The Equation that Changed Finance: The History and Impact of the Black-Scholes Model

Why You Need to Know

The right to buy or sell a stock at a set price on a set date is called an option. The set date is called the expiration date, and the set price is called the strike price. The right to buy is a call option, the right to sell is a put option, and an asset like a stock tied to an option is called the underlying asset. The market where these rights are bought and sold is the options market.

Yet how much should we price a future right that has not even happened yet today? This question long remained a realm of gut feeling and nerve. Option trading was considered closer to gambling than to respectable investing.

In 1973, this verdict was overturned. Fischer Black, Myron Scholes, and Robert Merton, who helped complete the theory together, introduced a formula to the world for calculating option prices. This Black-Scholes-Merton formula transformed option pricing from "gambling" to "scientific risk management." Soon after, option trading volume also exploded (DOI: 10.1177/29767032231211902). Half a century after its announcement, this model permeates risk management across every corner of the market, stretching from stocks, bonds, and foreign exchange to commodities and credit (DOI: 10.1142/s2010139225400063).

Yet the story of this equation is not solely one of triumph. A hedge fund in which two Nobel laureates who helped create this formula participated as partners collapsed in 1998. A hedge fund is a fund that pools money from investors and operates it with diverse strategies. The market has since witnessed moments, time and again, where the assumptions of this model diverged from reality.

This article follows, in chronological order, how a single mathematical idea was born, how it opened up a massive market, and why it exposed its own limits in the middle of a crisis. The starting point is the mathematics that handles the random movements of stock prices. Knowing Black-Scholes means knowing both the elegance of its formula and the vulnerability of its assumptions.

Starting point: How to price options

Even before Black-Scholes, people bought and sold options. According to an economic history study analyzing the interwar commodities options market, traders in the era before formulas may have also guessed "fair" prices in their own way. Long before the Black-Scholes-Merton model appeared, they may have intuitively adjusted prices when market conditions changed. Those prices were closer to the later theoretical values than previous studies suggested (DOI: 10.1111/ehr.12935). This article reads that as follows. The problem was not that "they had no idea of the price at all," but that "there was no principle to explain why it was that price."

The seed of that principle lay, unexpectedly, in a doctoral dissertation at the turn of the 20th century. In his 1900 thesis, French mathematician Louis Bachelier modeled stock price movements as random Brownian motion. Brownian motion is a random movement where it is impossible to predict which direction it will move next.

His life remained a lonely story (DOI: 10.37282/991819.19.84). It took another 70 years for his ideas to return to the heart of finance.

Turning Point 1973: Three People and a Single Equation

The core of the problem Black and Scholes wanted to solve was this. The value of an option depends on where the future stock price will go, but the future is unknowable. What can we hold onto, then?

Their breakthrough was the idea of no-arbitrage. Arbitrage is a trade that makes free profit without risk, and no-arbitrage is the premise that no such opportunities remain in the market. If you combine an option and its underlying asset (a stock) in a specific ratio, you can build a portfolio (a bundle of various assets) where risk instantly cancels out whether the stock price goes up or down. A risk-free portfolio should earn only the risk-free interest rate, which is the interest provided by a riskless asset. If it could earn more than that, a free arbitrage opportunity would arise, and the market would instantly wipe out that opportunity. This principle that "risk-free returns cannot exist" became the cornerstone of modern financial engineering (DOI: 10.1093/actrade/9780198787945.003.0003).

The technique of continuously adjusting this combination ratio over time to eliminate risk is dynamic hedging. Hedging is the act of blocking a potential loss on one side with an offsetting position. Dynamic hedging is also a self-financing replication strategy that "replicates" an option with stocks and bonds (DOI: 10.31410/eraz.2024.157).

To formulate this intuition into a rigorous equation, Itô's lemma was necessary—a core tool of stochastic calculus, which is math that handles randomly moving quantities as if differentiating and integrating them. It is a tool that allows us to handle randomly fluctuating stock prices as if differentiating them. Thanks to this tool, the partial differential equation (an equation describing relationships among rates of change) that option prices must satisfy and its solution, meaning the formula, were derived (DOI: 10.61173/b0jadz18; DOI: 10.47086/pims.956201).

Here, Robert Merton's contribution was decisive. Merton introduced continuous-time stochastic models (Itô calculus) into the theory of household consumption and investment decisions and formalized dynamic hedging. Continuous time is an approach that treats time as flowing seamlessly without breaks. In this way, Merton built a bridge between complete market equilibrium theory and the reality of personal financial planning.

Financial instruments whose value comes from the prices of other assets are called derivatives, with options being the prime example. In 1997, Merton and Scholes received the Nobel Prize in Economic Sciences for "a new method to determine the value of derivatives" (DOI: 10.1146/annurev-financial-100520-074656). (Fischer Black passed away in 1995, so he could not receive it together due to the rule that Nobel Prizes are not awarded posthumously.)

Coincidentally enough, the Chicago Board Options Exchange (CBOE) opened its doors in the exact same year the formula was published, 1973. From this point, standardized options began trading on an exchange for the first time. Theory and the market met in the same year.

Paradigm Shift: The Explosion of the Derivatives Market

What Black-Scholes changed was not just the price of a single option, but the very way finance was handled. Once option prices became objects of "calculation" rather than "gut feeling," trading could grow in scale with peace of mind. The model was not merely a dominant option pricing model. It fundamentally changed how financial theory, institutions, and corporate finance were handled, and its uses are ubiquitous in risk management across today's markets (DOI: 10.1142/s2010139225400063).

This idea expanded beyond options. Merton generalized option pricing theory to create Contingent Claims Analysis. Contingent claims refer to rights that can be received only when certain conditions are met. This analysis opened the way to evaluate all kinds of claims, such as corporate debt or government guarantees, "like options." This became the foundation of financial engineering, which designs new financial products (DOI: 10.1146/annurev-financial-100520-074656). The stochastic calculus and hedging concepts born during the process of deriving option prices led to the growth of a field in its own right, namely financial mathematics (DOI: 10.61173/b0jadz18).

The model also left behind an unexpected gift. Volatility is a value indicating how wildly a stock price fluctuates. By solving the formula backward, one can inversely calculate how much volatility the market price is currently "expecting." The volatility inversely obtained from market prices in this way is implied volatility. Implied volatility has established itself as a forecasting tool to gauge the future and a yardstick for risk and portfolio management (DOI: 10.54254/2754-1169/17/20231048).

An empirical study shows this difference in numbers. When pricing options using call option data from a U.S. company (EA), using volatility calculated from historical data resulted in an average error of 42.46%. Using volatility implied by the market sharply reduced the error to 9.17% (DOI: 10.54254/2754-1169/2024.18703). The market had now begun to read its own anxiety through the numbers spit out by the formula.

The First Crack: A Smile Where There Should Not Be One

The Black-Scholes formula stands on one core assumption: that the volatility of the underlying asset is the exact same single value for any option. If this assumption were correct, when we plot the implied volatilities of options with the same expiration date by strike price, we should get a flat straight line. But the graph in reality is not flat. It draws a smile with both ends raised, or a skew leaning to one side. This volatility smile is evidence that the formula's ideal assumptions clash with the real market (DOI: 10.54691/bcpbm.v38i.4314).

There is a dramatic turning point behind why this crack became clear: "Black Monday," the day in 1987 when the U.S. stock market crashed in a single day. After this event, a permanent change occurred in the global options market. The implied volatility skew, which was barely visible before then, began to appear.

Delta hedging is intertwined with this change. Delta is the degree to which an option value moves along when the underlying asset price moves. Delta hedging is a hedge that aims to eliminate risk from price direction by holding the underlying asset in the opposite direction matching this degree. One analysis explains this change as a shift in investors' perception of delta hedging. According to this analysis, before the crash, investors tacitly believed that "a delta-hedged portfolio is risk-free." After the crash made portfolio-insurance-style delta hedging fail miserably, the lesson that "even a delta-hedged portfolio is risky" was deeply imprinted, and this awareness appeared as a distorted volatility curve (DOI: 10.3934/qfe.2022021). Meanwhile, one paper argues that the volatility smile is not a simple calculation error. In that volatility and price changes are intertwined through market imbalance, it is a "real" phenomenon (DOI: 10.54946/wilm.11108).

The ways assumptions break down in the actual market are varied. One paper deals with the volatility smile through simulation using synthetic data, under the premise that discrepancies with market prices arise because of the model's ideal assumptions. This paper reports that the growth rate of stock returns can replace the smile (DOI: 10.54254/2754-1169/2024.18702).

Institutional constraints also break down assumptions. Short selling is a transaction where you borrow a stock you do not own, sell it first, and buy it back later to return it. A study comparing the Chinese market, where short selling is banned, with the U.S. and India showed that short-selling restrictions violate Black-Scholes premises (frictionless trading, free short selling) and generate systematic pricing errors (DOI: 10.54254/2754-1169/2026.bj30805). Frictionless trading refers to trading without transaction costs or restrictions.

In another study targeting Chinese retail (individual) investors, the volume-weighted mean absolute out-of-sample error was 12.03% of the option premium. Out-of-sample error is the error measured by applying the model to new data not used when building it, and option premium is the price of an option. This study warns that making investment decisions trusting only the model can adversely affect performance (DOI: 10.24135/afl.v11i.589).

Sometimes the model does not work in a market with low volatility. In a study applying the formula to 28 stocks on the Ghana Stock Exchange, the call option values of 18 stocks were calculated as 0, and there were only 7 where values came out for both call and put options. The authors conclude that 21 companies are difficult to become derivative underlying assets and attribute the cause to low volatility (DOI: 10.14738/abr.105.12350).

Crisis: The Collapse of the Nobel Laureates' Fund

The fragility of assumptions not remaining merely a matter of theory but becoming a real-world disaster happened in 1998. It was a hedge fund called Long-Term Capital Management (LTCM). Myron Scholes and Robert Merton themselves, who created the Black-Scholes-Merton formula, were participating as partners in this fund. It was just one year after they won the Nobel Prize.

LTCM squeezed profits out of tiny market price discrepancies by relying on sophisticated models and massive borrowing, namely leverage. Leverage is the practice of expanding investment scale with borrowed money, which magnifies both profits and losses. However, the 1997 Asian financial crisis and the 1998 Russian default (an event where a country declares it cannot pay back its debt) overlapped. Added to this was the excessive risk the fund took on, causing the fund to collapse (DOI: 10.58830/ozgur.pub483.c1987). This case is evaluated as showing that the risks of a fund with high leverage must be well managed.

A study that retraced the LTCM case from the perspectives of Keynes and Minsky summarizes the lesson of this event as follows. Model assumptions that determine the validity of a theory must be realistic. If this point is ignored, it is reasonable to expect that financial shocks will repeat (DOI: 10.21301/eap.v14i1.13).

We need to be honest here. This article considers it difficult to blame the 2008 global financial crisis solely on Black-Scholes. A working paper (2026) analyzing financial history points out the blind spots of financial models in general. Because models are often calibrated to fit stable and observable environments, they are least reliable during rare extreme events that heavily dictate outcomes (DOI: 10.2139/ssrn.6622980). Calibration is the task of adjusting a model's values to observed data.

Regarding the 2008 crisis, the Gaussian copula model used for pricing credit derivatives—products that buy and sell the default risk of loans or bonds—is often brought up. One commentary contrasts Black-Scholes as a "good" tool and the Gaussian copula as not. In doing so, it points out the difference between the maker and the user, and the importance of choosing the right tool for the job (DOI: 10.54946/wilm.12226).

Common Misconceptions and Honest Limitations

The biggest misconception surrounding Black-Scholes is the belief that "because it is a precise formula, it gives the right answer." The beauty of a formula and the realism of its assumptions are two different things. When you examine the premises the model stands on one by one, its limitations become clear.

The normal distribution assumption. The formula assumes that stock returns (the rate at which prices go up and down) follow a random walk and move roughly close to a normal distribution (DOI: 10.1177/29767032231211902). A random walk is movement where the next move is determined independently of the immediate past move, and a normal distribution is a bell-shaped distribution where most values cluster near the average and extremes are rare. However, actual market returns have fatter tails than a normal distribution. Having fat tails means that extremes like crashes and spikes happen more often than the theory predicts. There is an assessment that the collapse of LTCM we saw earlier is also connected to the problem that the model did not adequately reflect rare events.

The constant volatility assumption. The premise that volatility is a single value regardless of expiration and strike price is directly refuted by the volatility smile we saw earlier (DOI: 10.54691/bcpbm.v38i.4314). Reality's volatility changes over time and for each option.

Continuous hedging and frictionless market assumptions. The proof of the formula assumes that stocks and options can be bought and sold at any time without transaction costs (DOI: 10.1177/29767032231211902). In reality, there are transaction costs, the market is not open 24 hours a day, and during sudden market plunges, you cannot even sell at the price you want. Institutional frictions like short-selling bans also break this premise (DOI: 10.54254/2754-1169/2026.bj30805). Perfectly continuous hedging is only possible in theory.

What these limitations tell us is that Black-Scholes is not a physical law describing the "truth" of the market, but a useful approximation. An approximation means a value that is not exact but close enough to be useful. The moment you blindly trust a model like a physical law, especially the moment you treat rare extreme events as if they "do not happen," the risk quietly piles up and explodes all at once (DOI: 10.2139/ssrn.6622980). This article reads that the real danger of the formula lies not inside the formula itself, but in the overconfidence of the people using it.

Why It Is Still Used Anyway

Even with so many assumptions breaking down, Black-Scholes has survived as the industry standard for half a century (DOI: 10.1177/29767032231211902). Why is that? There are three reasons.

First, there are reports that the formula works quite well even where the assumptions break down. Although the original derivation relies on strong statistical assumptions, it yields good price estimates in practice. One paper presents a derivation where the only practical assumption is "scale-invariance". Scale-invariance is a symmetry where properties do not change even if units or scales are changed. The authors view this as the reason the equation works even without strong assumptions (DOI: 10.1142/s0218488520400012). Although it is still a pre-peer-review preprint, some papers claim that you get the same form of the formula even if you do not know the distribution of returns. The same is said to hold true even when allowing for cases where volatility and interest rates change randomly (stochastic volatility and stochastic interest rates) (DOI: 10.32388/tl11vf).

Second, in academic discussions, extensions dealing with these limitations have already been made. This involves plugging factors like stochastic volatility, jumps (movements where prices suddenly leap), and market imperfections into the model (DOI: 10.31410/eraz.2024.157). During crises, people also switch to alternative models. During the 2020 COVID crisis and the negative oil futures price incident in April of that year, major markets temporarily shifted to the Bachelier model, which can also handle negative strike prices (DOI: 10.1057/s41283-025-00160-0). This was Bachelier returning to the stage after roughly 100 years. However, this paper reports that there is no statistical difference in the predictive power of the two models for U.S. stock index options.

Third, and most fundamentally, this article reads Black-Scholes closer to a common language than an answer sheet. Implied volatility is used as a predictive tool to gauge the future and a yardstick for risk and portfolio management (DOI: 10.54254/2754-1169/17/20231048). Thanks to that, market participants can communicate "this option has this much uncertainty priced in" using the same scale. The fact that everyone converses using the same framework even while standing on wrong assumptions seems to be one reason this model has survived.

To Go Deeper

The trajectory of Black-Scholes converges on a single lesson: The more powerful a model is, the safer it is only in the hands of those who know when its assumptions break. Starting from Bachelier's forgotten paper (DOI: 10.37282/991819.19.84), made to treat gambling as science in the hands of three men in 1973 (DOI: 10.1177/29767032231211902), explosively increasing option trading (DOI: 10.1177/29767032231211902), and revealing its own limits in the 1987 smile and the 1998 collapse (DOI: 10.3934/qfe.2022021; DOI: 10.21301/eap.v14i1.13), the history of this equation is also the history of modern finance learning how much it should trust itself.

If you want to dig deeper into the relationship between models and markets, research discussing why financial firms should read risk from repeating structural patterns rather than statistical frequencies is a good next step (DOI: 10.2139/ssrn.6622980). If you are curious about why the formula works well even without strong assumptions, and what the mathematical root of that robustness is, symmetry-based derivation becomes the next step (DOI: 10.1142/s0218488520400012). Ultimately, the real legacy left by Black-Scholes is not a single formula, but the bold belief that "market uncertainty can be made into a calculable object" — and the humility, bought at a high price, regarding how far that belief can be pushed.

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Source Papers

• "The Black–Scholes–Merton OPM 50 Years Later," The Quarterly Journal of Finance (2025) — DOI: 10.1142/s2010139225400063 — 50 years since publication, a revolution that fundamentally changed financial theory, institutions, and corporate finance, its ubiquity in market-wide risk management, and CCA.
• "Quantum Uncertainty and the Black-Scholes Formula," Quantum Economics and Finance (2024) — DOI: 10.1177/29767032231211902 — Putting option pricing on a rational and objective foundation in 1973, a shift in perception from gambling to scientific risk management, an explosion in option trading, and the assumptions and flaws of random walk, constant volatility, and zero transaction costs.
• "A comparative analysis of option pricing models: Black–Scholes, Bachelier, and artificial neural networks," Risk Management (2025) — DOI: 10.1057/s41283-025-00160-0 — The industry standard since 1973, temporarily switching to the Bachelier model during the 2020 COVID crisis and April negative oil price event to accommodate negative strike prices.
• "Option Pricing Models: The Evolution of the Black-Scholes-Merton Model," ERAZ (2024) — DOI: 10.31410/eraz.2024.157 — Assumptions, advantages (volatility estimation, self-replicating hedging), limitations (constant volatility, complete market), and expansion to stochastic volatility, jumps, and market incompleteness.
• "A Study of Black-Scholes-Merton Model," Finance & Economics (2024) — DOI: 10.61173/b0jadz18 — Derivation from stochastic integration and Itô's lemma, driving the growth of the academic discipline of financial mathematics.
• "Robert C. Merton and the Science of Finance," Annual Review of Financial Economics (2020) — DOI: 10.1146/annurev-financial-100520-074656 — Continuous-time stochastic models (Itô calculus) and dynamic hedging, the 1997 Nobel Prize, contingent claim analysis (CCA), and financial engineering.
• "Commodity option pricing efficiency before Black, Scholes, and Merton," The Economic History Review (2019) — DOI: 10.1111/ehr.12935 — Commodity option traders between the World Wars intuitively approximating 'fair' values even before BSM.
• "On Louis Bachelier," Inference (2019) — DOI: 10.37282/991819.19.84 — Bachelier's only remaining contribution, his doctoral dissertation, and his lonely life.
• "Redde Rationem: Keynes, Minsky and the Collapse of Long-Term Capital Management Twenty Years Later," Issues in Ethnology and Anthropology (2019) — DOI: 10.21301/eap.v14i1.13 — The errors of modern financial theory seen through the bankruptcy of LTCM, the lack of realism in model assumptions, and repeating financial shocks.
• "Hedge Fonlar ve Long Term Capital Management Örnek Olayı" (2024) — DOI: 10.58830/ozgur.pub483.c1987 — The 1998 failure of the high-leverage hedge fund LTCM under the combined pressure of the 1997 Asian and 1998 Russian crises.
• "Financial market disruption and investor awareness: the case of implied volatility skew," Quantitative Finance and Economics (2022) — DOI: 10.3934/qfe.2022021 — Permanent changes in global option markets after the 1987 crash, the emergence of the implied volatility skew, and 'even a delta-hedged portfolio is risky'.
• "Keep on smiling: Market imbalance, option pricing, and the volatility smile," Wilmott (2023) — DOI: 10.54946/wilm.11108 — The volatility smile is a 'real' phenomenon where volatility and price changes are linked through market imbalance.
• "A Study of Option Pricing Models with Market Price Adjustments: Empirical Analysis Beyond the Black-Scholes Model," AEMPS (2024) — DOI: 10.54254/2754-1169/2024.18702 — The 1973 Black-Scholes formula, the cornerstone of financial theory, and the empirical discrepancy and volatility smile arising from idealized assumptions.
• "The Progress of Black-Scholes Model and Black-Scholes-Merton Model," BCP Business & Management (2023) — DOI: 10.54691/bcpbm.v38i.4314 — The 1973 proposal of BS and Merton's BSM modification, and the limitations (volatility smile) and extensions originating from idealized assumptions.
• "Limitations of the Black-Scholes Model in China: Short-Selling Constraints...," AEMPS (2025) — DOI: 10.54254/2754-1169/2026.bj30805 — Short-selling bans violate the frictionless, free short-selling premise, causing systematic pricing errors, with institutional comparisons to the US and India.
• "Is the Black–Scholes Model Good Enough for Retail Investors in China?," Applied Finance Letters (2022) — DOI: 10.24135/afl.v11i.589 — Out-of-sample absolute error premium of 12.03%, and the risks of relying solely on the model.
• "Assessing the Accuracy of Implied Volatility and Historical Volatility in the Black Scholes Merton Model," AEMPS (2024) — DOI: 10.54254/2754-1169/2024.18703 — Average error of 9.17% using implied volatility vs 42.46% using historical volatility.
• "The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange," Archives of Business Research (2022) — DOI: 10.14738/abr.105.12350 — Due to low volatility, call option values for 18 out of 28 stocks are 0, making them unusable as derivative underlying assets.
• "Applied Financial History: Expanding the Vision of Modern Finance" (2026) — DOI: 10.2139/ssrn.6622980 — Models calibrated to stable environments are most vulnerable in rare extreme events, bearing the structural blind spot of the gradual accumulation of risk (drift).
• "Why Black-Scholes Equations Are Effective Beyond Their Usual Assumptions: Symmetry-Based Explanation," IJUFKS (2020) — DOI: 10.1142/s0218488520400012 — Can be re-derived using scale invariance alone without strong statistical assumptions, explaining why it remains valid outside its assumptions.
• "Relaxing the Black-Scholes Assumptions Without Changing the Price Formula" (2023) — DOI: 10.32388/tl11vf — Maintaining the same form as the classical formula even when return distributions are unknown and stochastic volatility and interest rates are allowed.
• "The classical theory of option pricing," Mathematical Finance: A Very Short Introduction (2019) — DOI: 10.1093/actrade/9780198787945.003.0003 — No-arbitrage pricing theory, binomial and continuous-time models, the Black-Scholes formula, and the First Fundamental Theorem of Asset Pricing.
• "Derivation of Black-Scholes Equation Using Itô's Lemma," Proceedings of International Mathematical Sciences (2021) — DOI: 10.47086/pims.956201 — The most influential financial equation, eliminating risk through a hedging position, and derivation based on Itô's lemma.
• "Why the Black-Scholes Model is Good and the Gaussian Copula is Not," Wilmott (2026) — DOI: 10.54946/wilm.12226 — The difference between the 'maker' and the 'user', and the issue of using the right tool in the right place.
• "Research on the Development of Implied Volatility in Option Pricing," AEMPS (2023) — DOI: 10.54254/2754-1169/17/20231048 — The history of options and option pricing models, and the two functions of implied volatility (future forecasting and risk/portfolio management).