Philippe Alezard
With Bachelier, finance had found its founding insight: stock market prices can be thought of as random movements and options as rights whose value depends on the probability of future prices. With Wiener, this intuition is given a rigorous mathematical foundation: the Brownian motion becomes a continuous process, with independent and Gaussian increments, capable of giving shape to uncertainty. Finally, with Markowitz, uncertainty is no longer merely described; it becomes the subject of a rational decision, organised at the level of a portfolio. However, a decisive question remained: if chance governs prices, is it possible to give a rigorous price to a contract that relates precisely to this chance?
This is the question that Fischer Black, Myron Scholes and Robert C. Merton answered in the early 1970s. Their contribution is not just about producing a famous formula. It transforms the very nature of financial valuation. Before them, an option essentially appeared as a bet: the right to buy or sell an asset at a fixed price, in an uncertain future. After them, the option became an object that can be replicated, hedged and valued based on arbitrage reasoning. The price is no longer just an opinion about the future; it becomes the logical consequence of a dynamic hedging strategy.
The intellectual and financial context of the 1960s and 1970s is essential to understanding this breakthrough. In the United States, finance was then in the process of becoming an independent academic discipline. Business schools were moving closer to the markets, price databases were developing, and Markowitz’s portfolio theory had introduced variance and covariance into the language of investors, while the CAPM (Capital Asset Pricing Model) of Sharpe, Lintner and Mossin sought to establish a balanced relationship between expected return and systematic risk. At the same time, the options markets, long dominated by over-the-counter transactions, were undergoing an institutional change. In April 1973, the Chicago Board Options Exchange opened its doors and offered, for the first time, an organised market for standardised stock option contracts. The publication of the Black-Scholes model came in the same year, almost at the exact moment when the market needed a common language to price, compare and hedge these new instruments.
Fischer Sheffey Black was born in 1938 in Washington D.C. His career path was less linear than that of traditional academic economists. He first studied physics at Harvard before turning to applied mathematics and computer science. He obtained a doctorate from Harvard in 1964 in applied mathematics with a thesis devoted[1] to what we would now call artificial intelligence, at a time when this discipline was still in its infancy. Nothing in this initial career path would have predicted that he would become one of the founders of modern finance. However, this interdisciplinary background, spanning physics, calculus, logic and dynamic systems, undoubtedly explains his ability to view markets as formalisable mechanisms.
After his studies, Black worked, among other places, at Arthur D. Little, a consulting firm where he met Jack Treynor[2], one of the pioneers of modern portfolio theory and the equilibrium model of financial assets. This meeting was decisive. Treynor introduced Black to financial problems and encouraged him to think about the links between risk, return and market equilibrium. Black then developed a very personal approach to finance: he was less interested in institutions than in the abstract forces that must govern prices if arbitrage opportunities are eliminated. His mind is that of a theoretical engineer: he seeks the hidden constraint, the necessary relationship, the equation that must be true if the market is consistent.
Myron Scholes was born in 1941 in Timmins, Ontario, Canada, to a family with deep ties to the business world. In his Nobel autobiography, he emphasises the importance of this family environment: from a very young age, he was interested in trade, accounting, probability and risk. An eye operation during his adolescence disrupted his schooling and forced him to develop special working methods based on listening, memory and conceptualisation. This personal constraint would play an important role in his way of thinking: Scholes was not only a calculation technician, he was attentive to the economic structure of problems.
He continued his studies at McMaster University, then at the University of Chicago, where he obtained his doctorate in 1969. Chicago was then one of the most powerful centres of the new financial economy. Eugene Fama was working there on market efficiency, Merton Miller on corporate finance, Milton Friedman on monetary theory, and the university’s intellectual tradition valued equilibrium reasoning, price consistency and the discipline imposed by competitive markets. There, Scholes received an economic education that was very different from the more institutional European tradition: the objective was not only to describe the markets, but to deduce what prices should be in a world where agents exploit all the possibilities of arbitrage.
Robert Cox Merton was born in 1944 in New York. His father, Robert K. Merton, was one of the most influential sociologists of the 20th century, known in particular for his work on “self-fulfilling prophecy” and “unintended consequences”. Young Robert therefore grew up in an exceptional intellectual environment, but quickly chose a different path. He first studied mathematical engineering at Columbia in 1966, then applied mathematics at the California Institute of Technology, where he obtained his MS in 1967. He then joined MIT for his PhD in economics, under the supervision of Paul Samuelson. This point is crucial: Samuelson is one of Bachelier’s successors in American finance. Through him, Merton inherited a tradition in which mathematical physics, probability and economics could be brought together in a single analytical architecture.
Merton had a more advanced technical mastery of stochastic calculus than most financial economists of his time. Where Black and Scholes construct a highly powerful arbitrage intuition, Merton gives the model its mathematical generality. His 1973 article[3], “Theory of Rational Option Pricing”, published in the Bell Journal of Economics and Management Science, broadens the framework, clarifies the conditions of validity, establishes general restrictions on option prices and embeds the formula in a rational theory of contingent assets. This is why the term “Black-Scholes-Merton model” is now more accurate than the shorter term “Black-Scholes”.
The initial problem is seemingly simple. A call option gives its holder the right to buy a share at a price fixed in advance, called the strike price, at a future date. If, on that date, the share price is higher than the strike price, the option has a positive value; otherwise, it expires without value. This asymmetric structure makes the contract difficult to value. The buyer benefits from the upside potential while limiting their loss to the premium paid. The seller, on the other hand, collects this premium but is exposed to a potentially significant loss if the share price rises sharply. How can a fair price be determined for such a contract?
Before Black, Scholes and Merton, several approaches already existed. Bachelier had proposed an initial formula in 1900, based on an arithmetic Brownian motion. But his model implicitly allowed for negative prices and was not based on a modern theory of arbitrage. Other authors had tried to value options based on expected stock returns, investor preferences or subjective probabilities. The central problem remained: the price of an option seemed to depend on the expected return on the share, and therefore on the attitude of investors towards risk. However, this expected return is precisely one of the most difficult quantities to observe.
The revolution of Black, Scholes and Merton consists in showing that, under certain assumptions, the price of the option does not depend on the expected return of the share. It depends on the current price of the underlying, the strike price, the time remaining until maturity, the risk-free rate and volatility. The psychological or subjective parameter disappears. This disappearance is not a technical detail; it is the heart of the revolution. It means that an option can be valued without knowing investors’ return expectations, because a hedging strategy can be constructed that locally eliminates the risk.
The fundamental idea is that of the replicating portfolio. It assumes that, over a period of time, the share price follows a geometric log-normal Brownian motion:
dSₜ = μSₜ dt + σSₜdWₜ
The share price increases on average according to a trend index μ, the expected return, but it is disturbed by a random term, the volatility of the expected return, σS ₜ dW ₜ where W ₜ represents the Wiener process. An option on this stock itself has a value that varies with the price of the underlying and with time. By rigorously introducing the Itô lemma [4], Merton’s rigorous work, it becomes possible to express the infinitesimal variation of this option. Black, Scholes and Merton then show that by combining a position in the option and an appropriate position in the stock, it is possible to form a portfolio whose random part cancels out. This hedged portfolio becomes locally risk-free; it must therefore yield the risk-free rate. If it yielded more, arbitrage would be possible; if it yielded less, the reverse arbitrage would arise.
It is from this logic that the Black-Scholes-Merton differential equation [5] arises:
∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S − rV = 0
This equation expresses a very profound idea: the price of a derivative is not fixed by the subjective expectation of gain, but by the possibility of neutralising its risk by means of dynamic hedging. The term μ, i.e. the expected return on the share, has disappeared. What remains is the volatility σ. Modern finance here makes a major shift: it no longer seeks to predict the direction of the market, but seeks to measure the intensity of its uncertainty.
For a European call option that does not pay a dividend, the closed solution to this equation has become one of the most famous formulas in finance:
C = S₀N(d₁) − Ke⁻ʳᵀN(d₂)
where:
d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T
In this formula, S₀ is the current price of the share, K the strike price, T the time to maturity, r the risk-free rate, σ the volatility and N the distribution function of the normal distribution. The presence of the normal distribution recalls the direct connection with Bachelier and Wiener, but the economic architecture is now different. Brownian motion is no longer used only to describe a probable trajectory; it becomes the element to be neutralised in a hedging strategy.
The concept of delta plays a central role here. The delta measures the sensitivity of the option price to a change in the price of the underlying. In the case of the Black-Scholes-Merton European call, the delta is N(d₁). Hedging an option then amounts to holding a quantity of shares proportional to this delta, and then continuously adjusting this quantity over time and according to market movements. This is what is called dynamic hedging or the well-known delta management that all derivatives traders implement. The option ceases to be an isolated bet; it becomes a position within a living portfolio, regularly rebalanced.
This insight directly extends Markowitz’s work while going beyond it. For Markowitz, the portfolio is a combination of assets intended to organise the risk-return pair. For Black, Scholes and Merton, the portfolio becomes a replication instrument. It is no longer just a matter of reducing risk through diversification, but of building a combination of assets whose behaviour exactly reproduces that of the contract to be valued. Diversification dilutes risk; dynamic hedging neutralises it locally. It is a change in nature.
Merton’s contribution is decisive in this transformation. His work is not limited to confirming the formula. He generalises the method, extends the reasoning to other contingent assets, clarifies the role of the absence of arbitrage, and introduces a much broader vision of finance as the science of intertemporal valuation. In his later work, Merton applies these ideas to corporate debt, insurance, capital structure and risk management. A risky bond can thus be seen as a risk-free bond minus a default option, or, conversely, the shares of an indebted company can be interpreted as a call option on the value of the firm’s assets. Options theory then becomes a general grammar of corporate finance.
The practical scope of the model is considerable. As soon as organised options markets opened, the formula provided traders with a common benchmark. It makes it possible to compare observed prices with theoretical prices, to calculate sensitivities, to construct hedges and to manage option books in a systematic way. It gives rise to a whole language: delta, gamma, theta, vega, rho. These “Greek letters” are not mere technical refinements; they become the coordinates of risk. The trader no longer holds just an option; they hold a set of sensitivities to changes in the underlying asset, time, volatility and rates.
The model also transforms volatility itself. In the formula, volatility is the only essential parameter that is not directly observable in the same way as the share price, the strike price, the maturity or the risk-free rate. From then on, the markets will reverse the formula: from the observed price of an option, they deduce the implicit volatility. Volatility thus becomes a market price. It is no longer just a statistic calculated on past variations; it becomes the collective expression of anticipated uncertainty. This is one of the most powerful legacies of Black-Scholes-Merton: having helped transform uncertainty into a listed item.
This transformation paved the way for the rise of modern derivative finance. Stock options are just the beginning. The same logic extends to index options, currency options, interest rate options, caps, floors, swaptions, convertible bonds, structured products and credit instruments. Each time, the central idea remains: identify the underlying asset, describe its dynamics, construct or approximate a replication strategy, then deduce a price compatible with the absence of arbitrage. Even when the closed Black-Scholes formula is no longer applicable, its spirit remains present in the numerical methods, binomial trees, Monte Carlo simulations and partial differential equations used by banks.
However, the limits of the model must be measured with as much care as its power. The assumptions are strong: frictionless markets, no transaction costs, possibility of short selling, constant risk-free rate, constant volatility, continuous trading, perfect liquidity, no price jumps, lognormal distribution of the underlying. However, real markets violate almost all of these assumptions. Prices can jump, liquidity can disappear, transaction costs can become significant, correlations can break, and volatility can change abruptly. Dynamic hedging assumes a world where one can continually adjust one’s positions; crises remind us that the real time of the markets is never the continuous time of equations.
The most famous empirical anomaly is the volatility smile. If the model were perfectly true, all options written on the same underlying and with the same maturity should imply the same volatility. In practice, implied volatilities vary according to strike prices and maturities. Options that are very out of the money or very in the money often incorporate different premiums, in particular because the markets assign a higher probability to extreme events than the simple lognormal distribution assumes. The 1987 crash made this limitation particularly visible: the options markets began to incorporate a more structured fear of sudden falls, which the initial formula could not represent.
These limitations do not invalidate the model; on the contrary, they explain its longevity. Black-Scholes-Merton is not a complete snapshot of reality. It is a matrix. From it, we can understand what needs to be corrected: local volatility, stochastic volatility, jumps, transaction costs, incomplete markets, liquidity constraints, counterparty risks. Much of contemporary quantitative finance can be read as an ongoing dialogue with this model: either to use it as an approximation, or to extend it, or to show where its assumptions cease to work.
The story of the Nobel Prize also illustrates the collective and sometimes tragic nature of this discovery. In 1997, Myron Scholes and Robert Merton received the Bank of Sweden Prize[6] in Economic Sciences in Memory of Alfred Nobel “for a new method for determining the value of derivatives”. Fischer Black, who died in 1995, could not be recognised, as the prize is not awarded posthumously. His absence is a reminder that great scientific discoveries do not always allow themselves to be confined within the institutional rules of recognition. Black’s name, however, remains inseparable from the equation, to the point that common usage continues to refer to Black-Scholes, even though we know how much Merton contributed to the generalisation of the theory.
The reception of the model involves another irony. A few years after the Nobel Prize, Scholes and Merton would be associated with Long-Term Capital Management, a hedge fund whose collapse in 1998 would become a symbol of the dangers of over-reliance on models. It would be simplistic to conclude that Black-Scholes-Merton was wrong. Rather, the episode shows that a valuation model is not a complete model of the world. Knowing how to value an instrument under certain assumptions does not mean controlling debt, liquidity, mimetic behaviour or market reactions in a stress situation. Mathematical finance provides tools for thought; it never dispenses with judgement on the conditions under which these tools are used.
The lasting contribution of Black, Scholes and Merton therefore lies in a profound conceptual shift. Bachelier had shown that probability could be applied to speculation. Wiener had given a rigorous structure to continuous random motion. Markowitz had shown that the investor could rationally organise a portfolio exposed to risk. Black, Scholes and Merton took a further step: they showed that certain risks can be transformed into prices by dynamic replication. Finance no longer merely measures uncertainty; it learns to create it, to slice it, to transfer it and to hedge it.
Conclusion
The Black-Scholes-Merton equation has become one of the most powerful symbols of modern finance because it condenses a major intellectual transformation into a few lines. It does not just say how much a European option is worth in an idealised world. It states that the value of a contingent asset can be deduced from a hedging strategy and not from a simple forecast. In doing so, it replaces the psychology of anticipation with a logic of arbitrage.
Its legacy is everywhere: in trading rooms, pricing software, risk management, accounting standards, stress tests, volatility markets and the design of structured products. Even its limitations have become fruitful, as they have paved the way for stochastic volatility models, jump models, numerical methods and a more detailed consideration of liquidity and distribution tails.
But perhaps the most important lesson lies elsewhere. Black, Scholes and Merton did not eliminate chance. They showed that, under certain conditions, it could be locally neutralised by a financial construct. This is the whole ambivalence of quantitative finance: it makes uncertainty calculable without making it disappear. It gives the market a grammar of extraordinary power, but this grammar remains dependent on the world in which it is used. When this world remains close to the assumptions, the model provides clarity. When it moves too far away from them, the model can blind.
That is why their work belongs fully to a history of mathematicians and finance. It builds on Bachelier, Wiener and Markowitz, but it also marks a new threshold: the moment when the randomness of the markets becomes not only measurable and organisable, but replicable. From there, finance enters the age of derivatives, implicit volatility and risk engineering. The market is no longer just a place where securities are traded; it becomes a space where the very forms of uncertainty are constructed, broken down and reconstructed.
[1] He worked for 1 year at MIT on the subject with Marvin Minsky, considered the founding father of AI
[2] Jack Lawrence Treynor, American economist, considered by F. Black to be the father of the CAPM model https://people.duke.edu/~charvey/teaching/ba453_2006/french_treynor_capm.pdf
[3] Robert C. Merton, “Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science, vol. 4, no. 1, 1973, pp. 141-183.
[4] Ito Stochastic Differential Equations https://math.nyu.edu/~goodman/teaching/StochCalc/notes/drafts/l10.pdf
[5] https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf
[6] The Nobel Prize, “The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997”, press release and biographical notes on Myron Scholes and Robert C. Merton.