45 Years after the Publication of the Black-Scholes-Merton Model

David Moore

45 years after Fisher Black and Myron Scholes published their seminal model, it is time to shed some light on the background of the model's development and to tell Who Castelli was? What was the contribution of the Bachelier and Samuelson? How is Merton connected to the model? And who is Japanese mathematician who helped without knowing to develop the model? All of this and more in my article.

A New Breed of Science

In the recent 30 years, a new discipline called financial engineering has emerged in order to attempt to under s tan d finance using a scientific approach. Mathematicians, physicists and trader s work together in this discipline in order to incorporate the use of advanced mathematics with everyday finance.

Although financial engineering deals with many aspects of finance, the main application of this discipline is risk management within the stock market. Regardless of what type of stock market transaction one performs, risk is always present. However, it is the management of this risk that is studied by these “financial engineers”. People need a fast and reliable way to calculate and control the risk involved in all their stock trading.

This is where the Black- Scholes Option Pricing Model comes in. This ideas behind this formula, created by Prof. Robert C. Merton, Prof. Myron S. Scholes and the late Fisher Black, has been described by the CEO of IAVFA (the Israel Association of Valuators and Financial Actuaries), Mr. Roi Polanitzer as “the most successful theory not only in finance but in all of economics.”

Options

The functioning of the Black- Scholes Model is based on the use of stock options. Stock options are a form of financial derivative (an item that is not a stock in itself, but is an offshoot of one). It consists of a contract that gives one the right, but not the obligation , to buy stocks later at a fixed price (known as the exercise or strike price). The exercise price does not change, regardless of all changes in the stock’s value. These options are purchased at a fee known as the premium.

To illustrate, let’s say someone obtained the option to purchase 100 shares a year from now (a date known as the call date) for $100 each. If the stock were to rise to $120 by the call date, it would be feasible for this person to exercise his /her option, because the shares would still only cost you $100, even though they are worth $120.

However, if the value of the stock were $80 at the call date, then it would not be feasible to purchase these shares, because you would be paying $100 for shares that are worth $80 each (a loss of $20 a share). Thus, this person probably would not exercise his /her option, and would only lose the premium he/ she paid for the options a year ago.

Thus, stock options are a form of insurance policy. What makes stock options so appealing is that the purchaser knows that the limit of his /her losses can only be the premium price. However, there are no limits to his /her gains, because the limit of the value of the stock is theoretically limitless.

The question is, what is the fair price for an option on a particular stock? In other words, what is the option worth? When a stock has a call price of $100 and a value of $120, the option is worth at least $20 ($120 - $100 = $20). The value of the option clearly depends on the value of the stock. Thus, if there were a formula that could tell you the fair price for an option while taking into account all necessary factors, it would come of great use to the financial world. This is what the Black- Scholes Options Pricing Model does.

The Math Behind It

Option pricing requires five inputs: the option’s exercise price, the time to expiration, the price of the stock at the time of evaluation, current interest rates and the volatility of the stock. The only unreliable factor is the volatility of the stock. This number can be estimated from market data. The formula is as follows:

where the variable d is defined by:

According to this formula, the value of the option C, is given by the difference between the expected share price (the first term) on the right - hand side, and the expected cost (the second term) if the option is exercised. The higher the current share price S, the higher the volatility of the share price (Greek letter) sigma , the higher the interest rate r, the longer the time until the call date t and the lower the strike price L, the higher the value of the option C will be.

Limitations of the Model

As consistent as the model, there are limitations to the model. One limitation is that it assumes that the options can only be exercised on the call date. In other words, it cannot be exercised earlier. This model involves “European - Style” options, rather than “American - Style” options. “American - Style” options can be exercised anytime. American options are more flexible, thus more valuable. The model only takes European - style options into account. Thus, the model underes timates the value of options. Most options are not exercised until the call date anyways, but this law is not written in stone.

Another limitation is that it assumes that the interest rate (determined by the U.S. Government) is known and will remain more - or - less constant. Many researcher s have concluded that this is a safe assumpt ion to make. But there are times where the interest rate can change rapidly, thus putting the results of the model into question.

However, these limitations are considered insignificant, because they do not affect the value of the option unless in extreme circums tances (such as a sudden raise in interest rates or even a market crash).

The Birth of the Model

This formula did not create itself out of nowhere. Its roots lie deep in the branches of mathema tics known as probability and statistics. The combination of these two domains of mathema tics deals with the collection, organization, and analysis of numerical data in order to assist decision-making. In short, statistics let you “predict the future”, not with 100% accuracy, but well enough so that you can make a wise decision as to your next course of action.

It all started when Charles Castelli wrote a book called “The Theory of Options in Stocks and Shares” in 1877. Castelli’s book was the first to deal with the use of options. However, this book lacked the theoretical basis needed for actual application.

Twenty- three years later, a graduate student by the name of Louis Bachelier published his thesis paper “La Théorie de la Spéculation” (The Theory of Speculation) at the Sorbonne, in Paris. In this paper, Bachelier dealt with the “structure of randomness” in the market. He compared the behavior of buyers and sellers to the random movement s of particles suspended in fluids. Remarkably, this paper anticipated key insights developed later on by famed physicist Albert Einstein and future theories in the field of probability.

He created the first complete mathema tical model of options trading. He believed the movement s of stock prices were random and could never be predicted, but risk could be managed. He created a formula that yielded an output that could help protect market investors from excessive risk by means of pricing options. However, this formula contained financially unrealistic assumpt ions, such as the existence of negative values for stock prices and a zero interest rate. His paper was shelved and went unnoticed for decades.

It wasn’t until 1955 that the idea of options pricing resurfaced, when a professor at the Massachuset t s Institute of Technology named Paul Samuelson browsed through the Sorbonne library. He began developing a formula of his own. Other mathemat icians, such as Case Sprenkle and James Boness began toying with Bachelier’s ideas as well. But all of their efforts went fruitless.

A Revolution

Then in 1968, a 31- year - old independent finance contractor named Fisher Black and a 28- year - old assistant professor of finance at MIT named Myron Scholes began their work on options pricing. They were dissatisfied with all the formulas that had preceded them, because they were overly complicated and made assumpt ions that didn’t make sense. They wanted to find a formula that would calculate the fair price of an option at any moment in time just by knowing the current price of the stock, but they couldn’t see their way through the mass of equations they had inherited.

Then they decided to try something different. They decided to strip previously derived formulas to their bare - boned state. They dropped everything that represented something un- measurable. They were left with the vitals of calculating an option: the option’s exercise price, the time to expiration, the price of the stock at the time of evaluation, current interest rates and the volatility of the stock. But they were stuck with one problem: one couldn’t measure volatility, or in other words, risk.

So they decided if they couldn’t measure the risk of an option, they should make it less significant. Their solution to this problem was to become of the most celebrated discoveries of the 20th century. The solution was rooted in the old gamblers’ practice of hedging. When one makes a risky bet, one hedges his /her bet by also betting in the opposi te direction.

To illustrate, let’s say one was to bet that in the World Cup 2018 the favored France would beat Croatia in the final game. If one already bet $50 on France, one would hedge that bet by betting $45 on Croatia. Although France is favored, by hedging this bet with a slightly smaller bet on Croatia, we are protecting ourselves in the event of a Croatian upset. We minimize risk at the cost of lowering our possible winnings. However, since France is favored, the chances of winning $5 are substantial.

A more busines s - oriented example would be as such: Let’s say an Israeli company is expecting to make several large payment s in US dollars in a few months. They can hedge against a huge drop in the New Shekels (thus making it more expensive to buy US dollars) by purchasing options for US dollars on a foreign currencies market. Effective risk management requires that such options be correctly priced.

To hedge against risks in changes in share price, the investor can buy two options for every share he or she owns; the profit will then counter the loss. Hedging creates a risk free portfolio. As the share price changes over time, the investor must alter the composition of the portfolio, the ratio of number of shares to the number of options, to ensure that the holdings remain without risk.

They made up a theoretical portfolio of stocks and options. Whenever either fluctuated up or down, they tried to hedge against the movement by making another move in the opposite direction. Their aim was to keep the overall value of the portfolio in perfect balance. In other words, they tried to minimize risk.

They discovered that they could indeed reduce risk by creating a balance in which all movement s in the markets cancelled each other out. Black and Scholes had found a theoretical way to neut ralize risk. With risk now virtually eliminated from their equation, they had a mathematical formula that could give them the price of any option. This was a marvelous achievement.

There was a practical problem with their formula. It assumed that markets were always in equilibrium, that supply equals demand. They needed a way to instantly rebalance a portfolio of stocks and options to keep countering all their movements.

A Harvard graduate by the name of Robert Merton solved this problem by introducing the notion of continuous time. This idea is rooted in rocket science. A Japanese mathema t ician by the name of Kiyosi Ito theorized that when you plot the trajectory of a rocket, knowing where the rocket was second-by-second was not enough. You needed to know where the rocket was continuously. So he broke time down into infinitely small increments, thus smoothening the graphing of its path out until it became a continuum so that the trajectory could be constantly updated. Merton applied this idea to the Black- Scholes model so that the value of an option could be constantly recalculated and risk eliminated continually.

In 1997, Robert Merton and Myron Scholes were awarded the Nobel prize in economic sciences for their efforts. Their colleague Fisher Black had unfortunately passed away in 1994.

The History of the Model

In 1973, the Chicago Options Exchange was launched, one month before the Black- Scholes model was published. When these three men had published their paper in 1973 in the Journal of Political Economy, trader s, academics and economis ts marveled at its overwhelming power, despite the simple nature of its use.

Traders began using their ideas immediately. Texas Instrument s had incorporated their formula into their latest calculator, announcing their feature in the Wall Street Journal. The options market exploded soon after.

So overwhelming was the sudden mass use of the Black- Scholes Model, that when the stock market crashed in 1978, the influential business magazine Forbes put the blame squarely onto that one formula.

According to Capital Market Risk Advisors, about 20 percent of the $23.77 billion (US) in derivatives losses in the 90's are due to problems related to modeling. In 1997, however, model risk comprised nearly 40 percent of the $2.65 billion (US) in money lost.

At a conference in February 1998, an indust ry trade magazine called Derivatives Strategy sponsored a discussion group called “First Kill All the Models”. This group reflects the recent backlash against financial models. Many figures in the financial industry question whether models canmatch traders’ skill and gut intuition about market dynamics.

Derivatives make the news because, like an airplane crash, their losses can dramatic and chaotic. Enormous losses by Proctor & Gamble and Gibson Greetings and the bankruptcies of Barings Bank and Orange County, California have been attributed to the use of models.

However, Scholes says that it was not so much the formula itself that caused these losses, rather its misuse by market traders. Every statistician and mathemat ician knows you cannot predict the future with 100% accuracy. Laws as rigid as the laws of physics do not govern the market. I use to say: “Math doesn’t drive financial markets, people drive financial markets, and people are not predictable. We do not yet have a universal theory of human behavior or human motivation.”

It was not the model by itself that caused these losses, but the blind faith that market trader s put into it. They all jumped at the prospect of making money without risk. However, this formula cannot eliminate risk, it can only minimize it. Like many mathema tical models, it relies on imputs and assumes a functioning market. It is a powerful way to manage risk, but it’s not a crystal ball. Scholes says this equation should be used as a tool for making decisions, not a platform from which all decisions should be made.

Fisher says: “If a random bolt of lightning hits you when you’re standing in the middle of the field, that feels like a random event. But if your business is to stand in random fields during lightning storms, then you should anticipate, perhaps a little more robustly, the risks you’re taking on.” This formula is a method to calculate these risks, not a risk neutralizer.

Virtually, there is a danger of accepting models without carefully questioning them. Thus, the Black-Scholes is not the culprit for all derivative losses, but traders’ blind faith in them.

Numbers vs. Instinct

Many traders still use the ideas behind the Black-Scholes Options Pricing Model, if not the model itself. The fundament al ideas behind the equation forever changed the stock market. Today, trader s use many principles of the Black-Scholes Model as guides through the treacherous waters of the stock market. For this, Scholes and Fisher became Nobel laureates. But the lessons of putting all of one’s eggs in the same “Black-Scholes Model” basket have been learned. One cannot blindly put all his /her faith into the model and expect guaranteed financial success. Judgment is still required.

Samuelson says: “There is a tempting and fatal fascination in mathematics. Albert Einstein warned against it. He said elegance is for tailors, don’t believe in something because it’s a beautiful formula. There will always be room for judgment.”

Mr. Moore (Certified Portfolio Manager) IRA, QFV, is the owenr of "David Moore - Financial Advisory", a privately firm specializes in financial advisory and professional valuations for financial reporting, audit, statutory, regulatory, legal, tax and M&A purposes.

Mr. Moore is certified as an Investment Risk Actuary by IAVFA, Israel Association of Valuators and Financial Actuaries. In addition, Mr. Moore is the Head of Investment Risk Actuarial Science at the Israel Association of Valuators and Financial Actuaries.