From Hammurabi's disaster clause to Brownian motion, Black–Scholes and the race for microseconds
The two core activities of finance, trading time and trading risk, were there together from the start. Insurance was built into loan contracts, and states sold the right to collect their taxes.
If the harvest fails, that year's interest is waived.
The Code of Hammurabi not only capped interest on grain and silver loans but also waived a year's interest if storms or drought ruined the harvest. A kind of insurance, sharing the risk of natural disaster between creditor and debtor, was built into the loan contract.
If the ship sinks, the debt is forgiven.
Athenian merchants borrowed against their ships and cargo to go trading. If they returned safely they repaid the principal plus a steep 20–30% per voyage; if the ship was wrecked, they owed nothing. The high interest in effect included an insurance premium. Demosthenes's courtroom speeches preserve several lawsuits over such contracts.
Shares traded in the Forum.
The Roman Republic auctioned the collection of provincial taxes to private companies (publicani), and records suggest shares in these companies were traded near the Forum. Whether this was a true stock market is debated.
The Christian and Islamic bans on interest paradoxically became a driving force for financial innovation, because people had to find ways to get the same effect without calling it interest.
Not a loan, but a joint investment.
In Venice and Genoa, contracts spread in which a financier supplied the money and a merchant handled the voyage and trade, splitting the profit in fixed shares. Since both shared profit and risk, it did not count as interest. It was a prototype of the share, with almost the same structure as the mudaraba of Islamic finance, from which it very likely came.
Future money in today's terms.
Fibonacci's Liber Abaci contains problems that convert a series of future payments into their value today in order to compare them. On this basis the financial historian William Goetzmann argues that Fibonacci was the first to treat present value systematically. It is the beginning of "discounted future cash flows," the starting point of all financial pricing today.
The market rates the state's credit every day.
Venice forced wealthy citizens to lend it money, paid 5% a year in interest, and let them sell the bonds to others. It was the first government bond market, where the state's debt traded on a market. Bond prices rose when Venice won wars and fell when it lost.
Banking families steered state finances, and state defaults ruined financiers. The first building designed as an exchange appeared.
A bank run by the holders of government debt.
The Bank of Saint George, run directly by holders of Genoa's government bonds, was founded. The institution managing the state's debt became as powerful as the state itself.
A financier makes an emperor.
Jakob Fugger of Augsburg lent Charles V of the Habsburgs the money to bribe the electors and win the election for Holy Roman Emperor. But when the Spanish crown declared default several times, the Fugger family too eventually suffered heavy losses. The history of sovereign default begins here as well.
A building built as an exchange from the start.
In Antwerp, in today's Belgium, a building was designed from the outset as an exchange for trading goods and bills.
With the Dutch East India Company came the modern joint-stock company and stock exchange, and short selling, options and futures appeared almost at once. At the same time, Pascal and Fermat began probability theory, and finance and mathematics met in earnest for the first time.
Permanent capital, free transfer, limited liability.
Earlier trading companies dissolved after a single voyage, but the East India Company was permanent. Investors' money stayed in the company, but they could sell their shares at any time to get their money back, and they were liable only up to what they had invested. Maids and craftsmen bought shares. This structure is the foundation of every listed company today. The Amsterdam Stock Exchange arose to trade its shares.
Short selling appears as soon as the exchange does.
Isaac Le Maire, a founding investor in the East India Company, fell out with it and, with other investors, sold its shares short on a large scale and spread rumors to drive down the price. It was the first recorded organized short-selling attack, and the next year the Netherlands passed the first restrictions on short selling. Options and forward contracts already existed in Amsterdam by then.
A single bulb for the price of a house.
Tulip bulbs traded for the price of a house and then suddenly crashed. Famous as the first speculative bubble, recent research suggests the number of participants and the damage have been exaggerated. Interestingly, most trades were not for actual bulbs but for contracts to deliver bulbs later: forward contracts.
A perpetual bond with no maturity.
To raise money for dikes, Dutch water authorities issued perpetual bonds that pay interest forever with no maturity. A bond issued in 1648 and held by Yale University still collects interest today.
How should the stakes of an interrupted game be divided?
Pascal and Fermat answered that if a game is interrupted, the stakes should be divided according to each player's expected share, weighing the probabilities of every outcome that could still occur. This is expected value, the heart of all derivative pricing today. The fair price of an option, too, is ultimately the discounted expected value of what it might pay in the future.
Estimating survival from church records.
John Graunt, a London merchant, analyzed church records of deaths to estimate the probability of surviving to each age. It was the first analysis of population statistics.
Insurance in a coffeehouse, and the first book about the stock market.
Ship captains, merchants and insurers gathered at Edward Lloyd's coffeehouse to write marine insurance contracts, and it grew into Lloyd's of London, the world's largest insurance market. Because insurers signed beneath the contract, taking on insurance came to be called "underwriting." The same year in Amsterdam, Joseph de la Vega wrote a book describing options trading and the cycles of fear and greed on the stock market, the first book about the stock market.
Annuities are being sold too cheaply to the young.
The astronomer Edmond Halley (of Halley's comet) built a precise life table from the birth and death records of Breslau and calculated the fair price of life annuities. He showed mathematically that the English government, which sold annuities at the same price regardless of age, was selling them far too cheaply to the young.
The first great stock bubble burst and regulation followed. In Japan the world's first modern futures exchange opened, and mathematics began to explain life insurance and risk aversion. Government bonds that paid interest forever underpinned the British Empire.
Even a company no one knew the purpose of.
South Sea Company shares rose nearly tenfold in a few months and then collapsed. Legend has it that in the frenzy even "a company for carrying on an undertaking of great advantage, but nobody to know what it is" raised money. Newton sold at a profit, bought back in as the price kept rising, and lost heavily. Parliament passed the Bubble Act, banning joint-stock companies without a charter, and the law held back British joint-stock companies for more than a century.
Annuity mathematics from the discoverer of the normal distribution.
Abraham de Moivre, who first discovered the normal distribution, systematized annuity calculations. In 1762 Britain's first life insurance company to set premiums with mathematical life tables was founded, and actuarial science began.
The world's first modern futures exchange.
Samurai of the Edo period were paid stipends in rice, and Osaka merchants traded certificates of claim on rice not yet harvested. The Dōjima Rice Exchange, officially sanctioned by the shogunate, had standardized contracts settled on the books without delivery, margin deposits and even a clearinghouse.
A game with infinite expected value that no one will pay much to play.
Daniel Bernoulli examined a coin-tossing game whose prize doubles with each round until the first heads: its expected value is infinite, yet no one would pay much to play. His explanation was that people weigh not money itself but the expected satisfaction (utility) it brings, and that each extra unit satisfies less the more you have. It was a mathematical account of risk aversion and became the root of expected utility theory and portfolio theory 200 years later.
The bonds that built the British Empire.
Britain consolidated several government debts into consols, which paid 3% a year forever with no maturity. They were the most important financial asset in the world throughout the 19th century, and this low-rate bond market let Britain pay for the Napoleonic Wars.
Twenty-four men under a buttonwood tree.
Twenty-four stockbrokers met under a buttonwood tree on Wall Street and agreed on commissions and trading rules. It was the beginning of the New York Stock Exchange, and the first securities traded were US government bonds consolidating debts from the Revolutionary War.
Coffeehouse trading became formal exchanges, and the Rothschild family built a financial network across borders. Grain futures were standardized in Chicago, and the telegraph spread stock prices in real time.
From coffeehouse to formal exchange.
Stockbrokers thrown out of the Royal Exchange for rowdiness moved to Jonathan's Coffee-House, where they posted lists of prices and traded. This was the origin of the London Stock Exchange, formally established in 1801.
Five cities, five sons.
Mayer Amschel Rothschild sent his five sons to London, Paris, Vienna, Naples and Frankfurt, building a banking network across borders, and during the Napoleonic Wars they moved Britain's war funds to the Continent. The story that they made a fortune by learning the result of Waterloo first is closer to a later myth, but it is true that the family dominated Europe's government bond markets in the 19th century. In 1875 they also financed the British government's purchase of shares in the Suez Canal.
Farmers lock in prices, speculators take on the risk.
The Chicago Board of Trade was founded, and in the 1860s standardized grain futures contracts began trading. Farmers locked in prices before harvest to avoid the risk of a collapse, while speculators took on that risk in pursuit of profit. Chicago remains the center of the world's derivatives trading.
Seeing prices without going to the exchange.
The ticker, which printed stock prices in real time on paper tape via telegraph, was invented. Information began to reach prices faster.
The whole market in one number.
Charles Dow created an index showing the movement of the whole market as the average price of 12 leading companies. It was the first index to sum up the market in a single number.
A doctoral thesis modeled price changes as a random process and launched financial mathematics, only to be forgotten for 50 years. The crash produced securities regulation and value investing, and mathematicians laid the foundations of the probability theory that would later become finance's core tool.
Brownian motion, five years before Einstein.
Louis Bachelier, a student of Poincaré, studied the prices of options on government bonds at the Paris Bourse and modeled price changes as a random process: future changes cannot be predicted, and their spread grows with the square root of time. He derived an option pricing formula and, along the way, created the mathematics of Brownian motion five years before Einstein. But mathematicians looked down on finance, and the thesis was forgotten for 50 years until Paul Samuelson rediscovered it. Today Bachelier is called the father of mathematical finance.
Brownian motion made rigorous.
Norbert Wiener defined Brownian motion as a rigorous mathematical object. In 1933 Kolmogorov placed probability on an axiomatic footing in measure theory.
Twenty-five years to regain the peak.
The Roaring Twenties, when people bought stocks with borrowed money at tenfold leverage, ended with Black Thursday and Black Tuesday. The Dow fell about 89% from its peak by 1932 and did not regain its 1929 high until 1954.
Don't gamble on stocks with depositors' money.
Congressional hearings exposed banks dumping bad stocks on customers and manipulating prices. The Securities Act and the Securities Exchange Act required companies to disclose their finances and created the Securities and Exchange Commission (SEC), and the Glass–Steagall Act separated commercial from investment banking. Deposit insurance sharply reduced bank runs. The same year Alfred Cowles showed statistically that the recommendations of investment newsletters were no better than random picks.
Mr. Market and the margin of safety.
Having lost heavily in the crash, Benjamin Graham, with David Dodd, set out value investing: analyze a company's books and earnings, and buy only well below intrinsic value. He compared the market to "Mr. Market," a manic-depressive partner who quotes absurd prices every day depending on his mood. His student was Warren Buffett.
Pick the face others will think is prettiest.
Keynes compared investing to a newspaper beauty contest in which the prize goes to whoever picks the photos most chosen by everyone else. Entrants end up picking the face others will think prettiest, and beyond that, the face others think others will think prettiest. Stock prices, too, move not on a company's real value but on expectations about expectations.
Calculus for jagged paths.
Japan's Kiyoshi Itô created rules of integration and differentiation for paths, like Brownian motion, that are nowhere differentiable (the Itô integral; Itô's lemma came in 1951). Developed as pure mathematics, the theory became the key tool of the Black–Scholes formula 30 years later, and Itô is said to have been surprised to learn it was used on Wall Street. The same year von Neumann and Morgenstern put forward expected utility theory, which axiomatized Bernoulli's concept of utility.
Risk was measured in numbers for the first time, and the concepts of diversification and beta were born. The random walk and the efficient market hypothesis, which said markets cannot be predicted, were laid out, even as warnings came that large swings happen far more often than assumed. The mathematics of the casino moved to Wall Street.
Risk in numbers, for the first time.
A graduate student, Harry Markowitz, defined what investors care about as expected return (the mean) and risk (the variance), and showed mathematically the principle of diversification: as long as assets do not move in perfect lockstep, combining them lowers overall risk. What matters is not each asset's risk but its covariance with the others. At his thesis defense, Milton Friedman reportedly said this was not economics but mathematics.
As if a demon drew a random number each week.
The statistician Maurice Kendall analyzed changes in stock and commodity prices and showed that past changes cannot predict future ones. In 1959 the physicist M.F.M. Osborne, unaware of Bachelier, independently found that the logarithmic changes in stock prices move like Brownian motion.
The betting fraction that grows wealth fastest in the long run.
John Kelly of Bell Labs applied Shannon's information theory to gambling and derived a formula for what fraction of your wealth to stake on each favorable bet. It maximizes the expected logarithm of wealth, the same mathematics as Bernoulli's logarithmic utility.
The same cash flows must have the same price.
Modigliani and Miller showed that, without frictions, a firm's value is the same whether it finances itself with debt or with equity. The arbitrage argument used in the proof later became the core method of derivative pricing. The same year Tobin showed that adding a risk-free asset means every investor holds the same portfolio of risky assets and only adjusts the proportion.
From blackjack to Wall Street.
The MIT mathematician Edward Thorp, who with Shannon had even built a wearable computer to predict roulette, published a method of counting cards at blackjack to find moments when the odds favored the player and betting by the Kelly criterion, forcing casinos to change their rules. He soon turned his attention to Wall Street, "the biggest casino in the world."
Big crashes come far more often than we think.
Using more than a century of cotton price data, Benoit Mandelbrot showed that price changes are not normally distributed and that large swings occur far more often than the normal distribution predicts. Because the normal distribution was so convenient, mainstream finance ignored this warning for a long time and paid for it in 1987, 1998 and 2008.
Expected return is set by beta.
William Sharpe and others built the capital asset pricing model, starting from the idea that if every investor behaves as Markowitz describes, the common portfolio becomes the whole market. The conclusion: company-specific risk, which diversification can remove, earns no reward; only the degree to which an asset moves with the whole market (beta) is rewarded.
Randomness is evidence of efficiency.
Samuelson showed that if all information is already reflected in prices, future price changes respond only to unpredictable new information and so must be random. Eugene Fama formalized this as the efficient market hypothesis: because market prices reflect all available information, no one can consistently beat the market using public information (systematized in 1970).
The man who knew Black–Scholes first.
In Beat the Market (1967), Thorp calculated the fair price of warrants and laid out a hedging strategy to offset their risk. He already knew and was trading on a formula almost identical to Black–Scholes, but did not publish it. His fund, Princeton/Newport Partners, earned strong returns for 19 years without a single losing year.
After the Nixon shock, as exchange rates and interest rates swung, the need for ways to hedge risk grew, and the Black–Scholes formula arrived together with the opening of an options exchange. Theory became a product in the index fund, and securitization and junk bonds transformed finance. Then, on a day when everyone moved by the same model, the market fell 22.6% in a single session.
Bundling loans and selling them as bonds.
Ginnie Mae, a US government agency, issued the first mortgage-backed securities, bundling many home loans and selling the right to their payments as bonds. Instead of holding loans on their books, banks could now sell them and lend again. This securitization became the seed of the 2008 crisis.
Futures on money itself instead of grain.
As the era of floating exchange rates arrived, Leo Melamed of the Chicago Mercantile Exchange created currency futures, citing a report by Milton Friedman. Interest rate futures (1975) and stock index futures (1982) followed.
Expected return vanishes from the formula.
Black, Scholes and Merton showed that by mixing an option and its stock in the right proportions and adjusting continuously, you could build a portfolio with no risk at all, which, absent arbitrage, must earn exactly the risk-free rate. From this they derived a partial differential equation for option prices; change variables and it becomes the heat equation of physics that Bachelier had used. Remarkably, the stock's expected return drops out of the formula, so people with different forecasts can all agree on the same price. The world's first options exchange opened just before the twice-rejected paper was published, and traders walked onto the floor with calculators programmed with the formula. The sociologist Donald MacKenzie said the model did not describe the market but made it: "an engine, not a camera." Scholes and Merton received the Nobel Prize in 1997; Black had died before then.
Bogle's Folly.
If the efficient market hypothesis is right, it is better to buy the whole market as it is than to pay high fees trying to beat it. When John Bogle's Vanguard launched the first retail index fund tracking the S&P 500, the industry mocked it as "Bogle's Folly." Decades later, the data showed that most actively managed funds fail to beat the index over the long run. It is the biggest case of academic theory fundamentally changing the financial industry.
Losses hurt about twice as much as gains please.
Kahneman and Tversky published prospect theory: people judge outcomes as gains and losses relative to a reference point, and feel losses about twice as strongly as equal gains. The same year Harrison and Kreps established risk-neutral pricing: absent arbitrage, the price of any asset is its expected value computed in an artificial probability world where every asset grows at the risk-free rate, then discounted. Pascal and Fermat's expected value had returned in finished form.
Exchanging interest payments, and stock prices that swing too much.
Arranged by Salomon Brothers, IBM and the World Bank signed a large swap exchanging the interest on debts in different currencies. The swap market later grew to hundreds of trillions of dollars. The same year Robert Shiller showed that stock prices swing far more than changes in their real value, dividends, can justify.
Big swings follow big swings.
Robert Engle created the ARCH model, in which today's volatility depends on the size of past shocks, and in 1986 Tim Bollerslev generalized it as GARCH. These models, which explain how volatility clusters, are still basic tools of risk management, and Engle received the Nobel Prize in 2003.
Betting on the gap between two stocks that move together.
As physics research jobs dried up, many physicists and mathematicians went to Wall Street, where they were called "rocket scientists," and stochastic differential equations and Monte Carlo simulation became standard tools of finance. A team at Morgan Stanley systematized pairs trading: when the price gap between two stocks that tend to move together widens, sell the expensive one and buy the cheap one. In 1987 Engle and Granger's theory of cointegration became its statistical foundation.
22.6% in one day.
Portfolio insurance, which automatically sold futures as prices fell to limit losses, covered tens of billions of dollars. On October 19, as prices fell, the programs all sold futures at once, pushing prices lower still in a vicious circle, and the Dow fell 22.6% in a single day. It remains the largest one-day percentage drop in US history. Afterward the market began charging much more for options that protect against crashes, and implied volatility, which should be a single number, curved into a "volatility smile." The market had belatedly priced in Mandelbrot's fat tails.
A fund run by mathematicians, and companies bought with debt.
Renaissance Technologies, founded by Jim Simons, a differential geometer and former codebreaker, launched the Medallion Fund. The same year the private equity firm KKR bought RJR Nabisco for about $25 billion, financed by the high-yield junk bond market Michael Milken had built, becoming the symbol of the leveraged buyout boom. The Bank for International Settlements adopted the Basel Accord, requiring banks to hold more capital the riskier their assets.
Mathematicians and speech-recognition researchers earned unprecedented returns, while the limits of CAPM and human irrationality came to light. Managing risk with a single number became standard, and a fund run by Nobel laureates collapsed.
Beta alone doesn't explain it.
Fama and French published the three-factor model, adding firm size and value to market beta. The same year the momentum effect was reported, and factor investing became a huge market, but so did the multiple-testing problem: search the data enough and you will find factors that look significant by chance. Also that year Renaissance hired Peter Brown and Robert Mercer, who had worked on statistical speech recognition and machine translation at IBM. By finding tiny but statistically significant patterns in vast data and repeating them thousands of times, they drove the Medallion Fund to an unprecedented average return of about 66% a year before fees.
The most you could lose tomorrow with 99% confidence.
J.P. Morgan published the VaR methodology it had developed because its chairman wanted a one-page summary of the whole bank's risk by 4:15 every afternoon, and it became the standard of risk management. But VaR does not say how much you lose in the remaining 1% of cases. The same year J.P. Morgan created the credit default swap (CDS), which carves out and transfers only the default risk of a borrower. That anyone could buy one, even without owning the bond, was the crucial difference from insurance.
A fund of Nobel laureates, gone in weeks.
Long-Term Capital Management, with Black–Scholes's Scholes and Merton as partners, bet with enormous leverage on small price gaps between bonds that should in theory be equal. With about $4.7 billion in capital, it controlled more than $100 billion in assets. When Russia defaulted on its government debt and investors fled to safe assets, trades that were normally unrelated all moved in the same direction at once, and within weeks the fund lost over 90% of its capital. The New York Fed brought together 14 major banks to arrange a rescue of about $3.6 billion. The lesson: the market can stay irrational longer than you can stay solvent.
A wall gone after 66 years.
The wall between commercial and investment banking, built on the lessons of the 1929 crash, was torn down. Regulation follows a crisis, then loosens again with time: a recurring cycle.
After the internet bubble burst, home loans were securitized amid low interest rates and sold around the world, and formulas built on the normal distribution underestimated their risk. In 2008 the common formula of every crisis, leverage, liquidity and blind faith, brought world finance to the brink of paralysis.
The formula that killed Wall Street?
Shares of internet companies with no profits soared, then the Nasdaq fell about 78% by 2002. The same year David Li published a way to calculate simply the probability that many loans default together, using the Gaussian copula, which life insurers had used to model the correlation of spouses' deaths. It became the standard for pricing collateralized debt obligations (CDOs). The problem was that the model treated tail dependence, many loans defaulting together in extreme conditions, as almost zero, and was estimated on a short stretch of data when house prices were always rising.
Fleeting gaps between prices on different exchanges.
As trading became fully electronic and many exchanges competed, momentary gaps appeared in the price of the same stock across exchanges. High-frequency trading, which seizes those gaps first at microsecond speed, came to account for more than half of US stock trading volume.
Simultaneous default becomes reality.
During the housing boom of the low-rate years, lenders who could sell loans on immediately handed out even subprime loans to borrowers of doubtful ability to repay. These loans were bundled into mortgage-backed securities, CDOs and CDOs of CDOs and sold around the world, many rated AAA. When house prices fell, the simultaneous defaults the models said would almost never happen became reality; in September Lehman Brothers went bankrupt, and the government poured $85 billion into rescuing AIG, the insurer that had sold hundreds of billions of dollars of CDS. After the crisis, Hyman Minsky, who argued that stability itself breeds instability, was rediscovered.
Relearning the lessons of 1933, regulation returned, while trading raced at the speed of light. Chain reactions among algorithms crashed the market in minutes, and commission-free apps drew individual investors back in.
1,000 points in minutes.
The Dodd–Frank Act tightened bank capital rules, routed over-the-counter derivatives through central clearinghouses and barred banks from trading on their own account with depositors' money (the Volcker Rule). On the afternoon of May 6 that year, in the Flash Crash, the Dow plunged about 1,000 points in minutes and then recovered, with some stocks trading at a penny. One company spent about $300 million laying a new fiber-optic cable to shave a few milliseconds off the round trip between Chicago and New Jersey, and soon a race for microwave towers followed, exploiting the fact that light travels faster through air.
$440 million in 45 minutes.
A software error that accidentally reactivated old code made Knight Capital lose $440 million in 45 minutes, effectively bankrupting it. A risk of the algorithmic age.
The same question, opposite answers.
Fama, who says markets are efficient, and Shiller, who says markets get swept up in irrational bubbles, shared the Nobel Prize in Economics in the same year. It neatly captures the state of finance as a field.
Trading commissions drop to zero.
After Robinhood launched a commission-free trading app in 2013, major US brokerages cut commissions to zero one after another in October 2019. Robinhood made its money instead by routing customer orders to high-frequency trading firms in exchange for payment.
Retail investors on Reddit brought down hedge funds, and rumors spread on social media toppled a bank in hours. Index funds overtook active management, and alternative data and AI became new tools of quant investing.
The opposite of a short raid 400 years earlier.
Retail investors from a Reddit community collectively bought GameStop, a stock heavily shorted by hedge funds, and its price multiplied many times over in weeks, inflicting huge losses on the short sellers. It was the age of "meme stocks," Isaac Le Maire's short raid replayed in reverse. A few months later the hidden leverage of Archegos, a family investment firm that had used swaps, collapsed, and its banks lost more than $10 billion.
$42 billion in one day.
When news spread on social media among tightly networked startup depositors that rising rates had left Silicon Valley Bank with large losses on its bond holdings, about $42 billion in withdrawals poured in within a single day and the bank failed. In the age of smartphones and social media, a bank run happens not in weeks but in hours. Days later Credit Suisse, with a 167-year history, lost the market's trust and was taken over in an emergency deal by UBS.
Bogle's Folly wins.
In the United States, assets in index (passive) funds overtook those in actively managed funds. The same year major asset managers began experimenting with tokenization, issuing funds and government bonds as digital tokens on blockchains.
One policy reaches markets worldwide in hours.
In April, right after the United States announced sweeping tariffs, markets were badly shaken, with the S&P 500 falling about 10% in two days. Quant investing has expanded into alternative data, satellite images, payment data and news text, and machine learning, but in financial data, where noise vastly outweighs signal and patterns vanish the moment they are discovered, overfitting is the greatest enemy. Just as Renaissance once hired speech-recognition researchers, researchers in large language models are now moving to quant firms.