Abstract
Financial markets exhibit ‘wild’ randomness patterns that fractal geometry can model more accurately than conventional theories assuming ‘mild’ bell-curve variations.
- Financial markets follow three states of randomness—mild, slow, and wild—analogous to the three states of matter, with conventional financial theory incorrectly assuming only the simplest ‘mild’ pattern like coin tosses
- Real prices ‘misbehave very badly’ compared to the mild random walk model
- A multifractal model of wild price variation can create a more reliable financial theory
- The fractal view is ‘alone in facing the high odds of catastrophic price changes’
Prelude
Mandelbrot is introduced as an independent scientific maverick whose fractal geometry discoveries challenge established financial theory with controversial but important insights about market risk.
- Mandelbrot’s wartime experience taught him independence when his father escaped a German prison camp by leaving the main group of fleeing prisoners to walk alone through the forest, avoiding a Stuka dive-bomber attack that killed the others on the main road
- ‘It was the way my father behaved throughout his life. He was an independent man—and so am I’
- This early lesson in independence made Mandelbrot ‘aguerri, or war-hardened’ and shaped his contrarian scientific approach
- Mandelbrot calls himself a maverick who moves ‘orthogonally—at right angles—to every fashion’
- Mandelbrot’s 1962 work on market behavior was called ’the most revolutionary development in the theory of speculative prices’ since 1900 by MIT economist Paul Cootner, who then criticized its implications for making existing statistical tools obsolete
- Cootner warned: ‘If he is right, almost all of our statistical tools are obsolete—least squares, spectral analysis, workable maximum-likelihood solutions, all our established sample theory, closed distributions’
- The economics establishment finds Mandelbrot ‘intriguing’ but ‘bewildering’ and has ‘grudgingly adopted many of his ideas (though often without giving him full credit)’
- His approach treats a stock exchange as a ‘black box’ system studied with physics-based conceptual and mathematical tools
- Fractal geometry studies roughness and irregularity in nature, finding ordered patterns where others see only troublesome disorder, with applications from coastlines to computer graphics
- ‘Roughness is the uncontrolled element in life’ found in jagged metal fractures, rugged coastlines, phone line static, and irregular stock charts
- Mandelbrot’s manifesto ‘The Fractal Geometry of Nature’ (1982) became a scientific bestseller with T-shirts and posters of the Mandelbrot Set
- The Wolf Prize citation said Mandelbrot ‘changed our view of nature’
Part One: The Old Way
Risk, Ruin, and Reward
Financial markets are extremely risky with seemingly impossible events like the 1987 crash occurring regularly, revealing fundamental flaws in standard probability models that dramatically underestimate real market dangers.
- The August 1998 Russian financial crisis produced three major market drops that standard theory calculated as having odds of one in 500 billion for occurring in the same month, yet they happened anyway
- August 4: Dow fell 3.5%, August 21: fell 4.4%, August 31: fell 6.8%
- The final August 31 collapse had odds of ‘one in 20 million’ according to standard models—‘an event that, if you traded daily for nearly 100,000 years, you would not expect to see even once’
- The odds of all three occurring together were ‘about one in 500 billion’
- The October 19, 1987 market crash had a probability ’less than one in 10^50’ according to standard financial models—‘odds so small they have no meaning’ and represent ‘a number outside the scale of nature’
- The Dow fell 29.2 percent in one day—’the worst day of trading in at least a century’
- Standard theory suggests you could ‘span the powers of ten from the smallest subatomic particle to the breadth of the measurable universe—and still never meet such a number’
- Similar ‘impossible’ events: 1997 Dow fell 7.7% (odds: one in 50 billion), July 2002 had three steep falls in seven days (odds: one in four trillion)
- Mandelbrot’s life has been ‘a study of risk’ from wartime survival as a Polish refugee with false identity papers to founding fractal geometry that reveals hidden order in seemingly random financial and natural phenomena
- Lived in occupied France ‘with a borrowed identity and touched-up ration coupons, masquerading (badly) as a simple country boy’
- Fractal geometry has helped model weather, river flows, brainwaves, seismic tremors, galaxy distribution, Internet traffic, and movie animation
- His research goal: ‘help people avoid losing as much money as they do, through foolhardy underestimation of the risk of ruin’
By the Toss of a Coin or the Flight of an Arrow?
Financial markets can be described using probability theory despite seeming deterministic, but require understanding different types of randomness beyond the familiar bell curve patterns.
- Financial markets appear deterministic with identifiable causes but should be analyzed probabilistically as ‘black boxes’ because we cannot know all the complex psychological and anticipatory factors that drive prices
- IBM stock rose ‘$1 a share because the company announced it signed more computer-service contracts than expected, and so 5,218 real people…ordered 12,542,300 real IBM shares with $768,016,733 in real cash’
- Yet anticipation makes economics uniquely unpredictable: ‘A stock price rises not because of good news from the company, but because the brightening outlook for the stock means investors anticipate it will rise further’
- ‘Anticipation is the stuff of dreams and vapors’ and far harder to analyze than quantum mechanics
- Random processes can generate surprising complexity and structure, as demonstrated by Kolmogorov’s insight that ‘chance phenomena, considered collectively and on a grand scale, create a non-random regularity’
- A simple coin-tossing game over 10,000 tosses creates complex patterns with long up-and-down cycles and clustered ‘zero-crossings’
- Feller’s 1950 diagram of cumulative winnings from coin tosses resembles stock charts enough that chartists might mistake it for real market data
- Such patterns emerge from pure chance yet appear to show deliberate trends and cycles
- Three distinct states of randomness exist—mild (bell curve), slow (intermediate), and wild (power law)—with financial markets exhibiting the dangerous ‘wild’ form rather than the mild randomness assumed by standard theory
- ‘Mild randomness, then, is like the solid phase of matter: low energies, stable structures, well-defined volume’
- ‘Wild randomness is like the gaseous phase of matter: high energies, no structure, no volume. No telling what it can do, where it will go’
- Computer error noise (‘1/f noise’) exemplifies wild variation that ‘cannot be predicted or prevented; it can only be accommodated, with error-correcting software’
Bachelier and His Legacy
Louis Bachelier’s 1900 doctoral thesis laid the foundations of modern financial theory by applying probability mathematics to market prices, though his work was initially dismissed and only rediscovered decades later.
- Bachelier’s 1900 Paris doctoral defense before mathematician Henri Poincaré introduced probability theory to financial markets but was graded only ‘mention honorable’ rather than the top ’très honorable’ that would have launched his academic career
- Poincaré noted ’the subject chosen by M. Bachelier is a bit distant from those usually treated by our candidates’
- The thesis ‘Théorie de la Spéculation’ studied bond trading on the Paris Bourse—‘a thriving den of capitalism’ rather than respectable academic mathematics
- Bachelier spent ’the next twenty-seven years battling for recognition and tenure from the French academic establishment’
- Bachelier was not merely wrong but ‘difficult’ with extraordinary immodesty, describing his work as ’the renewal of a science’ that ‘surpassed the great treatise of Laplace’ and was ‘absolutely unique’ with no precedent
- A 1921 ministry letter called him ‘a malcontent’ who got positions only through political intervention ‘over the objections of other mathematicians’
- Students noted ‘he did not complete equations on the blackboard without reference to his notes’ and once ‘forgot the Greek alphabet’ after insisting students know it perfectly
- An education official observed: ‘He is not an eagle’
- Bachelier created the ‘random walk’ model by viewing bond markets as ‘fair games’ where prices have equal probability of rising or falling, adapting heat diffusion equations to calculate probability distributions for price movements
- ‘The factors that determine activity on the Exchange are innumerable’ making ‘mathematical forecasting’ impossible, but ’the laws of probability for price variation that the market at that instant dictates’ can be studied
- He found ‘a strange and unexpected analogy between the ‘diffusion’ of heat through a substance and how a bond price wanders up and down’
- Testing his equations against real options data, he calculated 40% odds of profit for a forty-five-day option; actual data showed 39% profitability
- The Efficient Market Hypothesis emerged from Bachelier’s ideas, asserting that security prices fully reflect all relevant information, making it impossible to consistently beat the market through any predictable strategy
- Eugene Fama elaborated Bachelier’s ‘fair game’ concept: ‘In an ideal market, security prices fully reflect all relevant information’
- Three ways to potentially beat the market all fail: chart reading (trends get arbitraged away), fundamental analysis (other analysts spot problems too), and insider trading (market notices suspicious behavior)
- Samuelson’s advice: ‘They also serve who only sit and hold’ because ‘most portfolio decision makers should go out of business’
The House of Modern Finance
Modern financial theory rests on three key innovations—Markowitz’s portfolio theory, Sharpe’s asset pricing model, and Black-Scholes options formula—all built on Bachelier’s probabilistic foundations but widely used despite fundamental flaws.
- The Capital Asset Pricing Model (CAPM) is used by 73.5% of U.S. Fortune 500 CFOs and 77% of European companies for crucial financial decisions, despite professors noting ‘it is not clear that the CAPM is a very good model’
- CAPM helps value assets from stocks to factories by calculating expected returns as the risk-free rate plus beta times the market risk premium
- Both utility regulators citing CAPM to raise electricity prices in New York and cut them in Northern Ireland shows it’s ‘a double-edged sword’
- The same model appears in corporate finance decisions, regulatory rate-setting, and investment portfolio construction worldwide
- Harry Markowitz revolutionized investing by recognizing that investors care about both risk and reward, not just profit, leading to his mathematical portfolio theory that optimizes the trade-off between expected return and volatility
- His ’eureka’ moment came in 1950 reading investment books: ‘I had not taken any finance courses, nor did I own any securities’
- Shakespeare understood diversification: ‘My ventures are not in one bottom trusted, Nor to one place’
- Markowitz’s math showed ‘if you mix some stocks that flip tails with others that flip heads, you can lower the risk of your overall portfolio’ without sacrificing much profit
- Sharpe simplified Markowitz’s complex calculations by showing that if everyone follows optimal portfolio theory, there would be just one market portfolio, leading to his CAPM formula that reduces portfolio math from millions of calculations to thousands
- His insight: ‘If everybody in the market plays by Markowitz’s rules’ there would be ’not as many efficient portfolios as people in the market, but just one for all’
- CAPM reduces calculations from ‘3.9 million with Markowitz, you prune to 2,801 with Sharpe’ for the entire New York Stock Exchange
- After publishing in 1964, ‘I sat by the phone. The phone didn’t ring’ for months, not realizing ‘how long it took people to read journals’
- Black-Scholes options pricing formula emerged from Fischer Black and Myron Scholes’ counterintuitive insight that option values depend only on current volatility and contract terms, not predictions of final stock prices
- Black’s key breakthrough: ‘When valuing an option, you do not need to know how the game will end—that is, what the stock price will finally be when the option expires’
- They tested their theory by buying National General warrants their formula identified as cheap, but ’lost their shirts’ when a takeover reduced warrant values
- Their paper was rejected twice before publication in 1973, with Black grumbling ‘One reason these journals didn’t take the paper seriously was my non-academic return address’
The Case Against the Modern Theory of Finance
Modern financial theory’s fundamental assumptions about rational investors, normal price distributions, and market continuity are systematically contradicted by real market behavior and trading practices.
- Real foreign exchange traders at Citigroup London explicitly reject efficient market theory, with analysts stating ‘The biggest edge you can have is the private information of who’s buying what’ and ‘We do not believe the market is efficient’
- Citigroup moves ‘about one-ninth of all the world’s internationally traded dollars, yen, euros, pounds, zlotys, and pesos’ daily
- Options traders focus on the ‘volatility surface’ that ‘should be flat as a pancake’ according to Black-Scholes but is actually ‘a wild, complex shape’
- A simple trend-following strategy of buying when dollar-yen rates rise above their 69-day moving average yielded ‘7.97 percent average annual return’ over a decade
- The assumption that people are rational wealth-maximizers is refuted by behavioral economics research showing systematic cognitive biases, such as people taking sure gains but gambling to avoid sure losses in equivalent situations
- Behavioral economics studies ‘how people misinterpret information, how their emotions distort their decisions, and how they miscalculate probabilities’
- Example: Most people choose a sure $100 over flipping for $200/$0, but gamble on losing $200/$0 rather than pay sure $100—‘sublimely irrational’ despite mathematical equivalence
- Computer simulations with just two investor types (fundamentalists vs. chartists) spontaneously generate ‘price bubbles and crashes’ through nonlinear interactions
- Price changes demonstrably do not follow the bell curve, with kurtosis values far exceeding the normal level of 3—such as 43.36 for the S&P 500 from 1970-2001, indicating dangerously fat tails
- Daily Dow variations from 1916-2003 show ‘fifty-eight days when the Dow moved more than 3.4 percent’ versus theoretical prediction of 1,001 such days
- Index swings beyond 7% ‘should come once every 300,000 years’ but ’the twentieth century saw forty-eight such days’
- Even removing the 1987 crash, the S&P 500 still shows kurtosis of 7.17—‘an uncomfortably hot dish’ compared to the ‘bland standards of the statistical kitchen’
- Multiple ‘anomalies’ systematically violate CAPM predictions, including the P/E effect, small-firm-in-January effect, and market-to-book effect, leading Fama and French to declare their findings ‘a shot straight at the heart of the model’
- Low P/E ratio stocks consistently outperform high P/E stocks, contradicting the theory that ‘Only beta, the degree to which a stock does or does not move with the rest of the market, should matter’
- Small company stocks outperformed large companies by 4.3% annually, with the effect concentrated in January due to tax-loss selling
- Fama and French’s ‘beta-is-dead paper’ found two factors (P/E and market-to-book ratios) ‘could account for most of what differentiated the profitability of one stock from another’
Part Two: The New Way
Turbulent Markets: A Preview
Financial markets exhibit the same mathematical patterns as turbulent fluid flow, with intermittent bursts of high volatility followed by calm periods, requiring multifractal analysis to model accurately.
- Turbulent wind flow exhibits the same intermittent, clustered pattern as stock market volatility, with both showing ‘bump, bump, bump’ alternations between wild motion and quiet activity concentrated in time
- In wind tunnels, turbulence creates ’eddies form; and on those eddies yet more, smaller eddies form. A cascade of whirlpools, scaled from great to small, spontaneously appears’
- Stock market volatility from the 20th century shows identical patterns with ‘Peak activity is 1929–1934, and again in 1987’
- The same ‘discontinuities’, ‘intermittency’, and ‘concentration of major events in time’ characterize both wind and financial data
- The October 27, 1997 stock market crash exemplified financial turbulence with cascading selling that forced trading halts, resembling Leonardo da Vinci’s description of flood waters ‘growing turbid with the earth from ploughed fields, destroying the houses therein’
- The Dow lost 554.26 points (7.18%) with trading halted twice as ‘cascades of selling washed across the exchange’
- In the final twenty-four minutes, ‘prices plummeted at an average rate of 0.10 percent a minute, or 6 percent an hour’—meaning ‘American business was falling $100 million a second’
- The fastest action concentrated into just three isolated minutes, demonstrating extreme temporal clustering of volatility
- Multifractal models can generate realistic ‘forgeries’ of financial charts by using fractal generators with trading time deformation, as demonstrated in a cartoon construction that mimics real price behavior
- The construction starts with ‘a box, one unit wide by one unit tall’ and a ‘zigzag shape called generator’ that creates increasingly complex price patterns
- By ‘scrambling the pieces’ of the generator randomly, ’the new chart starts to look real’ with realistic price variations
- The multifractal model ’takes normal clock time, deforms it into a unique form of ’trading time,’ and then generates a price chart from it all’
Studies in Roughness: A Fractal Primer
Fractal geometry studies the mathematics of roughness and irregularity in nature, using scaling patterns and fractional dimensions to quantify complex structures from coastlines to financial charts.
- Traditional geometry ignores the roughness that dominates nature, leading Mandelbrot to observe that ‘Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line’
- Euclidean geometry studies ‘smoothness in exquisite detail’ with ’lines, planes, and spheres’ but these are ‘concepts in men’s minds and works, not in the irregularity and complexity of nature’
- Scientists used to view irregularities as ‘minor imperfections from an idealized shape—like the slight fuzz on an otherwise perfectly smooth peach skin’
- ‘Roughness is no mere imperfection from some ideal, not just a detail from a gross plan. It is of the very essence of many natural objects’
- Fractals exhibit self-similarity where parts echo the whole at different scales, like cauliflower florets that are ’each a cauliflower in miniature’ or tree branches that are ‘complete smaller trees’
- Eugène Delacroix noted ’the lungs are made of a number of small lungs, the liver of small livers’ and ‘pieces of rocks are similar to larger rocks’
- The construction process uses an ‘initiator’ (starting shape), a ‘generator’ (template pattern), and a ‘rule of recursion’ (repetition instruction)
- Fractal patterns can be ‘self-similar’ (scaling equally in all directions) or ‘self-affine’ (scaling differently horizontally vs. vertically)
- Fractal dimension provides a new tool to measure roughness, with the British coastline having dimension 1.25 (more complex than a line but less than a plane), while smooth South African shores measure only 1.02
- Political borders show measurement subjectivity: ‘Spanish authorities reckoned their border with Portugal to be 987 kilometers long, whereas the plucky Portuguese counted 1,214 kilometers’
- U.S. rivers have fractal dimensions of 1.2 in the East but 1.4 in the ‘wilder West’—fitting intuition about the ‘rugged Colorado’ versus ‘placid Charles’
- Lung airways have dimension ‘very close to 3’ because the lining ‘is so convoluted and folded in upon itself that it partakes something of a three-dimensional nature’
The Mystery of Cotton
Mandelbrot’s 1963 analysis of cotton prices revealed that financial markets follow power-law scaling patterns rather than normal distributions, solving a century-old mystery of price variation through three converging mathematical insights.
- The cotton price mystery began when Mandelbrot found the same convex, V-shaped diagram on Harvard economist Houthakker’s blackboard that he used for income distribution, leading to the question of why wealth distribution and commodity prices would show identical mathematical patterns
- Houthakker had been studying cotton prices ‘for a while, getting nowhere’ because ’there were too many big price jumps and falls’
- He complained: ‘I try to measure the volatility. It changes all the time. Everything changes. Nothing is constant. It’s a mess of the worst kind’
- Houthakker gave Mandelbrot ‘cardboard boxes of computer punch-cards’ with over a century of cotton price data
- Zipf’s power law for word frequencies provided the first clue, with his uncle dismissively giving Mandelbrot a discarded review saying ‘This is for you. That’s the kind of silly stuff you like’
- Zipf found that word frequency follows a power law: ‘Small quakes are common while big ones are rare, with a precise formula relating intensity to frequency’
- The power law applies to earthquakes, word usage, and ‘all manner of human behavior, organization, and anatomy—even in the size of sexual organs’
- Mandelbrot improved Zipf’s formula to ‘quantify the richness of someone’s vocabulary and give it a numerical grade: high grade, rich vocabulary; low grade, poor vocabulary’
- Pareto’s income distribution provided the second clue, showing that wealth follows a power law where ‘a very few people are outrageously rich, a small number are very rich, and the vast bulk of people are middling or poor’
- Pareto gathered data from ’tax records of Basel, Switzerland, from 1454 and from Augsburg, Germany, in 1471, 1498, and 1512’
- His formula shows ’the odds of making more than ten billion once you make more than one billion are the same as those of making more than ten million once you make more than one million’
- Income curves are not symmetric bell curves but ‘social arrows’—‘very fat at the bottom where the mass of men live, and very thin at the top where sit the wealthy elite’
- Lévy’s stable distributions provided the mathematical framework unifying power laws, with cotton price analysis revealing the same α = 1.7 scaling parameter that governs both wild price swings and long-term dependence
- Paul Lévy was ’near-inaudible’ with ’long, gray’ appearance who ‘was largely ignored by other French mathematicians’ despite being ‘one of the greatest probabilists’
- Cotton prices formed straight lines on log-log paper with slope -1.7, showing ‘prima facie evidence of a power-law behavior’
- The scaling worked across time: ‘A month looks like a day, one set of days like another’—confirming Wall Street folklore that ‘all price charts look alike’
Long Memory, from the Nile to the Marketplace
Harold Hurst’s discovery of long-term dependence in Nile floods reveals that financial markets have ‘memory’ where past events continue to influence present price movements over extended periods.
- Harold Hurst spent sixty-two years in Egypt solving the mystery of Nile floods by discovering that river variations follow a three-quarters power law rather than the square-root law assumed by dam engineers
- Hurst earned the nickname ‘Abu Nil, or Father Nile’ for his work on ‘century storage’ to manage the vast resource of a river with 4,160-mile length covering ‘10 percent of the land area of the entire African continent’
- Annual Nile discharge ranged wildly ‘from 151 billion cubic meters in the wet year of 1878–1879, to 42 billion cubic meters in the drought of 1913–1914’
- His formula showed dam height requirements scale by time^0.73 rather than time^0.5, meaning ’the dam should be higher than’ conventional engineering predicted
- Hurst found the same three-quarters scaling pattern in fifty-one different natural phenomena worldwide, from ‘Lake Huron and the Truckee River near Lake Tahoe’ to ’tree rings in Flagstaff pines and Sequoia—even sunspot numbers’
- The pattern appeared in ‘rainfall measurements from Adelaide, Australia, to Washington, D.C.’ and ’temperature readings from St. Louis to Helsinki’
- Skeptical engineer F. A. Sharman sarcastically noted that claiming ‘a common thread from tree rings to sun spots to mud layers’ was ‘a sensational step towards finding the single universal law of nature’
- Despite criticism, Hurst’s formula works: New York needs reservoirs ‘105 inches, or two and a half years’ supply’ for century-scale drought protection
- Financial markets exhibit long memory where IBM’s 1982 decision to work with Intel and Microsoft ‘continues to reverberate today in IBM’s stock price’ thirty years later, and the 1911 Standard Oil breakup still affects descendant companies
- ‘The dependence there is about thirty years long’ for IBM-Intel-Microsoft relationships that shaped entire industries
- John D. Rockefeller’s Standard Oil Trust breakup ‘in 1911 continues to affect its surviving children today, ExxonMobil, ConocoPhillips, ChevronTexaco, and BP Amoco’
- ‘In such a world, it is common sense that events in the distant past continue to echo in the present’ through complex economic interdependencies
- The Hurst exponent H measures dependence strength: H = 0.5 indicates independence (Brownian motion), H > 0.5 shows persistence (trends continue), and H < 0.5 reveals anti-persistence (trends reverse)
- ‘Interest rates on loans from banks to brokers’ had H = 0.7 showing ’long, persistent trends’ while ‘Wheat and U.K. bonds were about 0.5: independent’
- Edgar Peters found high-tech stocks had high H values: ‘Apple, 0.75, Xerox, 0.73, and IBM, 0.72’ while boring stocks were lower: ‘Anheuser-Busch, 0.64, Texas State Utilities, 0.54’
- The field remains ‘more intricate than any one simplistic test could resolve’ with competing methodologies yielding different H estimates for the same markets
Noah, Joseph, and Market Bubbles
Financial markets exhibit two forms of wildness—the Noah Effect (abrupt price jumps) and Joseph Effect (long-term trends)—which interact to create rational bubbles through overshooting and crash dynamics.
- The Noah Effect represents abrupt market changes like the 1987 crash that arrive ‘without warning or convincing reason’ and seem ’like the end of the financial world,’ analogous to the biblical flood that was ‘catastrophic, but transient’
- Market crashes follow ‘a hierarchy of turbulence, a pattern that scales up and down with time’ with varying severity of financial storms
- Even ‘a great bank or brokerage house can seem like a little boat in a big storm’ during extreme market events
- The Noah Effect manifests as ‘discontinuity’ where prices jump rather than move smoothly
- The Joseph Effect represents almost-cycles and long memory where ‘a big 3 percent change in IBM’s stock one day might precede a 2 percent jump another day’ as past price moves continue to echo in future trading
- Like Joseph’s interpretation of Pharaoh’s dream about ‘seven fat cattle’ followed by ‘seven lean kine,’ markets show persistent periods of growth or decline
- Joseph became ’the first international arbitrageur’ by stockpiling grain during good years to profit during the predicted famine
- The effect appears as ’long-range dependence in an otherwise random process—or, put another way, a long-term memory’
- Investment bubbles arise rationally from the interaction of Joseph and Noah effects, as demonstrated by Cisco Systems whose stock soared when investors extrapolated earnings growth trends but crashed when growth expectations proved unsustainable
- Cisco managed ’extraordinary 53 percent’ annual revenue growth from 1995-2000 leading to stock gains of ‘101 percent a year throughout the 1990s’
- Credit Suisse First Boston called Cisco ‘Potentially The First Trillion Dollar Market Cap Company’ at the bubble peak
- The crash came when ’earnings flattened’ in 2000 and Cisco ‘reported its first quarterly loss’ in 2001, showing how ‘overshooting and crash’ patterns repeat ‘incessantly’
- The rescaled range (R/S) statistical test separates Noah and Joseph effects by comparing data before and after reshuffling, since ‘The Joseph Effect depends on the precise order of events, while the Noah Effect depends on the relative size of each event’
- The test measures whether ‘whatever dependence was originally present must have been negligible’ or if ’the precise sequence must have been important’
- Under certain circumstances ‘H is simply equal to 1/α’ creating ‘a dual relationship’ between long memory and fat tails
- The two effects can interact: ‘an ‘almost-trend’ emerging in a stock price’ will ’eventually break up’ but ‘can do so rapidly. A sudden lurch downward, perhaps. A discontinuity’
The Multifractal Nature of Trading Time
Multifractal models capture financial market behavior by deforming time itself—expanding during dramatic price changes and contracting during quiet periods—to generate realistic price charts.
- Trading time operates differently from clock time, expanding during high volatility when ‘fortunes are won or lost’ and ‘Time flies,’ while contracting during quiet periods when ‘Time hangs heavy’ and traders ‘might as well go for a long lunch’
- Currency traders experience subjective time changes: fast markets with ‘Scores of news items flitting across the electronic crawl’ versus slow times with ‘No news, only tired reports’
- The multifractal model ’takes normal clock time, deforms it into a unique form of ’trading time,’ and then generates a price chart from it all’
- ‘Time does not run in a straight line, like the markings on a wooden ruler. It stretches and shrinks, as if the ruler were made of balloon rubber’
- Multifractals extend beyond simple fractals by having ‘more than one scaling ratio in the same object—some parts of the object shrink quickly, others slowly,’ like how gold ore clusters unevenly on Earth’s surface
- Unlike simple fractals that are ‘black and white,’ multifractals involve ‘halftones, shades of gray’ because ’the world is not black and white’
- Examples include ’the way gold ore clusters here and there on the surface of the earth’ and ’the way the velocity of the wind on a stormy day comes ‘intermittently,’ in clusters of high gusts’
- Multifractals manifest both Noah Effect (sharp price changes) and Joseph Effect (long-term dependence) in ‘a Noah Effect that can vary over a broad range of differing degrees’
- The multifractal model uses a ‘Baby Theorem’ where two parent generators—a ‘father’ that transforms clock time into trading time and a ‘mother’ that generates price movements—combine to create realistic financial charts
- The father generator ’takes clock time and transforms it into trading time’ while the mother ’takes clock time and changes it into a price’
- Combined, ’the baby takes the father’s trading-time and converts it into a price by the rules the mother provides’
- Like genetics, ‘Each parent contributes one half of a chromosome to the baby’ creating fractal charts with both wild jumps and long-term memory
- Testing on dollar-Deutschemark exchange data from 1992-1996 showed the multifractal model accurately captures market scaling across time periods from ’two hours to 180 days—an unusually long zone of regularity’
- Analysis used ‘1,472,241 prices in all’ from tick-by-tick records gathered by ‘Zurich consulting firm, Olsen & Associates’
- The model shows ‘crossovers’ at boundaries: below two hours ‘market microstructure’ dominates, above 180 days ’the Noah Effect is fading’
- Tests on other assets confirmed multifractal behavior: ‘Archer Daniel Midlands, Lockheed, Motorola, and UAL were textbook multifractals’
Part Three: The Way Ahead
Ten Heresies of Finance
Mandelbrot challenges ten fundamental assumptions of modern finance, arguing that markets are turbulent, discontinuous, and far riskier than conventional theory suggests.
- Markets are turbulent systems with the same mathematical patterns as fluid dynamics, exhibiting ’the tell-tale traces of turbulence’ in price charts through scaling, long-term dependence, and discontinuous jumps
- Like turbulent submarine recordings from Puget Sound, market data shows ‘stretches of high-frequency noise interrupted by low patches’ that scale fractally
- Turbulence sources include scaling in natural phenomena: ‘oil fields,’ ‘gold, uranium, and diamond mines in South Africa,’ and ‘storms and earthquakes scale’
- ‘Every event, no matter how remote or long ago, echoes across all other events’ creating the long-term dependence characteristic of turbulent systems
- The Equity Premium Puzzle arises because ‘real investors know better than the economists’ and ‘instinctively realize that the market is very, very risky, riskier than the standard models say,’ justifying higher stock returns as compensation
- Economists Rajnish Mehra and Edward Prescott found stock premiums of ‘4.1 percent and 8.4 percent’ over T-bills, which ‘conventional formulae say the risk premium should not exceed 1 percent or so’
- Japanese households keep ‘53 percent of their financial assets in cash, and barely 8 percent in shares’ while Europeans keep ‘28 percent in cash, 13 percent in shares’
- People focus on ’extremes of profit or loss that matter most’ rather than averages, since ‘Just one out-of-the-average year of losing more than a third of capital…would justifiably scare even the boldest investors’
- Market timing matters because ‘big gains and losses concentrate into small packages of time,’ with nearly half the dollar’s decline against the yen from 1986-2003 occurring on just ten out of 4,695 trading days
- ‘46 percent of the damage to dollar investors happened on 0.21 percent of the days’ showing extreme temporal concentration
- ‘40 percent of the positive returns from the Standard & Poor’s 500 index came during ten days—about 0.5 percent of the time’ in the 1980s
- George Soros ‘famously profited about $2 billion by betting against the British pound’ in just ’two turbulent weeks in 1992’
- Alexander’s ‘filter method’ trading strategy failed because prices jump discontinuously rather than move smoothly, causing investors to buy at 5.5% rather than the intended 5% trigger level, destroying theoretical profits
- IBM president Albert Williams asked Mandelbrot to investigate Alexander’s claim of ‘36.8 percent a year’ profits from systematic trend-following
- ‘It made the difference between a 36.8 percent profit and a loss of as much as 90 percent of the investor’s capital’
- Alexander retracted, admitting ‘The big, bold profits of Paper 1 must be replaced with rather puny ones’ and ‘I must admit that the fun has gone out of it somehow’
In the Lab
Current fractal finance experiments include Olsen’s currency trading platform, Bouchaud’s statistical arbitrage funds, and various academic research projects, though practical applications remain limited.
- Richard Olsen’s Oanda.com operates as a ’test reactor’ for fractal finance theory, providing a currency trading platform with 10,000 customers trading $1 billion daily while generating unique market research data
- Olsen calls his approach ‘quantum theory for finance’ studying customer behavior with complete transaction visibility unlike other firms
- His ‘heterogeneous markets’ theory sees trading as ‘small explosions’ where ‘quotes stutter’ rather than smooth molecular collisions
- Olsen Investment Corp. funds returned between ‘3.15 percent’ and ‘21.05 percent’ in 2003 using computerized quasi-fractal models
- Jean-Philippe Bouchaud’s Capital Fund Management uses multifractal ’tail chiseling’ for risk management while running €725 million in hedge funds employing statistical arbitrage strategies
- Their Ventus fund gained ‘28.1 percent’ in 2002 when ’the market overall had fallen by a third’ but only ‘3.32 percent’ in 2003, showing ‘With statistical arbitrage, there are ups and downs’
- Tail chiseling ‘minimizes the odds of too many of the assets in a portfolio crashing at the same time’ using power-law assumptions
- They focus on crash protection since ‘it is not small declines that wipe an investor out, it is the crashes’
- Four major practical problems need resolution before fractal finance can be widely applied: analyzing investments, building portfolios, valuing options, and managing risk
- Investment analysis suffers from ‘primitive state of natural history three centuries ago’ with limited tools that ‘frequently confound species’
- Options valuation shows ‘wide error range’ with Black-Scholes producing volatility estimates that should be ‘just one flat line’ but instead show ‘Rococo structure’ patterns
- Risk management using Value at Risk ‘consistently understated the basic market parameter, β’ by an average of ‘6 percent’ for Paris Bourse stocks
- Mandelbrot challenges financial leaders to allocate just ‘5 percent’ of the $432.5 million Wall Street fraud settlement to fund fundamental market research, arguing ‘Is not understanding the market at least as important to the economy as understanding solid-state physics is to IBM?’
- The challenge targets ‘Alan Greenspan, Eliot Spitzer, and William Donaldson—Federal Reserve chairman, New York attorney general, and SEC chairman’
- ‘If we can map the human genome, why can we not map how a man loses his livelihood?’
- The 1953 Dutch flood analogy: after flooding killed 1,800 people, ’the pragmatic Dutch did not waste time arguing about the math. They cleaned up the damage and rebuilt the dikes higher and stronger’