The Midas Formula: How One Equation Built Wall Street — And Almost Burned It Down

17 Equations That Changed The World | Chapter 17 | Black–Scholes | Ian Stewart

Apr 25, 2026

“The Black–Scholes equation changed the world by creating a booming quadrillion-dollar industry; its generalisations, used unintelligently by a small coterie of bankers, changed the world again by contributing to a global financial meltdown.” — Professor Ian Stewart

The Surface Answer: What Is The Black–Scholes Equation?

In 1973, two economists named Fischer Black and Myron Scholes published a paper that Wall Street treated like a divine transmission from the financial heavens.

It was an equation for pricing options — financial contracts that give you the right, but not the obligation, to buy or sell something at a fixed price on a future date.

Think of it like this: you’re standing outside a bakery. You pay the baker 10 kronor today for the right to buy a loaf of bread next month at today’s price — whether bread costs 5 kronor or 50 kronor by then. That contract has a value. The problem, before 1973, was that nobody agreed on exactly what that value was.

Black and Scholes solved it. Or so the story goes.

Their equation looks like this:

∂V/∂t + ½σ²S²(∂²V/∂S²) + rS(∂V/∂S) − rV = 0

Before your eyes glaze over — here is the translation into plain Swedish-grandmother language:

“Given how wildly a stock tends to move, and given how much time remains before the contract expires, what is the contract worth right now?”

That’s it. That’s the whole question.

The equation answered it with mathematical precision. Suddenly, risk had a price tag. And Wall Street — never slow to monetise a revelation — went absolutely berserk.

The Word You Need to Know

Derivative (noun): A financial instrument whose value is derived from something else — a stock, a currency, a commodity, or even another derivative. Options are derivatives. So are futures contracts, swaps, and approximately nine thousand other financial instruments invented since 1973, each more abstract than the last.

Volatility (noun, from Latin “volatilis” — able to fly): In finance, the tendency of an asset’s price to fluctuate. The more wildly something moves, the more valuable an option on it becomes — because uncertainty is, paradoxically, something you can profit from if you price it correctly.

Arbitrage (noun, from French “arbitre” — judge): The simultaneous buying and selling of equivalent assets in different markets to exploit price differences. Black–Scholes was built on the assumption that arbitrage opportunities are always eliminated by the market. Spoiler: they are not always eliminated before they detonate.

Layer Two: What They Left Out

Here is where we need to pay attention, because this is the part that matters.

The Black–Scholes equation is built on a set of assumptions. They are stated clearly in the original paper. Most people using the equation later apparently skimmed that part.

The equation assumes:

Markets move smoothly. No sudden jumps. Price changes follow a “random walk” — small, continuous, normally distributed steps, like a drunk person wandering home who nevertheless always stays within a predictable distance of the path.
Volatility is constant. The wildness of a stock’s movement does not change over time.
No transaction costs. You can trade as often as you like, for free.
You can always borrow money at a fixed, risk-free rate. No credit crunch. No liquidity crisis. No bank refusing to pick up the phone.
No Black Swans. Extreme events — the kind that happen roughly never, according to the model — stay nicely in the tails of the bell curve where they belong and do not come crashing into the middle of a Tuesday in October.

On 19 October 1987 — Black Monday — the world’s stock markets lost more than 20% of their value in a single day.

Under Black–Scholes assumptions, this was not merely unlikely. It was, in mathematical terms, essentially impossible. The kind of event that should not occur once in the entire history of the universe.

The universe was apparently not consulted.

The Monty Python Sketch That Actually Happened

Allow me to describe the founding of Long-Term Capital Management (LTCM), without embellishment, because embellishment would only make it less absurd.

In 1994, a group of the most credentialed financial minds on the planet — including Myron Scholes and Robert Merton, who would go on to win the Nobel Prize in Economics in 1997 — founded a hedge fund based on the mathematical principles derived from Black–Scholes.

They were considered geniuses. Not regular geniuses. Quantitative finance deity geniuses.

The fund generated 40% annual returns in its first two years. They treated other financial institutions with barely concealed contempt. Their models told them exactly how much risk they were carrying. The models said: not very much.

Nassim Nicholas Taleb, who would later write The Black Swan, described their strategy as “picking up pennies in front of a steamroller.”

In the summer of 1998, Russia defaulted on its debt. This event was, technically, outside the parameters of their models. The models had not assigned meaningful probability to it. The models were wrong.

LTCM lost $4.6 billion in under four months.

The Federal Reserve had to organise an emergency bailout by a consortium of banks to prevent the collapse from spreading into the broader financial system. The Nobel laureates required rescuing by the central bank.

The steamroller won.

Then Came 2008

The story did not end in 1998. Humanity, being humanity, took notes and learned nothing.

By 2007, Ian Stewart reports in 17 Equations That Changed The World, the international financial system was trading derivatives valued at one quadrillion dollars per year. That is a one followed by fifteen zeros. That is ten times the total worth, adjusted for inflation, of everything manufactured by the world’s industries over the previous century.

The instruments driving this number — credit default swaps, collateralised debt obligations — were priced using descendants of Black–Scholes. The assumptions underneath those instruments were still the same assumptions. Markets were still assumed to be smooth. Volatility was still assumed to be manageable. Extreme events were still assumed to be extremely rare.

The housing market, it turns out, is not a smooth Gaussian process.

When the sub-prime mortgage market turned, the derivatives built on top of it — priced with formulas that assumed they were safe — turned with it. The equation Stewart had called the “Midas Formula” became, in his own words, “the Black Hole equation — sucking money out of the system.”

Warren Buffett, who had previously described derivatives as “financial weapons of mass destruction,” wrote in 2008:

“The Black–Scholes formula has approached the status of holy writ in finance... If the formula is applied to extended time periods, however, it can produce absurd results.”

He noted, with characteristic dry wit, that Black and Scholes themselves almost certainly understood this. Their devoted followers, however, had apparently decided the caveats were optional.

Layer Three: The Reframe — What This Is Really About

Here is the dimensional view. This is not a story about a bad equation.

At the individual level: Black–Scholes is an extraordinary intellectual achievement. It answered a genuine question with mathematical rigour and opened up an entirely new field of risk management. Airlines use derivatives to hedge fuel costs. Exporters use them to hedge currency exposure. Small businesses use them to protect against interest rate swings. This is genuinely useful. This is the equation working as intended.

At the institutional level: The equation was taken from its context, stripped of its caveats, and used as a talisman — Stewart’s word, and a precise one. A talisman is not a tool. A tool you use with skill and judgment. A talisman you wave at problems and expect protection. Traders and risk managers stopped asking “does this assumption hold?” and started asking “what number does the formula give me?” The difference between those two questions is the difference between engineering and magic.

At the civilisational level: This is a story about what happens when we confuse the map for the territory. Black–Scholes is a map of financial risk. A very good map — for certain terrain, in certain conditions. But maps have edges. And when you walk off the edge of the map while believing the map is reality, you do not fall off a table. You fall off an economy.

The $846 trillion derivatives market that existed as of June 2025 — the largest since before 2008 — is operating in an intellectual universe built in 1973. The foundations of that universe include assumptions that have been empirically violated, repeatedly, in living memory. Nobody has yet invented a better map that the market will actually use.

The Missing Links — What Stewart’s Chapter Points Toward But Doesn’t Say

Connect this to what we have been tracking across this publication:

The Epstein/Black connection. Leon Black, founder of Apollo Global Management, ran one of the world’s largest private equity firms — a firm whose financial engineering relied on precisely the kind of complex derivative instruments Black–Scholes made possible. His decade-long financial relationship with Jeffrey Epstein — $158 million paid to Epstein between 2012 and 2017 — has never been satisfactorily explained. The opacity of complex derivatives is not unrelated to the opacity of complex financial relationships. Complexity, in finance as in human behaviour, is frequently used as a feature, not a bug.

The thermoeconomic frame. We covered Warwick Powell’s thermoeconomics work in this publication. His thesis: modern economies are not primarily monetary systems. They are energy transformation systems. Black–Scholes is a formula that treats money as the fundamental unit of reality. It has no term for energy cost. No term for ecological consequence. No term for what happens to a worker in Uddevalla, or a farmer in Kenya, when a derivatives cascade rewrites the price of the crop they just planted. The equation is internally consistent and externally blind.

The Ubuntu principle. I am because we are. Black–Scholes prices risk for the individual holder of a contract. It does not price risk for the system. It has no variable for what happens when everyone uses the same hedge and the hedge fails simultaneously. This is called systemic risk — and it was precisely what 2008 demonstrated. A formula that tells every individual player they are safe can still produce collective catastrophe. This is not a flaw in the mathematics. It is a flaw in what the mathematics was asked to measure.

Facts, No Spin

Fischer Black and Myron Scholes published their model in 1973. Robert Merton published a companion paper the same year and is often co-credited.
Scholes and Merton received the Nobel Prize in Economics in 1997. Fischer Black died in 1995 and was therefore ineligible — the Nobel is not awarded posthumously.
LTCM, the hedge fund co-founded by Scholes and Merton, lost $4.6 billion in 1998 and required a Federal Reserve-organised bailout.
By 2007, global derivatives trading reached approximately $1 quadrillion per year in notional value.
As of June 2025, the Bank for International Settlements reports outstanding OTC derivatives at $846 trillion — up 16% year-on-year, the largest annual increase since before 2008.
The Black–Scholes equation is still used. Every trading day. Globally. Often correctly. Sometimes not.

Consequences — With Cautious Optimism

The cautious optimism here is not naive. It is structural.

The equation still works. The assumption that the equation is sufficient — that is what fails. And the failure is not mathematical. It is cultural. It is the culture of finance that decided a good model was a perfect model, that caveats were for academics, that complexity was indistinguishable from safety.

Cultures change. They change slowly, and usually only after they break something expensive. 2008 broke something expensive. Whether the lesson was actually learned is a different question — one the next Black Swan will answer.

The real innovation we need is not a better pricing formula. It is a better question: “What are we not measuring?”

Black–Scholes answers the question it was given with impressive precision. The problem was always the question.

COGNITIVE-LOON

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