The Formula That Ate the Market
How Black, Scholes and Merton transformed option pricing, and what happened when an elegant model met a market that could move beneath it.
Chicago, 1973. A new exchange opens for business with standardised options on just sixteen stocks. A few weeks later, the Journal of Political Economy publishes an eighteen-page paper called The Pricing of Options and Corporate Liabilities.
The timing is remarkable.
Options have existed in various forms for centuries, but there has never been an agreed way to determine what one should be worth. Traders rely on experience, rules of thumb and negotiation. Two people can look at the same contract and arrive at very different prices because the future payoff depends on several uncertainties at once.
Fischer Black and Myron Scholes offer something different. Under a defined set of conditions, they show how the value of a European call option can be derived from a small set of inputs: the current share price, the exercise price, the time remaining, the risk-free interest rate and the volatility of the underlying share.
The formula does not tell the trader where the share price will finish. Its deeper achievement is to show how the option’s payoff can, in theory, be replicated by continually adjusting a position in the share and borrowing or lending at the risk-free rate.
If two positions produce the same future payoff under the model’s assumptions, they should have the same price today. Otherwise, a trader could buy the cheaper one, sell the dearer one and lock in a profit without accepting risk.
This is the principle of no-arbitrage. It gives finance a new way to price uncertainty without first having to forecast the asset’s expected return.
The formula is elegant, practical and enormously influential. It helps turn options from specialised contracts into instruments that can be priced, compared and hedged on an industrial scale.
It also demonstrates a danger that will recur throughout this series. A model can solve the problem it was given while leaving out the conditions that determine whether the solution can be used safely.
The formula made the invisible visible.
It also made the catastrophic appear impossible.
THREE MEN, ONE BREAKTHROUGH
Fischer Black did not begin as a conventional economist. He studied physics and applied mathematics, worked in consulting, and approached finance as a problem in systems and relationships.
The question that drew him towards options was deceptively simple. If an option derives its value from a share, what relationship must hold between the two prices?
Black initially explored the problem through the capital asset pricing model. Myron Scholes, then at MIT, was working on related questions. Together they found a way to remove the unknown expected return of the share from the pricing problem.
Robert Merton developed the continuous-time argument with greater mathematical precision and showed how widely the method could be applied. His work helped establish what became known as the Black–Scholes–Merton framework.
The breakthrough was not merely a neat equation. It was a method.
Imagine an option whose price rises by roughly fifty cents when the underlying share rises by one dollar. A trader who has sold the option might buy an appropriate fraction of the share to offset a small movement in the option’s value. As the share price changes, that fraction changes too. The hedge must therefore be adjusted repeatedly.
In the model’s continuous world, the adjustment is continuous. The option and its changing hedge become a replicating portfolio. Once the cost of that portfolio is known, the option has a theoretical value.
The method is powerful because it replaces one difficult question with another that appears more manageable. Instead of asking what return the share will earn, it asks how much the share is likely to move and how the hedge must respond.
That is where volatility enters the formula.
Volatility here is not risk itself. It is an estimate of the scale or dispersion of price movement. A higher estimate makes the option more valuable because it increases the range of possible outcomes while the option holder’s loss remains limited to the premium paid.
The distinction matters. The formula uses volatility as a pricing input. It does not establish that volatility captures every way a position, a portfolio or an institution can fail.
THE WORLD INSIDE THE FORMULA
Every model creates a smaller world in which its reasoning can operate.
In the standard Black–Scholes setting, trading can occur continuously. Transaction costs and taxes are absent. Borrowing and lending are available at the risk-free rate. The underlying share can be traded in any required amount. Volatility and the risk-free rate remain constant over the option’s life. The share price follows a continuous process, so it does not jump from one level to another without passing through the prices between them.
These are not claims that real markets literally behave this way. They are simplifying conditions that make the pricing argument possible.
The assumed price process is geometric Brownian motion. This is related to the mathematical tradition begun by Louis Bachelier, but it is not the same model. Bachelier allowed prices themselves to follow an arithmetic Brownian motion. Black and Scholes used a process in which proportional price changes behave regularly enough for the continuous hedge to be derived and prices remain positive.
Under those conditions, prices over a specified interval are lognormally distributed, continuously compounded returns are normally distributed, volatility remains constant and extreme moves become very rare.
Real markets are less accommodating.
Volatility changes. Prices jump. Trading costs widen under pressure. Liquidity can vanish. A hedge that appears available on paper may become expensive or impossible to execute at the assumed price. Most importantly, the act of hedging can affect the market when many participants attempt similar trades at the same time.
None of this makes the formula useless. A map does not become useless because it leaves out every stone and tree. The problem begins when the map’s omissions are mistaken for features that do not exist.
The formula made the invisible visible. It also made the catastrophic appear impossible.
WHAT THE MARKET ADDED
The formula requires a volatility input, but volatility cannot be observed in advance. It must be estimated.
A trader can use historical movement or insert a forecast. There is also another possibility. Begin with the option’s market price and work backwards through the formula to find the volatility that makes the formula reproduce that price. This is implied volatility.
If the standard model described the market perfectly, options on the same underlying asset and with the same expiry would imply roughly the same volatility across exercise prices.
They do not.
Before 1987, the pattern across many equity options was often relatively subdued, although it was never perfectly flat. After the crash, the difference became much more pronounced. Lower-strike equity options, which provide protection against large falls, tended to carry higher implied volatility than options nearer the current market price.
This pattern is often loosely called a volatility smile. In equity markets it is usually better described as a skew or smirk because the curve is not symmetrical. The market places a particularly high price on protection against a severe decline.
The shape contains information that one constant volatility number cannot.
It reflects several realities: return distributions have heavier tails than a simple normal model suggests; large falls and rising volatility often arrive together; supply and demand for protection are uneven; and market participants know that liquidity and hedging conditions can deteriorate during a sell-off.
The market did not discard Black–Scholes when these differences became obvious. It adapted the formula.
Traders began using it as a common language. An option quoted at “thirty volatility” is not a declaration that the underlying asset will move with a constant volatility of thirty per cent. It means that inserting thirty per cent into the formula produces the observed option price.
The model became a translation device. Its failure as a literal description of the world did not prevent it from remaining extremely useful.
OCTOBER 1987
On 19 October 1987, the Dow Jones Industrial Average fell 22.6 per cent in one session.
The size of the decline sat far into the tail of any normal distribution calibrated to ordinary daily market movement. Giving it an exact probability is misleading because the answer depends on the period and volatility estimate chosen. The important point is simpler. A model built around continuous price movement and stable volatility had no convincing account of what occurred.
The crash also exposed the problem of feedback.
Portfolio insurance had become popular among large institutional investors during the 1980s. Its purpose was to limit losses without requiring the investor to buy an exchange-traded put option. The protection could instead be created synthetically by adjusting exposure as the market moved.
When prices fell, the strategy called for exposure to be reduced, often through sales of stock-index futures. If the market fell further, more exposure had to be sold.
For one investor in a deep and liquid market, that logic might be manageable. When many large investors followed related instructions at once, it produced a different result. Falling prices generated sell orders. Those orders added pressure to a market already short of willing buyers. Further declines then called for further selling.
The hedge was no longer responding to an independent market. Collectively, the hedgers had become part of the process moving it.
Portfolio insurance did not single-handedly cause Black Monday. The market had risen sharply earlier in the year, economic and currency concerns had unsettled investors, global markets were already falling, and weaknesses in trading, clearing and settlement systems made the disruption worse. Official and academic accounts identify portfolio insurance as an important accelerator, not a complete explanation.
That distinction strengthens the lesson.
The crash did not reveal one defective formula controlling the market. It revealed a system in which similar rules, crowded positions, limited liquidity and mechanical responses could interact in ways that no participant intended.
The danger came from the combination.
THE SCALE PROBLEM
Dynamic hedging is easiest to imagine when the trader is small relative to the market.
The model assumes that shares can be bought and sold at the prevailing price without the hedge itself changing that price. It also assumes that the next adjustment can be made when required.
Real markets place limits on both assumptions.
A small order in a liquid market may have little visible effect. A large order will usually move through the available bids or offers and receive progressively worse prices. If many traders need to make the same adjustment, the market impact can be much larger. The observed price process then partly reflects the strategies responding to it.
There is also a directional asymmetry. A trader who has sold options may need to buy as the market rises and sell as it falls. That response can reinforce movement. Other option positions can create the opposite behaviour. The effect depends on who holds which exposure, how quickly they hedge and whether the market can absorb the required trades.
This is why scale and liquidity belong inside any serious discussion of risk.
A strategy can be sound for one participant yet destabilising when adopted by many. A hedge can reduce an institution’s exposure while transferring pressure into another market. A position can appear manageable at yesterday’s price and become dangerous when everyone seeks the same exit.
The formula does not conceal these effects maliciously. They sit outside the problem it was designed to solve.
WHAT SURVIVED THE CRASH
Black–Scholes survived because its central insight was stronger than a literal reading of all its assumptions.
The framework gave traders a disciplined way to connect an option price with the underlying asset, time, interest rates and volatility. It supplied measures such as delta and gamma that describe how the option’s value changes as conditions change. It allowed exposures across different contracts to be compared and helped markets become deeper and more transparent.
The 1987 crash did not erase those achievements. It changed how sophisticated users interpreted them.
Implied volatility came to be represented as a surface rather than a constant. Models were extended to allow changing volatility, jumps and other features. Traders applied adjustments for transaction costs, discrete hedging and liquidity. Scenario analysis and stress testing became essential companions to formula-based valuation.
Yet the central limitation remained.
A model can describe the sensitivity of a position to specified changes. It cannot guarantee that the next crisis will arrive through one of the changes specified. It can estimate how a hedge should behave if trading remains possible. It cannot create the liquidity required to execute that hedge.
The calculation and the capacity to survive are different things.
Fischer Black understood that market prices contain noise and that models are approximations. He died in August 1995. When Myron Scholes and Robert Merton received the Nobel Memorial Prize in Economic Sciences in 1997, the Nobel committee explicitly acknowledged Black’s contribution.
The award was deserved. Their method transformed the valuation of derivatives and opened an extraordinary field of financial research and practice.
Less than a year later, Scholes and Merton would be involved in a crisis that brought the distinction between price and survival into sharp focus.
THE FORMULA AND THE WORLD
The lesson of Black–Scholes is not that mathematics failed.
The mathematics did exactly what mathematics should do. It stated a problem, made its assumptions visible and derived a result. It allowed those assumptions to be tested, challenged and improved.
The failure occurs when users forget the conditional nature of the answer.
For an Outlier Hunter, the practical response is not to abandon models. We use models constantly. ATR helps place different markets on a comparable scale. Rules define entries and trailing exits. Historical tests show how those rules behaved across conditions that actually occurred.
But measurement does not turn the future into a known distribution.
The position must remain small enough to survive being wrong. The portfolio must extend across genuinely different markets rather than several expressions of the same crowded exposure. The exit must be executable in the world that exists, not merely at the price printed by the model. Profitable positions must retain room to grow because the event that matters most may sit beyond the range the past made comfortable.
Black–Scholes made uncertainty easier to price.
It did not make the market continuous, liquid or obedient.
The formula revealed a relationship.
It did not contain the world.
Next: Article 5, The Genius and the Abyss
Greenwich, Connecticut, 1994.
John Meriwether is assembling an exceptional team. Myron Scholes and Robert Merton will join it, along with former Federal Reserve Board vice-chairman David Mullins and specialists drawn from trading, economics and mathematics.
Their fund will search for small pricing differences that its models suggest should eventually converge. For several years, the results will appear to confirm the method.
Then Russia will default, markets will move together, liquidity will retreat and positions designed to converge will widen instead.
The models will still describe value.
The fund will run out of room to wait.
Richard Brennan writes on systematic trading, complex adaptive markets, and the philosophical foundations of trend following at atstradingsolutions.com. His books include The Fractals of Finance, Complex Adaptive Markets, Carved by Impossibility and The Aussie Turtles Trend Following Guide.
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