Introduction

In 1973, a research paper that had been rejected by two academic journals was published, and within months its central formula was being sold to traders on a handheld calculator. Few people today recognize the name of the formula: Black-Scholes. Yet, the equation at the center of that paper now underpins financial markets worth several hundred trillion dollars in notional value, and it influences the cost of retirement plans, mortgages, and even paychecks. To see how such an influential equation reaches our everyday lives, let’s begin with retirement.

Many people approach retirement by holding a promise: a pension that pays a set amount each month, or an annuity that guarantees a minimum income no matter how the stock market performs. These guarantees seem like simple commitments, but financially they behave like options, contracts that give their holder the right, but not the obligation, to buy or sell a stock at a set price at a set date. Because an option eventually expires, its value also depends on how much time remains, also known as maturity.

Options can be used in two ways. A call option gives the right to buy, and a put option gives the right to sell. Suppose an investor pays $10 for a call option to buy a $100 stock at $100 in one year. If the stock rises to $130, the investor earns $20 after the fee, but if it falls to $70, the investor lets the option expire and loses only the $10. On the other hand, a put option works in the opposite way: it pays off when the prices fall. Generally, this is how a retirement guarantee functions. Whoever makes the promise must cover the shortfall if markets drop below the promised level.

Although this is now a part of our everyday lives, this same concept raised a question that troubled traders for centuries: what is such a right actually worth? In 1973, the Black-Scholes equation provided the first widely accepted answer. By combining five inputs -- the stock's current price, the strike price (the fixed price at which the option holder can buy or sell the stock, agreed upon when the option is created), the time remaining, the interest rate, and the stock's volatility (how widely its price tends to swing) -- the equation produces a fair price for the option. The rest of this article follows the equation in two parts. First, its origins, which originate from a student's thesis in 1900. Then, its adoption, which soon propagated from the trading floor into pensions, mortgages, and paychecks.

Where the Equation Came From

Although the Black-Scholes Equation is most used in financial markets today, its roots actually lie in physics. In 1900, Louis Bachelier, a student of the physicist Henri Poincaré, studied the Paris stock exchange and proposed that a stock is equally likely to rise or fall at any given moment, much like the flip of a coin. Over time, the possible prices spread into a bell-shaped distribution, similar to the pattern formed when many balls fall through a Galton board – a wall of pegs.

Figure 1: Visualization of Bell-Shaped Distribution on a Galton Board

Bachelier recognized that this pattern matched the mathematics of how heat spreads, and he used it to price options. Unfortunately, few people paid attention to his work at the time.

The problem he addressed did not disappear, however. Several decades later, the physicist Ed Thorp, known for developing card-counting strategies in blackjack, turned to the stock market and applied a technique known as hedging, which reduces risk by taking an offsetting position.

Here’s an example of what hedging might look like: A trader who sells a call option can buy some of the underlying stock so that gains on one position offset losses on the other, adjusting the holding as prices move. 

These two ideas were combined in 1973 when Fischer Black and Myron Scholes reasoned that a portfolio combining a stock and its options, hedged until it carried no risk, should earn only the rate paid by a safe government bond. This removed the need to predict where the stock was headed, which had limited earlier pricing formulas. Robert Merton published his own derivation the same year, and his name is now attached to the model as well. Even so, the paper was rejected by two journals before it was finally published.

The Equation’s Impact

The timing proved fortunate, as the Chicago Board Options Exchange opened the same year, creating a market for the Black-Scholes formula. Trading grew from 911 contracts on its first day to 100,000 by 1977, and within six months of the paper's publication, Texas Instruments was selling a calculator preprogrammed with the formula. Today, derivatives, which are financial contracts whose value depends on another asset, are estimated at several hundred trillion dollars. 

Although its main use is on the trading floor, the equation's influence spread into the same retirement guarantees discussed earlier. Some annuities promise that your savings will grow when the stock market rises but will never shrink when it falls. To keep that promise, the insurance company buys a contract that pays off if the market goes up. Because these promises can last 20 to 30 years while the contracts protecting them last only about 10, the insurer has to keep adjusting its protection along the way.

This same idea shows up in everyday life. A fixed-rate mortgage lets you refinance if interest rates drop, and economists treat that right as an option that can be given a price. Many employees are paid partly in stock options, and companies need a fair way to value them, which is why one company’s filing with the SEC shows it using Black-Scholes. Furthermore, even seemingly unrelated industries, such as the airline industry, use options. They can buy options that pay off when oil prices rise, which helps cover the cost of more expensive fuel. 

Although options can be very lucrative, they are not risk-free. When many investments fall at the same time, they can make a market drop even worse.

Conclusion

The wide reach of the Black-Scholes equation reflects a broader feature of modern finance: many of the promises individuals depend on, including pensions, annuities, and mortgages, are priced by models most people are unaware of. Understanding how those prices are determined, and the assumptions behind them, is therefore relevant not only to traders but to anyone whose financial security rests on a promise about the future.