The development of the Black-Scholes formula: Theory, research and practice

If we look back over the history of modern financial markets, one of the most influential developments was the Black-Scholes option pricing formula. However, there are a number of misconceptions around its origin and the evolution of derivatives. To begin with, although Fischer Black, Myron Scholes, and Robert Merton’s seminal contribution to options pricing was published in 1973, ideas concerning the rationale and approaches for options valuation were already circulating in the mid- to late-1960s.

Speaking at the Bloomberg Quant (BBQ) Seminar, Peter Carr of NYU’s Tandon School of Engineering, suggested that we turn back the clock 50 years and evaluate what was and what was not known about options on November 18, 1969.

One of the first areas of possible confusion involves the three faces of the Black-Scholes work: there is the Black-Scholes partial differential equation (PDE), the Black-Scholes option pricing model, and also the set of Black-Scholes option pricing formulas. The PDE was, in fact, derived in June 1969 by Black alone, using the Capital Asset Pricing Model (CAPM). The option pricing formulas followed that same year, with Black and Scholes using a formula originally published by Sprenkle in 1961. And finally, the Black-Scholes model was actually a term introduced by Robert Merton, who showed that the CAPM was not needed to derive the PDE or the formulas. The formulas themselves were focused on valuing European call and put options. Black and Scholes found that by setting the expected return for the option and its underlying stock equal to the risk-free rate, the formula for the call valuation satisfied the PDE and boundary conditions.

Delving into options theory and practice: 1960s and 1970s

One of the key principles behind the Black-Scholes formula, risk-neutral valuation, was originally explored by De Finetti, Ramsey, Savage, and Arrow-Debreu, but did not achieve prominence in financial research until the 1970s. Other influences on the development of the formula and model include Bachelier (1900) and Boness (1964).

As a practical consideration, prior to 1973, calls and puts were not traded on exchanges; calls were introduced in 1973, with puts following in 1977. However, there was a considerable amount of activity around warrants, which were issued by companies on their own stock.

In 1967, Edward Thorpe and Sheen Kassouf came out with their book, Beat the Market, which focused on warrants. Just two year earlier, Paul Samuelson had published “Rational Theory of Warrant Pricing” and in 1968, he produced a critical review of Thorpe and Kassouf’s book in the Journal of the American Statistical Association.

So, even though the honors for the formula culminated a Nobel Prize in Economics for Scholes and Merton in 1997, Thorpe and Kassouf were using formulas in practice and Samuelson was coming very close to the same insights that led to the discovery of the PDE and the option pricing model.

Actuarial valuation vs. risk-neutral valuation

One of the significant differences in the approaches lay in the use of actuarial valuation as opposed to risk-neutral valuation. Actuarial valuation is a time-honored technique that values each security individually. Implementing actuarial valuation requires time-series stationarity, without necessarily requiring cross-sectional pricing consistency. Risk-neutral valuation, on the other hand, does require cross-sectional pricing consistency, without necessarily requiring time-series stationarity. Much of the earlier work on options pricing, including Samuelson’s, took the actuarial path, and he only addressed the case of call options, thereby missing out on the insights made possible when put call parity is imposed. Further, Samuelson restricted his assumptions by relying on consistency in time series under stationarity. If he had not been operating with those assumptions and constraints, Samuelson might have come up with the more complete option pricing formula himself.

The work of 1969 had strong merits, but in 1970, Merton found an alternative way to derive the Black-Scholes PDE and developed the put and call option pricing formulas based on delta-hedging continuously over time, and made a major contribution to the field, in spite of the fact that the famous formula is most often cited without his name attached. The story behind the development of the Black-Scholes formula highlights the interplay between economic theory, financial research, and market practice. And while some individuals win the highest accolades, the debates and dialogues across industry and academia shaped the thinking and the financial landscape for years to come.

Lightning Talks

Following a short Q&A session, Bruno Dupire of Bloomberg L.P., the host of the event, kicked off a series of “lightning talks,” 5-minute presentations where industry experts, researchers, and academics present a wide range of subjects to stimulate fresh thinking and interaction between various disciplines. Each talk examines a way that the industry is evolving and serves as an essential exploratory aspect of the Bloomberg Quant Seminar series.

In this session, Bruno Dupire presented his simple solution to a very old problem, namely: the isoperimetric problem. This work involves determining the maximum area that can be enclosed by a loop of a given perimeter. And the answer is… a circle.

In addition, Miroslav Visic of GulfSec explained how to have better relationships through financial engineering, Frederic Siboulet of Deloitte declared that Black-Scholes is dead and extolled the use of machines. Julien Guyon of Bloomberg L.P. outlined how new voting rules could finally resolve Brexit, Philip Lasser of Juilliard demonstrated the audio tapestry of Bach and Bobby Shackleton of Bloomberg L.P. offered insights on trading on weather data. Finally, Jessica Yu of Wells Fargo commented on deep learning for derivatives pricing.

About the Bloomberg Quant Seminar Series

The Bloomberg Quant (BBQ) seminar series takes place in New York and covers a wide range of topics in quantitative finance. Each session is chaired by Bruno Dupire, head of Quantitative Research at Bloomberg LP, and features a keynote speaker presenting his or her current research. This presentation is followed by several “lightning talks” of five minutes each in quick succession. This format gives the audience the opportunity to be exposed to a wider variety of topics.  Sign up to receive invitations to future events in this series.