1
April 26, 1973. Chicago.
Inside the Chicago Board of Trade (CBOT) building, a room that had once been a smoking lounge was buzzing with activity since early morning. The stale smell of cigarettes had soaked into the walls. For decades, traders had drifted in during their lunch breaks to exhale clouds of smoke. Someone had cleared out the room, built eight octagonal trading posts from raw lumber, and bolted CRT monitors to the walls. Green characters blinked on black screens. This was cutting-edge technology in 1973.
Illinois Governor Dan Walker and SEC Chairman Brad Cook were in attendance. But the protagonists of that day were not the politicians — they were the traders in their brightly colored mesh jackets. Vivid yellow, red, blue — jacket colors identified their firms. Large badges with names and abbreviations were pinned to their chests, so that in the crowd, anyone could tell who was shouting what.
The Chicago Board Options Exchange (CBOE). America's first standardized options exchange opened its doors that day. Launched as a subsidiary of the Chicago Board of Trade, the CBOE was, strictly speaking, the offspring of the grain exchange next door. Chicago, where futures on wheat, corn, and soybeans had long been traded, was already a city of derivatives. From a place that traded the future prices of agricultural commodities, the city now began trading the future volatility of stocks. From the farmer's uncertainty to the investor's uncertainty — the subject had changed, but the essence remained the same.
Sixteen stocks were listed on opening day. All were names traded on the New York Stock Exchange: AT&T, Xerox, McDonald's, Ford, Polaroid, Texas Instruments, Merck, Eastman Kodak, Motorola, Northwest Airlines, Sperry Rand, Upjohn, Gulf & Western, Loews, Atlantic Richfield, and Pennzoil. A cross-section of American industry in 1973 was compressed into those sixteen names. AT&T, the monopoly giant of telecommunications. Xerox, leading an office revolution with its photocopiers. McDonald's, the symbol of fast food's spread. Ford of the automobile industry. Polaroid of instant photography. Texas Instruments of semiconductors. Oil, pharmaceuticals, film, airlines — the industries that undergirded postwar American prosperity were arrayed across those sixteen tickers.
The first trade is said to have been in Xerox. In an era when photocopiers were transforming the office, it was the copier company's option that opened the new market's inaugural transaction. Why these sixteen stocks? The CBOE's intent was clear — large-cap stocks with high trading volume and sufficient volatility. To launch a new product called options, the underlying assets had to be actively traded. No one would buy options on a stock that no one was buying in the first place.
The subsequent trajectories of these sixteen companies are telling. Polaroid, overtaken by digital photography, went bankrupt in 2001. Eastman Kodak, unable to adapt to the digital camera era, filed for bankruptcy protection in 2012 — ironic, given that Kodak itself had invented the digital camera. Northwest Airlines was absorbed by Delta and vanished. Gulf & Western survived the corporate raider era, reinvented itself as Paramount, and was then merged into Viacom. Sperry Rand merged with Burroughs to become Unisys. A significant number of the companies that had represented American industry in 1973 did not survive the next half century. Yet the market created to trade uncertainty about these companies never stopped, even after the companies themselves disappeared. In fact, the very process of their disappearance — bankruptcy, merger, acquisition — represented yet another trading opportunity for the options market.
First-day trading volume was 911 contracts. In the grain pit next door, tens of thousands of futures contracts were being executed each day through shouts and hand signals. By comparison, 911 contracts was negligible. Futures traders who wandered over from next door during lunch — later known as the "McDonald's Lunch Bunch" — participated half out of curiosity.
Yet in that musty old smoking lounge, the grammar of finance was about to change.
From its opening-day total of 911 contracts, the CBOE's first full year saw 1.1 million contracts traded. A decade later, in 1983, volume reached 82 million contracts. By 2008, it exceeded one billion. In 2024, total U.S. listed options volume reached 12.2 billion contracts. Companies disappeared and industries were reshaped, but the market for trading uncertainty itself grew to 12.2 billion contracts in half a century.
The detonator of this explosion was not a gun or a bomb. It was mathematics on paper — a single formula.
2
Fischer Black had studied physics as an undergraduate. He earned a doctorate in applied mathematics. Finance was not his original domain.
In 1965, Black began financial research at the consulting firm Arthur D. Little. Acorn Park in Cambridge — the research campus along the Charles River where Arthur D. Little was headquartered — was a peculiar space, neither university nor bank. Chemists, engineers, and management consultants mingled in the hallways, and smoke rose from every office. That was the way of offices in the 1960s. Ashtrays were standard fixtures on desktops, and the smell of pipe tobacco drifted up from stacks of printed documents. It was in this office that Black began applying the tools of physics to the unfamiliar territory of finance. The problem he dug into was easy to state and brutal to solve: what should the price of an option be?
Options had existed for a very long time. According to Aristotle's Politics, Thales of Miletus used astronomical observations to predict a bountiful olive harvest the following year, and during the winter he secured the rights to olive presses at a low price. When harvest season came and demand for presses surged, Thales resold the rights at a premium. This is the earliest recorded transaction resembling an options trade. Options trading also appeared during the Dutch tulip bubble of the seventeenth century. Speculators bought and sold the right to purchase tulip bulbs at specified prices before the flowers had even bloomed. Yet for more than 2,400 years after Thales, no one had found a way to calculate the "fair price" of an option. Options existed, but the very idea that their value could be expressed mathematically had never taken hold.
Why was it so difficult? A stock is something that exists here and now. The price of a single share of Samsung Electronics can be checked on the market instantly. That price is a number agreed upon every moment by thousands of buyers and sellers. But an option concerns the future. What is the price of "the right to buy a share of Samsung Electronics at 70,000 won three months from now"? It depends on where Samsung's stock price will be in three months — and no one knows. Over those three months, semiconductor prices could collapse, or a new chip could become a blockbuster. War could break out. Interest rates could rise. Every future possibility is compressed into this single small contract. It is the buying and selling of uncertainty itself.
In 1969, Black and Myron Scholes began working together in earnest, both orbiting MIT. Both men were digging into the same problem, but their starting points were different.
Black approached finance like a physicist. Having earned a bachelor's in physics at Harvard and a doctorate in applied mathematics, he treated finance as just another physical system. He set up the premise — "if the market is efficient" — and followed the conclusions that logically flowed from it. Just as physics begins with "if there is no friction," Black began with "if there are no arbitrage opportunities." Having come to finance through Arthur D. Little, he was unburdened by the conventions of Wall Street or the traditions of academic economics. What mattered was mathematical consistency. If the premise was correct, the conclusion must be correct — even if the conclusion defied intuition. Black was also an outsider with no formal academic affiliation. He was not an economics professor but a consultant. This may have given him a certain freedom — unencumbered by the inertia of established schools of thought.
Scholes was an economist and a pragmatist. His interest lay not in theory for its own sake but in how theory operated in the real world. He later told a Stanford University reporter: "Quality is important, but quantity is also important." That sentence captured his temperament. He was a man who wanted a formula he could use, not merely one that was beautiful.
Alongside them, Robert Merton served as a sparring partner in extended debates. Merton came from an intellectual family. His father, Robert K. Merton, was a towering figure in sociology — the man who coined the concept of the "self-fulfilling prophecy." The son would witness his father's intuition come to life in finance — for the Black-Scholes formula would itself become a kind of self-fulfilling prophecy. Merton had been independently working on the same problem as Black and Scholes, and in 1973 he published his own paper in the Bell Journal of Economics and Management Science. His paper generalized and extended the formula, presenting a broader mathematical framework that encompassed cases involving dividends, changing interest rates, and varying structures of the underlying asset. In some academic circles, the question of whether to call it the "Black-Scholes-Merton model" lingered for years. Merton never publicly complained about the omission, but the murmur never quite died.
The conversation among these three men produced the critical breakthrough. The rigor of the physicist, the pragmatism of the economist, the generalizing power of the mathematician. Three distinct temperaments converged into a single formula.
Imagine the scene of that convergence. The late 1960s, MIT Sloan School of Management. In the seminar room at the end of the corridor, a green chalkboard is packed with partial differential equations, chalk dust lodged between fingers. Paper cups of coffee cool on the table, and smoke rises from an ashtray. After a long silence, Black delivers a single remark. Scholes translates it into the language of practice. Merton identifies the mathematical gap. Through the window, the Charles River is visible. Anti-Vietnam War protests were roiling the campus, but the three men in this room were preparing a different kind of revolution. No one yet knew that the symbols on the chalkboard would transform Wall Street.
Yet the most poignant passage of this breakthrough comes much later.
Fischer Black was diagnosed with throat cancer in 1993. He continued his research through his remaining time. As a Goldman Sachs partner and financial theorist, the subject he pursued to the end was "noise" — the meaningless fluctuations in markets that are indistinguishable from signal. On August 30, 1995, Black died. He was fifty-eight. Two years later, in 1997, the Nobel Prize committee in economics awarded the prize for options pricing theory. The recipients were Scholes and Merton. Because Nobel Prizes are not awarded posthumously, Black's name could not appear on the list of laureates. The committee settled for a separate mention of Black's "pioneering contributions" in the award's press release.
If only Black had lived two more years. But "what if" is the language of options — and options have expiration dates. The man who created the formula for pricing uncertainty could not know when the most certain reward of his life would arrive.
The core of what Black and Scholes discovered was this: do not try to calculate the price of an option directly. Instead, construct a portfolio that perfectly offsets the option's risk.
An analogy helps. When you take out fire insurance on your home, the insurance company does not know the exact probability that your specific house will burn down. But if it insures thousands of houses, it can statistically predict how many will burn overall. The insurance company does not predict individual events; it diversifies risk to offset it.
Black and Scholes' idea had a similar structure. By combining a stock and an option in the right proportions, one can construct a combination whose value does not change whether the stock price goes up or down. A portfolio whose risk has been perfectly offset must earn the risk-free rate of return — the equivalent of a bank deposit. If it did not, someone could earn money for free, and markets do not permit such opportunities. From this logic, the price of the option is determined as a single value.
To restate the core insight: there is no need to predict the future. As long as risk can be offset, the price emerges on its own.
This was revolutionary because it was a declaration that, in the face of uncertainty, human intuition and experience were unnecessary. The Medici branch manager weighing a merchant's reputation on his mental scales; the savings bank loan officer debating pre-sale assumptions for two hours — all those processes of human judgment could be replaced by mathematics. At least within the domain of options.
3
What is an option? Let us start with the concept.
Begin with the most intuitive analogy. Imagine you have signed a contract allowing you to purchase an apartment in Seoul's Gangnam district for 1.5 billion won (roughly $1.1 million) six months from now. The deposit is 30 million won. If the market price rises to 1.8 billion won in six months, you buy at 1.5 billion and pocket a 300-million-won gain. After subtracting the 30-million-won deposit, you still clear 270 million won. But if the market price falls to 1.3 billion won in six months? There is no reason to buy at 1.5 billion. You simply walk away from the contract. All you lose is the 30-million-won deposit. Potential gains when the price rises are virtually unlimited, while losses when the price falls are capped at the deposit. This is the essence of an option.
Translate it to stocks. A call option is "the right to buy." Say Samsung Electronics stock is currently trading at 70,000 won. If someone sells "the right to buy this stock at 75,000 won three months from now," that is a call option.
Why would anyone buy such a right? If the stock price reaches 90,000 won in three months, the right to buy at 75,000 won is worth 15,000 won per share. But if the stock price falls to 60,000 won in three months, there is no reason to buy at 75,000 won — you simply let the right expire. Your loss is limited to the amount you paid for the right in the first place — called the premium.
A put option is the reverse — "the right to sell." A structure that profits when the stock price falls.
This is, in effect, insurance. Taking out fire insurance on your home is buying "the right to receive compensation if the house burns down." You pay the premium, collect if disaster strikes, and if it does not, the premium simply becomes a sunk cost. A put option works the same way: insurance against a stock price decline. The premium is the insurance payment; if the price drops, you receive a payout; if it does not, you forfeit the premium.
But how is the price of insurance determined? Fire insurance premiums are calculated from decades of accumulated fire statistics. Plug in the variables — region, building type, usage — and a price emerges. Option pricing, however, was not so straightforward.
Stock prices are fundamentally different from fire probabilities. A fire is an independent event. The house next door burning down does not raise the probability of your house catching fire. But in the stock market, a sharp drop in one stock can trigger sharp drops in others, and panic begets panic. Events are interconnected. Moreover, stock prices change every moment, and the rate of change itself changes. Volatility can be high or low, and there is even volatility of volatility. When interest rates shift, option prices shift too, and the mere passage of time toward expiration alters the price. Variables move, variables influence one another, and all of it operates simultaneously.
Before 1973, option prices were set by experience and gut feeling. In the over-the-counter (OTC) market, dealers negotiated prices one-on-one with counterparties. Without standardized benchmarks, the same option could trade at different prices depending on the dealer. If you wanted to buy an AT&T call option, you had to phone a put-call dealer and ask for a quote. The dealer quoted a price based on personal experience and market instinct. Call a different dealer and you might get a different price. There was no basis for comparison.
When the SEC investigated the options market in 1959, it found a closed OTC ecosystem dominated by a handful of dealers. A small number of brokers affiliated with the Put and Call Dealers Association controlled the market. Contract terms were not standardized, and price transparency was virtually nonexistent. Options existed, but it was difficult to call what they traded in a "market."
The Black-Scholes formula offered a single answer to this chaos. Feed in five ingredients — the current stock price, the strike price, the time remaining until expiration, the risk-free interest rate, and volatility — and out comes the fair price of the option. Like a recipe: put in the ingredients, get the dish. There was one catch: the fifth ingredient, volatility, could not be directly observed and had to be estimated. One of the recipe's ingredients was something as nebulous as "intensity of flavor." This blind spot would become important later.
4
The formula was beautiful. The problem was using it in the real world.
Black and Scholes' paper was initially rejected by academic journals.
First attempt: submitted to the Journal of Political Economy. Rejected. It was one of the premier economics journals, published by the University of Chicago. Second attempt: submitted to the Review of Economics and Statistics. Rejected again. Scholes later explained that one reason for the rejection was that the initial manuscript had deferred some material to a subsequent paper, making the article appear incomplete.
But there was a more fundamental issue. According to the Nobel Prize award ceremony speech, "the conclusion that no risk premium was needed was, at the time, too unfamiliar." The claim that an option's fair price could be determined independently of investors' risk preferences ran directly counter to the prevailing wisdom of economics. In economics, risk and return were an inseparable pair. Bear more risk and you should expect higher returns — this was a foundational premise of financial theory. Yet Black and Scholes were saying that in the domain of options, this premise was unnecessary. There were also reports that the paper was considered "too mathematical" for economics.
After two rejections, the turning point came. Eugene Fama and Merton Miller at the University of Chicago took an interest in the paper. Fama was the originator of the efficient market hypothesis; Miller was a towering figure in corporate finance theory. Both would later receive Nobel Prizes of their own. Their recognition of the paper's value and their intervention with the editorial board proved decisive. The Journal of Political Economy, which had previously rejected the paper, agreed to reconsider it, and after revisions, publication was confirmed in 1973.
Black later reflected on the process with characteristic restraint. He sent the paper, it was rejected, he sent it again, and it was rejected again. More than two years had elapsed. During those two years, the formula had already been spreading through unofficial academic channels. People had read the manuscript as it circulated in working-paper form, and discussions had taken place at MIT seminars. Before the formula was officially published, it already existed unofficially.
That same year, Merton published his paper generalizing the formula in the Bell Journal of Economics. And that same year, the CBOE opened. The Nobel committee's background documentation recorded the CBOE's opening as occurring "one month before the formula's official publication." It was coincidence, but the timing was perfect. The formula and the market were born simultaneously.
The CBOE's old smoking lounge became the site of a collision between academia and the marketplace. The language of people who had written doctoral dissertations on partial differential equations was thrown into the world of people who made money through shouts and hand signals.
The makeup of the early traders was eclectic. In the recollection of Joe Sullivan, they were "a motley bunch." There were experienced futures traders, but also newcomers who barely understood what an option was.
Follow one of them through a day. Six in the morning. He wakes up in a southern Chicago suburb and hurries to knot his tie. A thirty-minute drive to LaSalle Street. Walking from the parking garage to the CBOT building, the wind off Lake Michigan hits his face — Chicago's wind is not kind. Inside, he pulls out his mesh jacket. Bright yellow: the color that identifies his firm. He slips it on and pins on his badge. In his inside pocket, folded, is a value sheet printed the previous night at the computer center. The large continuous-feed paper must be folded four times to fit in a pocket; the creases sometimes wear through into holes. Stepping into the pit, noise engulfs him. Dozens of traders shout and gesture simultaneously. Buying: palm facing inward. Selling: palm facing outward. Numbers are indicated with fingers. His voice goes hoarse, but he cannot stop shouting — stop and you miss the trade. Lunch is a sandwich eaten standing in the hallway beside the pit. When the closing price prints, he removes his jacket and sorts through his trading cards. He tallies the day's profit and loss. His shirt is soaked with sweat. After work, a beer at a bar on LaSalle Street. The man on the next stool was a competitor who had been shouting in the same pit. Outside the pit, they are colleagues. This was the daily life of a CBOE trader in 1973.
Explaining the world of partial differential equations and stochastic calculus to these men was impossible. But the output of the formula — numbers — could be conveyed.
Fischer Black himself had created something. It was called a "value sheet" — a printout of Black-Scholes formula results computed by stock price and volatility and laid out in tabular form. The large computer printout paper was folded again and again until it was small enough to slip into a jacket pocket. Traders stood in the pit with these sheets in their pockets.
Trading worked like this. Check the current price of the underlying stock on the CRT monitor. Pull the value sheet from your pocket and find the theoretical price corresponding to that stock price. Then shout. The pit's method was open outcry — calling out bids and offers through voice and hand signals.
The output of advanced mathematics reduced to a folded sheet of paper, carried into a former smoking lounge where men shouted — this scene was Chicago's reality in 1973. The union of abstraction and flesh. Partial differential equations meeting sweat.
By around 1975, the Black-Scholes model began to be loaded onto Texas Instruments (TI) calculators. Programmable handheld calculators had arrived. Scholes himself later recalled that the spread of these calculators accelerated the formula's adoption in practice. Instead of paper value sheets, a few presses of calculator buttons could produce a theoretical price. Value sheets, based on the previous day's closing prices, inevitably lagged intraday price movements, but calculators could accept real-time prices and spit out theoretical values on the spot. When the tool changed, speed changed, and when speed changed, the market changed.
Then something fascinating happened. In the early days, actual CBOE option prices showed significant deviations from Black-Scholes theoretical prices. This was less a matter of the market being "wrong" than of most traders not yet using the formula. The majority relied on experience and instinct; the formula belonged only to the few with academic backgrounds.
But as value sheets and TI calculators spread, market prices began gradually converging toward theoretical prices. The mechanism was straightforward. Traders who used the formula bought "underpriced" options and sold "overpriced" ones. When this strategy consistently generated profits, others took notice. When someone pulled a folded sheet of paper from a pocket, checked a price, and called out a bid with confidence, the trader relying on gut feeling found himself increasingly at a disadvantage. As those who used the formula made money, those who did not began to follow suit, and eventually nearly everyone priced options using the same formula.
Donald MacKenzie of the University of Edinburgh gave this phenomenon a name: "performativity." The model had not described reality; the model had created reality. The Black-Scholes formula was not adopted because it accurately predicted the market. Rather, as market participants adopted the formula, the market aligned itself with the formula.
MacKenzie explained this with the metaphor of "not a camera but an engine." A camera captures the world as it is; an engine moves it. The Black-Scholes formula was built as a camera of the options market, but in practice it became an engine. Before the formula was adopted, the market did not match it. After adoption, the market converged toward it. Theory did not describe reality — theory reconstituted reality. In science, this kind of reversal is rare. Whether or not people believe in Newton's law of gravity, the apple falls. But in finance, participants' beliefs change reality itself. When everyone uses the same formula, the market truly begins to move according to that formula.
5
Black-Scholes changed more than the options market.
The formula's real legacy was a single idea. Risk can be priced. Uncertainty is not something to be feared but a commodity that can be traded. This idea overturned the entire playing field of finance.
Options are part of a larger category called derivatives. Contracts "derived" from an underlying asset. Options derived from stocks, swaps derived from interest rates, CDS derived from credit — all are variations on the same logic. Slice up future uncertainty, attach a price, and buy and sell it.
Once Black-Scholes provided a method for pricing options, the same logic began extending into other domains. The risk of rising interest rates? Trade it with an interest rate swap. The risk of exchange rate fluctuations? There are currency options. The risk of a counterparty going bankrupt? Transfer it with a credit default swap. Whatever the type of risk, the notion that it could be isolated, priced, and passed on to someone else permeated every corner of finance. The Nobel committee explicitly stated that this methodology had been extended to the valuation of "derivatives, insurance contracts, guarantees, and the flexibility of real investments."
Let us revisit the trajectory of the flow that began with 911 contracts on the CBOE's opening day. In 1983, the CBOE alone saw 82 million contracts annually. In 1993, 140 million. In 2006, 500 million. In 2008, over one billion. In 2024, total U.S. listed options volume reached 12.2 billion contracts. In half a century, trading volume had grown more than 13 million-fold.
This was not simply a matter of numbers getting larger. It was a transformation in the very way risk was perceived.
Before 1973, risk was something to be borne or avoided. A farmer bore the risk that next year's wheat price might fall. A bank prepared for the risk of loan defaults by setting aside reserves. Risk was a cost — an unavoidable one.
After 1973, risk became a tradable commodity. A farmer who did not want exposure to falling wheat prices could sell that risk to a speculator willing to take it on. A bank that did not want the risk of loan defaults could transfer it to another investor. Risk had transformed from a cost into a commodity. And once risk became a commodity, it could be sliced: a single risk could be divided into multiple pieces and sold to different investors, and conversely, multiple types of risk could be bundled together to create new products. Risk was no longer something fixed; it had become something fluid.
Here, the form of the question changed.
The Medici bank's branch manager had asked: "Can this merchant repay the money?" The Bank of England's board of directors had asked: "Can the state pay the interest on these bonds?" The loan review committee at a Korean savings bank had asked: "Can this developer successfully sell out the units?"
Black-Scholes changed the very form of the question.
"What is the price of this uncertainty?"
It was no longer a binary judgment of whether repayment was possible, but a question of whether the probability of default was reflected in the price. Judgment began to be replaced by calculation, intuition by formula, humans by mathematics. From a world where a gatekeeper stood at the door and declared "you may enter, you may not," to a world where anyone could enter so long as they paid a price commensurate with their risk.
This transition proceeded slowly but irreversibly. It was not only options traders who used the formula. Bond traders began applying the same logic to interest rate models. Insurance companies adopted derivative techniques for risk assessment. Corporate treasurers began using options to hedge against exchange rate fluctuations. The Black-Scholes formula was a single tool, but the idea it planted — that uncertainty can be quantified, and quantified uncertainty can be traded — became a philosophy that spread across the entirety of finance.
In 1997, Scholes and Merton received the Nobel Prize in Economics. Without Black. The committee assessed the pair's achievement as having "made possible economic valuations in many areas of the economy through a new methodology." It was the moment when the method of pricing uncertainty received humanity's highest intellectual honor.
At the time of the Nobel ceremony, Scholes' affiliation was listed as "Long-Term Capital Management (LTCM)." The co-creator of the formula for pricing uncertainty was working at a hedge fund that applied that formula in practice. It seemed the perfect union of theory and action. At the celebration, no one knew what that affiliation would come to mean just one year later.
6
The Black-Scholes formula was beautiful. It was concise, logical, and practical. Feed in five ingredients and a single answer emerged. Traders in the smoking lounge made money with it, and the Nobel committee bestowed upon it humanity's highest intellectual honor.
But every formula rests on assumptions.
Unpacked, the assumptions of Black-Scholes are these. Trading can occur at any time — there are no transaction costs. Stock prices move smoothly and continuously — they do not suddenly jump or crash. Volatility is constant — the character of the market does not abruptly change. These assumptions were mathematically necessary. For the formula to work, the world had to behave this way.
Real markets do not behave this way.
Transaction costs exist. There is always a gap between the bid and the ask, and slippage occurs the moment an order is executed. Stock prices do not move smoothly. They jump. When a company reports earnings, its stock can leap 10% in an instant, and when war breaks out, the entire market can plunge in a heartbeat. Volatility is not fixed but shifts moment by moment — dormant during calm periods, exploding during crises. In the Black-Scholes world, a stock price rocks gently like a boat on a placid lake. In reality, stock prices sometimes become a raft in a storm.
Yet from 1973 to 1987, these assumptions appeared to hold more or less intact. Markets grew steadily. The Dow Jones Industrial Average surged more than threefold, from its 1982 low of roughly 800 to 2,700 in August 1987. The options market expanded explosively. Black-Scholes became the industry standard, and TI calculators and value sheets became fixtures of the trading floor. The formula worked, the market moved according to the formula, and few questioned the formula's assumptions. Fourteen years of success were hardening assumptions into truths.
An analogy with a thermostat is useful here. Delta hedging, the core technique of Black-Scholes, involves making micro-adjustments to a portfolio each time the market moves in order to offset risk. Like a thermostat holding a room at 22 degrees, the strategy recalibrates the portfolio every time the market ticks. The thermostat works well when temperature changes are gradual. If the temperature hits 23 degrees, the cooling kicks in; if it drops to 21, the heating starts.
But what if a fire suddenly breaks out in the room? What if the temperature leaps from 22 to 200 degrees in an instant? The thermostat cannot cope. It has exceeded its design parameters.
The Black-Scholes formula rested on a premise: markets move continuously. If yesterday's price was 100, today's will be 101 or 99, but it will not suddenly become 60. On October 19, 1987, when the Dow Jones Industrial Average plunged 22.6% in a single day, that assumption shattered.
And from the wreckage, the market itself fashioned a new form.
The morning after Black Monday, traders returned to the CBOE pit. The usual shouting was absent. They put on their jackets and stood in the pit, but no one was the first to call out a price. The value sheets in their inside pockets had lost all meaning — those numbers had been calculated from the previous day's closing price. On a day when 22.6% of that close had simply evaporated, those numbers were waste paper. The green characters on the CRT monitors blinked, but no one could tell where the figures on the screen would stop. The realization that the world the formula had assumed — a world where prices moved smoothly — had collapsed in a single day pressed down on the pit. New calculations were needed. New assumptions were needed. But that morning, it was not even clear what needed to change.
One of Black-Scholes' assumptions was that volatility is constant. Regardless of the strike price, regardless of the expiration date, options on the same underlying asset should have identical implied volatility. This was a premise demanded by the formula's mathematical structure. Before Black Monday, actual markets had broadly conformed to this assumption. Plot implied volatility by strike price on a graph, and the line was nearly flat. It was a time when the formula described the world and the world was aligned with the formula.
Black Monday crumpled that flat line.
After 1987, a peculiar pattern emerged in the options market. The implied volatility of put options with strike prices far below the current stock price — so-called out-of-the-money puts, which only gain value if the market crashes — rose to abnormally high levels. Traders had begun assigning higher prices to extreme declines. A market that had experienced a 22.6% single-day drop no longer accepted the formula's assumption that such a drop was "virtually impossible." When implied volatility was plotted against strike price, the resulting curve rose at both ends, and financial scholars called it the "volatility smile."
The name was ironic — there was nothing to smile about. What the curve meant was plain: market participants no longer believed the Black-Scholes assumption that extreme events almost never happen. In the mathematical world of Black-Scholes, stock price movements follow a normal distribution, and the probability of a 22% single-day decline is effectively zero — an event that might occur once in a span longer than the age of the universe. Yet that event occurred on October 19, 1987. The formula had been underestimating tail risk — events that are low in probability but catastrophic when they occur. Black Monday proved that tail risk could become reality regardless of what the mathematics predicted, and traders began reflecting that lesson directly in prices. Mark Rubinstein at Berkeley systematically documented this phenomenon in a 1994 paper, and numerous scholars subsequently explored the meaning of the smile.
Here lay the second phase of performativity. The Black-Scholes formula had created the market — that was the first phase. The formula was adopted by the market, and the market aligned with the formula. But now the market had begun correcting itself at precisely the points where the formula was wrong. The participants of the market the formula had created were, in the language of prices, resurrecting the possibility of extreme events that the formula had ignored. The formula was wrong, but the market it had created was transcending its creator's limitations. The student had begun to surpass the teacher.
The volatility smile was also a warning. A warning that the market knew more than the formula. A warning that fears uncaptured by mathematics were alive and breathing inside prices. And those fears came from memory. The memory of 1987. The memory that everything could collapse in a single day.
To understand where that memory began, one must go to the New York Stock Exchange on the morning of Monday, October 19, 1987.