The Arrival of Black-Scholes

This history helps traders who use the formula to price options and protect positions explain where the formula came from.

What happened

The paper "The Pricing of Options and Corporate Liabilities", by Fischer Black and Myron Scholes, appeared in the May–June 1973 issue of the Journal of Political Economy¹.

The paper presented a formula for the theoretical price of an option. The formula is called the Black-Scholes model.

In April of the same year, the Chicago Board Options Exchange (CBOE) opened².

The derivation and use of the formula are covered in The Black-Scholes Model.

Option valuation before the paper

Attempts to find the value of an option with a formula go back to 1900. The French mathematician Louis Bachelier valued a stock option in his doctoral dissertation at the Sorbonne. His assumptions did not fit reality: he set the interest rate at zero and allowed the stock price to move in ways that could make it negative³. Bachelier's approach is now called the Bachelier model⁴.

Later, Case Sprenkle, James Boness, and Paul Samuelson revised the formula. They assumed a stock price that cannot become negative (a lognormal distribution). They also added a non-zero interest rate. The formula Boness proposed in 1964 came close to the Black-Scholes formula³.

Even so, the methods before 1973 shared one difficulty. They worked out the expected value of the option at expiry and discounted it back to the present. The discount rate requires a risk premium (the extra return investors demand for taking on risk). Because this extra return reflects investors' attitude toward risk, it was hard to observe³.

Timeline

Figure description

The timeline moves from earlier work on valuing options, to the year the formula appeared with the opening of CBOE and the publication of the paper, then to use on the trading floor, the crash, and the prize decision. The filled circles mark the opening of CBOE and the publication of the paper. Spacing on the figure is not scaled to real elapsed time, which differs between the early research and the later events.

The idea in the paper and Merton's contribution

The research of Black and Scholes, and of Merton, answered the question of how to set the risk premium. The three showed that valuing an option does not require a risk premium. The risk premium does not disappear; it is already included in the stock price⁷.

The core of the idea is that combining the stock and the option creates a position without risk while the stock price makes small moves. If no risk remains in the position, its return should equal the risk-free interest rate. Otherwise, arbitrage would produce a sure profit. When the stock price or the remaining time changes, the mix of the two must change as well⁷.

Robert C. Merton was working on option valuation at the same time as Black and Scholes were writing the paper. Black later wrote that Merton "pointed out that if you assume continuous trading in the option or the stock, you can maintain a relation between them that is literally riskless". The final version of the paper derived the formula that way. In the same year, 1973, Merton presented the formula in his own paper and extended it. His extension showed that valuation does not need an assumption of market equilibrium; no-arbitrage (the absence of opportunities to earn a sure profit through arbitrage) is enough⁸.

How the formula was used

CBOE opened one month before the paper appeared. By 1975, its traders were using the formula on specially programmed calculators to price options and protect their positions². The three were not directly involved in founding the exchange, but the model gave traders a way to find values by calculation rather than by feel⁹. By 1997, thousands of traders and investors in markets around the world used the formula every day².

Of the inputs to the formula, only volatility cannot be observed directly. When an option's price is known, you can solve the formula backward to find the volatility the market expects¹⁰. The value found this way is implied volatility.

Later relaxation of the formula's assumptions

The original derivation of the formula rested on somewhat restrictive assumptions. Later research has relaxed them. It has treated random changes in interest rates or volatility, price jumps, and trading costs. All of these use the method of analysis that Black, Merton, and Scholes set out (building a riskless position to value the option)¹¹.

After the 1987 stock market crash

Before the stock market crash of October 19, 1987, the implied volatility of equity index options, lined up by strike price, was relatively flat. After the crash, the Black-Scholes formula came to greatly underprice S&P 500 put options that had short maturities and strike prices well below the current level. According to a research paper from the Federal Reserve Bank of Chicago (the Chicago Fed), the change lasted a long time. Rubinstein (1994) and others recorded this shape in S&P 500 option prices⁶.

The shape that appeared after the crash, in which implied volatility differs by strike price, is called the volatility smile¹². How it looks and how it is handled in valuation are covered in Volatility Smiles and Surfaces.

The 1997 Nobel Prize in Economic Sciences

On October 14, 1997, the Royal Swedish Academy of Sciences decided to award that year's Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (the Nobel Prize in Economic Sciences) to Merton (Harvard University) and Scholes (Stanford University). The reason given was "a new method to determine the value of derivatives"¹³.

The Academy explained that the two had developed a formula for the valuation of stock options in collaboration with "the late Fischer Black"¹³. Black had died in August 1995, and the award decision was for the two of them.

Notes

1. Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities", The Journal of Political Economy, Vol. 81, No. 3 (May-Jun. 1973), pp. 637–654 (the bibliographic details on the JSTOR cover page). ↩

2. The Royal Swedish Academy of Sciences, "Advanced information" (October 14, 1997), "Practical importance". The paragraph covers the start of CBOE trading, the use of calculators by 1975, and the thousands of traders and investors who use the formula every day. ↩ ↩² ↩³

3. The Royal Swedish Academy of Sciences, "Advanced information" (October 14, 1997), "The history of option valuation". The paragraphs cover Bachelier's dissertation and its assumptions, the later improvements, Boness's 1964 formula, and the difficulty of the methods before 1973. ↩ ↩² ↩³

4. 服部孝洋, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to Swaptions (Model Edition): The Normal (Black) Model and the SABR Model), Ministry of Finance, "ファイナンス" (October 2021), page 3 (in Japanese). It states that the normal model is also called the Bachelier model. ↩

5. Cboe Global Markets, "The Creation of Listed Options at Cboe", the paragraph beginning "On April 26, 1973". The opening date is the same as in "The Development of Futures Exchanges". ↩

6. Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein, "Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?", Federal Reserve Bank of Chicago, WP 2011-11, "1 Introduction" (page 1, page 4 of the PDF). ↩ ↩²

7. The Royal Swedish Academy of Sciences, "Advanced information" (October 14, 1997), "The Black-Scholes formula". The paragraphs cover why no risk premium is needed, the idea of building a riskless position from the stock and the option, arbitrage, and the need to keep changing the position. ↩ ↩²

8. The same "The Black-Scholes formula". The paragraph quotes Black's paper "How We Came Up with the Option Formula" (1989), and states that Merton's 1973 paper "Theory of Rational Option Pricing" extended the formula. ↩

9. Cboe Global Markets, "The Creation of Listed Options at Cboe", the paragraph containing "Black, Scholes and Merton weren't directly involved". ↩

10. The Royal Swedish Academy of Sciences, "Advanced information" (October 14, 1997), the paragraph after the formula in "The Black-Scholes formula". Only sigma cannot be observed, and it has to be estimated from market data; if the option's price is known, the market's expectation of sigma can be found from the formula. ↩ ↩²

11. The Royal Swedish Academy of Sciences, "Advanced information" (October 14, 1997), "Pricing of derivatives", second paragraph. ↩

12. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University), page 4 of the PDF, section "volatility surface". It states that, with maturity held fixed, the tendency for implied volatility to be higher at lower strike prices is called the volatility skew or smile. ↩

13. The Royal Swedish Academy of Sciences, "The Prize in Economic Sciences 1997 - Press release" (October 14, 1997), the opening announcement of the award and the following "pioneering formula" sentence. Advanced information gives August 1995 as the date of Black's death. ↩ ↩²