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The Fractalist
** by
Benoit Mandelbrot

Albert Einstein, Benoit Mandelbrot, Brownian motion, business cycle, Claude Shannon: information theory, discrete time, double helix, Georg Cantor, Henri Poincaré, Honoré de Balzac, illegal immigration, Isaac Newton, iterative process, Johannes Kepler, John von Neumann, linear programming, Louis Bachelier, Louis Blériot, Louis Pasteur, mandelbrot fractal, New Journalism, Norbert Wiener, Olbers’ paradox, Paul Lévy, Richard Feynman, statistical model, urban renewal, Vilfredo Pareto

Admiral Brard Recommends Caltech In the real world of Paris and Carva in 1947, the most obvious person to ask for advice was neither Szolem nor Paul Lévy, but rather the professor of applied mathematics, Roger Brard (1907–77). A naval engineer, he held the rank of admiral and headed a large bassin des carènes—the lovely old-fashioned term for water tunnel. He had no office at Carva, so we met in his car. I still recall the make: Matford. A sign of the times, there were so few cars in town that he always found a parking space near the school. In the 1930s, when the lovely SS Normandie, touted by Popular Mechanics as the latest “giant of the sea,” took a trial cruise, a resonance was revealed between the hull and the propellers; Brard helped with the diagnosis and the cure. Although his numerous papers in probability theory are no longer quoted, Carva viewed him as very practical (contrary to Paul Lévy) and put him in charge of all topics in applied mathematics.

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At that point he was overthrown, and the political situation resumed the course that soon returned de Gaulle to power. Close to home, Mendès took every opportunity to promote the sciences and bemoan their weakened state in France. He was widely heard, which may be why—before Britain or Germany—a wild and uncontrolled enrollment surge hit the French universities and the academic market flipped from puny to wide open. In a short time, this would have a major impact on me. Paul Lévy Getting to know Paul Lévy was one of my few academic accomplishments in 1954–55. He never had a formal disciple, I never had a formal teacher, and I never thought of becoming his clone or shadow. Yet much of probability theory has long consisted of filling logical gaps in his works, and in a real, though indirect, fashion, he was the teacher of several members of his family, and also mine. He documented his life, thoughts, and opinions at length in a book well worth reading because of his lack of any attempt to appear better or worse than he was.

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Years later, on a kind of “wall of honor” in his Paris study, Szolem hung a photograph that his mentor, Jacques Hadamard, had dedicated to him as his “spiritual son.” Hadamard had spent most of his working life as professor at the Collège de France, an ancient and famed postgraduate institution. In 1937, Szolem succeeded him in that chair. In 1973, Szolem was elected to the Académie des Sciences to a chair held by the great scientist Henri Poincaré, then for a long time by Hadamard, followed briefly by Paul Lévy (to be introduced in due time). A record of Szolem’s brilliance and the reasons for his being quickly and widely accepted is found in a letter dated August 28, 1924, from Paul Jouhandeau to Max Jacob—one very well-known French literary figure to another. The words that follow fit the look of a more or less contemporary photograph. [I met a] mathematician of genius who revealed mathematics to me.

pages: 364 words: 101,286

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The Misbehavior of Markets: A Fractal View of Financial Turbulence
** by
Benoit Mandelbrot,
Richard L. Hudson

Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John Meriwether, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, Myron Scholes, new economy, paper trading, passive investing, Paul Lévy, Paul Samuelson, plutocrats, Plutocrats, price mechanism, quantitative trading / quantitative ﬁnance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, Vilfredo Pareto, volatility smile

His rival for the chair, Georges Cerf, was a brilliant young mathematician with all the right connections in Paris and an ally in Dijon, Maurice Gevrey, a sitting math professor. Gevrey appears to have taken a passionate dislike to Bachelier. Scouring the latter’s work, Gevrey soon spotted a glaring mathematical error. When the academic committee met to decide the professorship, Gevrey brandished a letter from the eminent French probabilist, Paul Lévy in Paris, confirming the fault. Result: “Bachelier was blackballed,” as Lévy ruefully recalled years later, in correspondence with me. By then, Lévy regretted the incident. He had read only the passage highlighted by Gevrey rather than the entire treatise; and in the full context of Bachelier’s work the error appears benign. Lévy later apologized to Bachelier that “an impression, produced by a single initial error, should have kept me from going on with my reading of a work in which there were so many interesting ideas.”

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And he let fly a fusillade of more than four hundred vitriolic words at Lévy. He called Lévy’s critique “violent and unjustified” and based on total ignorance of his work. The Parisian, who had just finished a book on probability, had not even bothered “opening my book” on the subject before writing his own, Bachelier complained. He concluded with an insinuation typical of the time: “Without doubt, it is inconceivable that M. Paul Lévy had wanted, by a sort of last-minute trick, to favor un coreligionnaire.” Lévy was a Jew. Given Bachelier’s temper, it is remarkable that he ever won the security of a professorial chair—which he ultimately did, at Besançon. But that was twenty-seven years after his doctoral thesis, the work for which he is so well remembered today. The Coin-Tossing View of Finance The Bourse, the bustling Paris exchange, was at that time a world capital of bond trading.

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Denying them would create an additional and totally unnecessary risk. Same formula, different result because of one change in the parameters. It is all exquisitely versatile. Clue No. 3: The Laws of Exceptional Chance The last hint in the cotton mystery goes back again to my student days. After the war, I was at the École Polytechnique, one of France’s “Ivy League,” the Grandes Écoles. One of my professors was Paul Lévy, a well-known mathematician, and the same man who had unintentionally played so decisive a role in Bachelier’s life story. Lévy was independently wealthy, the scion of a Jewish merchant and academic family. To students at the back of his lecture hall—as I was—he was near-inaudible and his long, gray, and well-groomed figure bore an odd resemblance to the somewhat peculiar way he had of tracing the long “?”

pages: 338 words: 106,936

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The Physics of Wall Street: A Brief History of Predicting the Unpredictable
** by
James Owen Weatherall

Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business cycle, butterfly effect, buy and hold, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative ﬁnance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, Vilfredo Pareto, volatility smile

An event that occurred toward the end of Bachelier’s career, in 1926 (the year before he finally earned his permanent position), cast a pall over his final years as a teacher and may explain why he stopped publishing. That year, Bachelier applied for a permanent position at Dijon, where he had been teaching for several years. One of his colleagues, in reviewing his work, became confused by Bachelier’s notation. Believing he had found an error, he sent the document to Paul Lévy, a younger but more famous French probability theorist. Lévy, examining only the page on which the error purportedly appeared, confirmed the Dijon mathematician’s suspicions. Bachelier was blacklisted from Dijon. Later, he learned of Lévy’s part in the fiasco and became enraged. He circulated a letter claiming that Lévy had intentionally blocked his career without understanding his work. Bachelier earned his position at Besançon a year later, but the damage had been done and questions concerning the legitimacy of much of Bachelier’s work remained.

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He quickly confirmed Houthakker’s most troubling findings: it appeared that there was no “average” rate of return. The prices looked random, but they weren’t explained by the standard statistical tools or Bachelier’s and Osborne’s theories. Something weird was going on. Mandelbrot had seen unusual distributions before. In addition to studying Zipf’s and Pareto’s work, he was familiar with a third kind of distribution, discovered by one of his professors in Paris, Paul Lévy. It was Lévy who, upon reading a small section of one of Bachelier’s papers, concluded that Bachelier’s work was plagued with errors. Much later, Lévy would recognize his own mistake and apologize to Bachelier. Part of what made Lévy return to Bachelier’s work was a renewed interest in random walk processes and probability distributions. Ironically, this later work of Lévy’s received far less attention than his earlier work, leaving Lévy alienated and obscure at the twilight of his career.

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As described below, Mandelbrot would later argue that the distributions of rates of return for financial markets do have finite means, but not variances. However, it can often be difficult to calculate the mean for a Lévy-stable distribution — in cases where variance is undefined, the average value calculated from any finite data set takes a long time to converge to the mean — which accounts for why Mandelbrot and Houthakker originally believed that the mean did not exist. “. . . discovered by one of his professors in Paris, Paul Lévy”: Mandelbrot offers some biographical background on Lévy in Mandelbrot (1982) and describes his interactions with him in Mandelbrot and Hudson (2004). “. . . a class of probability distribution now called Lévy-stable distributions”: They are also called α-stable distributions. Throughout the text (and in Mandelbrot’s popular writing), “wildness” is code for “α < 2.” For a Lévy-stable distribution with 1 < α < 2, the mean is defined, but the variance is not; if α ≤ 1, neither mean nor variance is defined.

pages: 695 words: 194,693

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Money Changes Everything: How Finance Made Civilization Possible
** by
William N. Goetzmann

Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, banking crisis, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bretton Woods, Brownian motion, business cycle, capital asset pricing model, Cass Sunstein, collective bargaining, colonial exploitation, compound rate of return, conceptual framework, corporate governance, Credit Default Swap, David Ricardo: comparative advantage, debt deflation, delayed gratification, Detroit bankruptcy, disintermediation, diversified portfolio, double entry bookkeeping, Edmond Halley, en.wikipedia.org, equity premium, financial independence, financial innovation, financial intermediation, fixed income, frictionless, frictionless market, full employment, high net worth, income inequality, index fund, invention of the steam engine, invention of writing, invisible hand, James Watt: steam engine, joint-stock company, joint-stock limited liability company, laissez-faire capitalism, Louis Bachelier, mandelbrot fractal, market bubble, means of production, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, new economy, passive investing, Paul Lévy, Ponzi scheme, price stability, principal–agent problem, profit maximization, profit motive, quantitative trading / quantitative ﬁnance, random walk, Richard Thaler, Robert Shiller, Robert Shiller, shareholder value, short selling, South Sea Bubble, sovereign wealth fund, spice trade, stochastic process, the scientific method, The Wealth of Nations by Adam Smith, Thomas Malthus, time value of money, too big to fail, trade liberalization, trade route, transatlantic slave trade, tulip mania, wage slave

The “high priest” of non-normality before Nassim Taleb ever started to trade or write about extreme events was Benoit Mandelbrot, the creator of fractal geometry, a mathematician who both carried the mantle of French mathematical finance and who also believed he had discovered its fatal flaw. Mandelbrot was a student of Paul Lévy’s—the son of the man who gave Bachelier bad marks at his examination at the École Polytechnique in 1900. Lévy’s research focused on “stochastic processes”: mathematical models that describe the behavior of some variable through time. For example, we saw in Chapter 15 that Jules Regnault proposed and tested a stochastic process that varied randomly, which resulted in a rule about risk increasing with the square root of time. Likewise, Louis Bachelier more formally developed a random-walk stochastic process. Paul Lévy formalized these prior random walk models into a very general family of stochastic processes referred to as Lévy processes. Brownian motion was just one process in the family of Lévy processes—and perhaps the best behaved of them.

pages: 374 words: 114,600

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The Quants
** by
Scott Patterson

Albert Einstein, asset allocation, automated trading system, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Bernie Sanders, Black Swan, Black-Scholes formula, Blythe Masters, Bonfire of the Vanities, Brownian motion, buttonwood tree, buy and hold, buy low sell high, capital asset pricing model, centralized clearinghouse, Claude Shannon: information theory, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Donald Trump, Doomsday Clock, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Gordon Gekko, greed is good, Haight Ashbury, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, index fund, invention of the telegraph, invisible hand, Isaac Newton, job automation, John Meriwether, John Nash: game theory, Kickstarter, law of one price, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, merger arbitrage, money market fund, Myron Scholes, NetJets, new economy, offshore financial centre, old-boy network, Paul Lévy, Paul Samuelson, Ponzi scheme, quantitative hedge fund, quantitative trading / quantitative ﬁnance, race to the bottom, random walk, Renaissance Technologies, risk-adjusted returns, Robert Mercer, Rod Stewart played at Stephen Schwarzman birthday party, Ronald Reagan, Sergey Aleynikov, short selling, South Sea Bubble, speech recognition, statistical arbitrage, The Chicago School, The Great Moderation, The Predators' Ball, too big to fail, transaction costs, value at risk, volatility smile, yield curve, éminence grise

While praising Louis Bachelier, a personal hero of Mandelbrot’s, the mathematician asserted that “the empirical distributions of price changes are usually too ‘peaked’ relative to samples” from standard distributions. The reason: “Large price changes are much more frequent than predicted.” Mandelbrot proposed an alternative method to measure the erratic behavior of prices, one that borrows a mathematical technique devised by the French mathematician Paul Lévy, whom he’d studied under in Paris. Lévy investigated distributions in which a single sample radically changes the curve. The average of the heights of 1,000 people won’t change very much as a result of the height of the 1,001st person. But a so-called Lévy distribution can be thrown off by a single wild shift in the sample. Mandelbrot uses the example of a blindfolded archer: 1,000 shots may fall close to the target, but the 1,001st shot, by happenstance, may fall very wide of the mark, radically changing the overall distribution.

pages: 461 words: 128,421

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The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street
** by
Justin Fox

activist fund / activist shareholder / activist investor, Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, bank run, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Bretton Woods, Brownian motion, business cycle, buy and hold, capital asset pricing model, card file, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discovery of the americas, diversification, diversified portfolio, Edward Glaeser, Edward Thorp, endowment effect, Eugene Fama: efficient market hypothesis, experimental economics, financial innovation, Financial Instability Hypothesis, fixed income, floating exchange rates, George Akerlof, Henri Poincaré, Hyman Minsky, implied volatility, impulse control, index arbitrage, index card, index fund, information asymmetry, invisible hand, Isaac Newton, John Meriwether, John Nash: game theory, John von Neumann, joint-stock company, Joseph Schumpeter, Kenneth Arrow, libertarian paternalism, linear programming, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market design, Myron Scholes, New Journalism, Nikolai Kondratiev, Paul Lévy, Paul Samuelson, pension reform, performance metric, Ponzi scheme, prediction markets, pushing on a string, quantitative trading / quantitative ﬁnance, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, road to serfdom, Robert Bork, Robert Shiller, Robert Shiller, rolodex, Ronald Reagan, shareholder value, Sharpe ratio, short selling, side project, Silicon Valley, Social Responsibility of Business Is to Increase Its Profits, South Sea Bubble, statistical model, stocks for the long run, The Chicago School, The Myth of the Rational Market, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Kuhn: the structure of scientific revolutions, Thomas L Friedman, Thorstein Veblen, Tobin tax, transaction costs, tulip mania, value at risk, Vanguard fund, Vilfredo Pareto, volatility smile, Yogi Berra

Bachelier went on to a modestly successful career as a math professor, and published a well-received popular treatise on games, chance, and risk (Le jeu, la chance et le hasard). When he died in 1946, one year before Irving Fisher, no one on the trading floor was making use of his ideas. His colleagues, meanwhile, were nonplussed by his interest in markets. On a bibliography of Bachelier’s writings found in the files of the great French mathematician Paul Lévy is scrawled the complaint, “Too much on finance!”8 IRVING FISHER WAS ABLE TO go where Bachelier did not because he had more than just mathematics and probability theory at his disposal. He was an economist. He was able to go where other economists did not because he, unlike all but a handful of them at the time, was a mathematician. And he was able to do something tangible with his insights because he was a wealthy resident of a country where, in the early decades of the twentieth century, financial markets were just beginning to grow into the vast bazaars that would steer the economy for the rest of the century and beyond.

pages: 733 words: 179,391

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Adaptive Markets: Financial Evolution at the Speed of Thought
** by
Andrew W. Lo

"Robert Solow", Albert Einstein, Alfred Russel Wallace, algorithmic trading, Andrei Shleifer, Arthur Eddington, Asian financial crisis, asset allocation, asset-backed security, backtesting, bank run, barriers to entry, Berlin Wall, Bernie Madoff, bitcoin, Bonfire of the Vanities, bonus culture, break the buck, Brownian motion, business cycle, business process, butterfly effect, buy and hold, capital asset pricing model, Captain Sullenberger Hudson, Carmen Reinhart, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, corporate governance, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, cryptocurrency, Daniel Kahneman / Amos Tversky, delayed gratification, Diane Coyle, diversification, diversified portfolio, double helix, easy for humans, difficult for computers, Ernest Rutherford, Eugene Fama: efficient market hypothesis, experimental economics, experimental subject, Fall of the Berlin Wall, financial deregulation, financial innovation, financial intermediation, fixed income, Flash crash, Fractional reserve banking, framing effect, Gordon Gekko, greed is good, Hans Rosling, Henri Poincaré, high net worth, housing crisis, incomplete markets, index fund, interest rate derivative, invention of the telegraph, Isaac Newton, James Watt: steam engine, job satisfaction, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Meriwether, Joseph Schumpeter, Kenneth Rogoff, London Interbank Offered Rate, Long Term Capital Management, longitudinal study, loss aversion, Louis Pasteur, mandelbrot fractal, margin call, Mark Zuckerberg, market fundamentalism, martingale, merger arbitrage, meta analysis, meta-analysis, Milgram experiment, money market fund, moral hazard, Myron Scholes, Nick Leeson, old-boy network, out of africa, p-value, paper trading, passive investing, Paul Lévy, Paul Samuelson, Ponzi scheme, predatory finance, prediction markets, price discovery process, profit maximization, profit motive, quantitative hedge fund, quantitative trading / quantitative ﬁnance, RAND corporation, random walk, randomized controlled trial, Renaissance Technologies, Richard Feynman, Richard Feynman: Challenger O-ring, risk tolerance, Robert Shiller, Robert Shiller, Sam Peltzman, Shai Danziger, short selling, sovereign wealth fund, Stanford marshmallow experiment, Stanford prison experiment, statistical arbitrage, Steven Pinker, stochastic process, stocks for the long run, survivorship bias, Thales and the olive presses, The Great Moderation, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, theory of mind, Thomas Malthus, Thorstein Veblen, Tobin tax, too big to fail, transaction costs, Triangle Shirtwaist Factory, ultimatum game, Upton Sinclair, US Airways Flight 1549, Walter Mischel, Watson beat the top human players on Jeopardy!, WikiLeaks, Yogi Berra, zero-sum game

In other words, beating the market was mathematically impossible. Unfortunately, Bachelier’s work languished for years, and the reasons for this neglect are unclear. His thesis, Théorie de la Spéculation, was eventually published in 1914. It was commended by the French scientific establishment, but not extravagantly so. Bachelier was denied tenure at the University of Dijon due to a negative letter of recommendation from the famous probability theorist Paul Lévy, after which Bachelier spent the rest of his career at a small teaching college in the town of Besançon in eastern France.16 Most likely, Bachelier’s work slipped through the cracks because it was too avant-garde for the times—too much like finance for the physicists, and too much like physics for the financiers. The story of the rediscovery of Bachelier’s work is almost too implausible to be true.