distributions, only ran-dom draws from an unspecified gen-erator. So we need data to figure outthe probability distribution. How dowe gauge the sufficiency of the size ofthe sample? Well, from the probabili-ty distribution. If at the same timeone needs data to figure out the prob-ability distribution, and the probabili-ty distribution to figure out if we haveenough data, then we have a severecircular epistemological problem.

Note here that fat tails are conta-gious. If you combine two randomvariables each following a power lawdistribution but with different expo-nents, the result is a power law dis-tribution with, for tail exponent, thelower of the two. Here we have twoprocesses, one of finite, the other ofinfinite variance; accordingly theinfinite variance will prevail.

A traditional philosophical wayto deal with the regress argument, ifone follows the epistemological tra-ditions, would be to either 1) putyour hands up and bemoan theProblem of Induction, and find theo-logical arguments to have someunquestioned belief or 2) proceed toa systematic layering: One can pose ameta-distribution, one that would

take into account the probability ofthe candidate distribution being thewrong one. You can use priors andprobabilize with series of meta-prob-abilities. Neither handy, nor con-vincing, and it implies as ElieAyache2 put it in this magazine “try-ing to find a random generatorbehind the random generator”. Andit does not escape the attacks by clas-sical Pyrrhonian skeptics: we seem tobe either 1) justifying belief with ref-erence of other belief, itself justifiedby other belief, all the way up untilsome unargued dogma, which couldbe fragile (in this case some “known”distribution or generator for thetime series) 2) justifying belief some-where in the loop with another pre-viously derived belief and fallingback into severe circularity; finally 3)the regress may never end and westay at the beginning.

Note that the quantitative-statis-tical literature is not thoughtfulenough or self-critical “to be evenwrong” on the subject. How?Conventional tests of normalitystudy the square errors from aGaussian and use a Gaussian-inspired distribution (a special caseof the Gamma distribution, the Chi-Square, which is the distribution of

Figure 2: A dataset of 2,501 prices. What is the informational increase?

58 Wilmott magazine

BENOIT MANDELBROT

Point 1: The slowness in the rateof convergence makes a cubic αvery seriously NonGaussian.If we accept that α is approximately3, “outside the Lévy regime”, we arestill in trouble with respect to theconvergence to the Gaussian. Finitesecond moment implies conver-gence under aggregation, but weneed to remember that with α < 4have an undefined 4th moment. Theimplication is rather serious.Consider that the 4th moment is thevariance, corresponds to the error ofthe measurement in the variance(what we option traders call the“Vvol”). It will be infinite! Thisimplies a quite nasty rate of conver-gence. There will always be aNonGaussian jump in the extremetail to make the tail scalable.

Another way to view it is that theobservations that we are adding arelikely to be biased towards the mid-dle of the distribution, making itconverge in the body but much moreslowly in the tails. We can examinethis quantitatively. Take α = 3. It iseasy to show7 that, in standard devia-tion terms, outside (Log (n), with nthe number of observations, we stayin a scalable regime. Even if you addup 1 million days, the Gaussian

under aggregation. There have beenseries of papers6 disagreeing withMandelbrot’s early work and its con-clusions. Researchers tend to be“skeptical” about the Lévy regimehypothesis producing, for more thana quarter century now, “evidence” tothe effect that Mandelbrot’s earlycharacterization of infinite varianceis wrong – people seem to very badlyneed a Gaussian in order for them tooperate with the current academicframework. Their methodology isbased on two arguments, first, the“observation” of α > 2 and, second,the examination of the behavior ofthe data when they lengthen thetime observation period.

These studies are either inconse-quential or wrong in their infer-ences. First, it does not make muchdifference whether or not we are in aLévy regime since we don’t reallystay in the Gaussian regime in theparts of the distribution that matter.Second, we do not “have evidence”that we are not in a Lévy regime.Third, we need to go beyond the“Lévy regime” and consider theMandelbrot regime by lifting thetoo-restrictive assumption of inde-pendent increments. I will get intothe details of the arguments next.

Figure 3: The regime densities.

may never care about the size of theprobability. It is so obvious that weshould work to control L – or, if wecan’t, to only enter transactionswhere such L can be controlled.

Now the question: what if wecan’t insulate ourselves from suchdistributions? The answer is “dosomething else”, all the way to find-ing another profession. Risk man-agers frequently ask me what to do ifthe commonly accepted version ofValue-at-Risk does not work. Theystill need to give their boss somenumber. My answer is: clip the tailsif you can; get another job if youcan’t. “Otherwise you are definingyourself as a slave”. If your boss isfoolish enough to want you to guessa number (patently random), gowork for a shop that eliminates theexposure to its tails and does not getinto portfolios first then look formeasurement after. Indeed if like meyou think that Modern PortfolioTheory is charlatanism (as con-firmed by my trader’s observationsand empirical research, andMandelbrot’s work), use portfoliosthat do not depend on their meas-urements. It is so easy to avoid traps.

The asymmetric masqueradeproblemA power law (as we saw in thethought experiment) can easily mas-querade as a Gaussian but not thereverse (at least not easily). We canreject the Gaussian more easily thanwe can accept it. More generally, adistribution with fat tails can showmilder tails than its “true” proper-ties, except, of course, when it is toolate. It will even tend to do so. Thesmall sample properties of theseprocesses are such that we are notlikely to encounter large moves inthem. We can call that problem an“epistemic headwind”. To answers

some questions put to me in thismagazine about skepticism andasymmetric knowledge4, I will usethe argument that it is always easierto figure out what the distribution isnot than what it is. Compare that tothe attributes of humans: a criminalcan masquerade as an honest citi-zen; an honest citizen cannot as easi-ly fake being a criminal. Many exten-sions of this point are accepted inmany fields: one single event consti-tutes a catastrophe; one needs manydays without an event to pronouncean environment as catastrophe-free.

This asymmetry is at the core ofskeptical empiricism: our body ofknowledge is more readily increasedby negative observations than byconfirming ones. Remarkably, wecan do something with this; it leadsus to a ranking of the robustness ofresults. And remarkably, it is becauseI elect to behave operationally as ifthe market followed a Mandelbrot-stable process that I can build portfo-lios that I am comfortable with. AMandelbrot-stable variable is simplyhere what is called a Levy stable, butwith non-serially independent draws(what BM calls multifractal). We willreturn to the situation.

The α problemTake X a random variable, we have apower law P [X > x0] ∼ O(x−α

0 ).Clearly we are told that if the firstand second moments of the distribu-tion are defined, i.e., α > 2, then,under aggregation the seriesbecomes Gaussian so we can use theconventional tools of analysis. Notehere that this only holds if we haveindependent increments.

BM came up with papers in the1960s5 showing cotton prices withtail α < 2, in other words implyingLevy-stability; the distribution has fattails and does not become a Gaussian

Wilmott magazine 59

BENOIT MANDELBROT

■ E. Ayache,2004a, The Back of Beyond,

Wilmott, 26-29

■ E. Ayache,2004b, A Beginning, in the end,

Wilmott, 6-11

■ M. Blyth, R. Abdelal and Cr. Parsons,2005,

Constructivist Political Economy, Preprint,

forthcoming, 2006: Oxford University Press.

■ J.-P. Boucheaud and M. Potters, 2003,

Theory of Financial Risks and Derivatives

Pricing, From Statistical Physics to Risk

Management, 2 nd Ed., Cambridge

University Press.

■ X. Gabaix, P. Gopikrishnan, V.Plerou &

H.E. Stanley, 2003, A theory of power-law

distributions in financial market fluctuations,

Nature, 423, 267-270.

■ B. Mandelbrot, 1963, The variation of cer-

tain speculative prices. The Journal of

Business, 36(4):394–419.

■ B. Mandelbrot, 1997, Fractals and Scaling

in Finance, Springer-Verlag.

■ B. Mandelbrot, 2001a, Quantitative

Finance, 1, 113-123

■ B. Mandelbrot, 2001b, Quantitative

Finance, 1, 124-130

■ R. R. Officer 1972 J. Am. Stat. Assoc. 67,

807–12

■ D. Sornette, 2003, Why Stock Markets

Crash : Critical Events in Complex Financial

Systems, Princeton University Press

■ D. Sornette ,2004, 2 nd Ed., Critical

Phenomena in Natural Sciences, Chaos,

Fractals, Self-organization and Disorder:

Concepts and Tools, (Springer Series in

Synergetics, Heidelberg)

■ H.E. Stanley, L.A.N. Amaral, P.

Gopikrishnan, and V. Plerou, 2000, Scale

invariance and universality of economic fluc-

tuations, Physica A, 283,31-41

■ N N Taleb and A Pilpel, 2004, I problemi

epistemologici del risk management in:

Daniele Pace (a cura di) Economia del ris-

chio. Antologia di scritti su rischio e deci-

sione economica, Giuffrè, Milano.

■ R. Weron, 2001,Levy-stable distributions

revisited: tail index > 2 does not exclude the

Levy-stable regime.International Journal of

Modern Physics C (2001) 12(2), 209-223.

1 See Taleb and Pilpel, 2004. See also Elie

Ayache , 2004a

2 Ayache, 2004b.

3 We can extend this convexity argument to

the philosophy of probability in general.

Take the subjectivist concept of probability

as degree of belief attributed to De Finetti.

Probability is held to be the price I am willing

to fix in such a way as I would equivalently

buy or sell a state of the world, making sure I

remain consistent and avoid the “Dutch

book” problem. Well, you do not have to put

yourself in a situation which you have to

trade.

4 see Ayache 2004.

5 Mandelbrot (1963) See also Mandelbrot

(1997).

6 See Officer (1972), Stanley et al (2000),

Gabaix et al (2003)

7 See Sornette (2004) for the proof. See

also Bouchaud and Potters (2003).

8 Mark Spitznagel brought this to my

attention.

9 Weron (2001).

10 See Sornette (2004) for the argument.

11 Mandelbrot (2000a, 2000b).

REFERENCESFOOTNOTESregime stops at 3.7 sigmas! Typicalpenny-wisdom since the conse-quences of outside such moves aredisproportionately large. Figure 3shows the two-regime densities.

The situation is reminiscent ofthe value at risk problem. The tail ofthe distribution is where our errorscompound. That is where ironicallypeople like the precision.

Point 2: Absence of Evidence isNot Evidence of Absence: TheSmall Sample Bias ProblemMeasuring an α > 2 does not implywith any confidence that the “true”α is not <2. I avoid the discrepancieshere in the measurement resultsfrom the various estimators,whether Hill and Log-Log linearregression. It just takes time (anddata) for these distributions to revealthemselves. Simulate a series of sym-metric random draws with α = 1.9and you will recover an α close to 3with 106 samples. This is an argu-ment well known to many traders8

and discussed in Weron (2003).9

As we saw with the 2,500 dayproperties in the thought experi-ment, matters can be even morecomplex with a mixed process. Inshort, the fat tailed process tend toshow the underestimation of theobserved volatility.

Point 3: Where the AggregationFattens the Tails. Many of theseinferences and indeed much ofthe mathematics we are used toassumes that we have independ-ent drawsNow consider the following intu-ition: very bad moves generate verylarge up or down moves. And alsoconsider that this may only happenin extreme circumstances, when themoves exceed a given threshold.Intuitively, a large loss might gener-

ate series of self-causing liquida-tions. What would that do to thescalability?

Well, in such mechanism, theaggregation fattens the tails. Such isthe observation made by Sornetteconcerning the events leading up tothe crash of 1987,10 prompting himto analyze the properties of “draw-downs” independently.

This brings us to the Mandelbrotmultifractal generalization11 thatshows that the process can have 1< α< ∞. Indeed much of the work on sta-ble distributions is restrictive –obsessively relying on the assump-tion of independence.

Final note and consequences forFinancial Engineering andQuantitative Finance. I conclude by saying that to many ofus the field of finance seem to beintricately linked to modern portfo-lio theory. I showed that it does nothave to be so. And it does not takemuch to fix the problem.

© Copyright 2005 by N. N. Taleb.

W