Black-Scholes Method for Pricing

Hedging is an increasingly popular practice, offering investors an opportunity to mitigate potential losses by counterbalancing with futures contracts. However, the predictability of the potential returns on such contracts has long proven nebulous, due to the uncertainty surrounding long-term worth and value. In 1973, however, financial analyst Fischer Black and finance professor Myron Scholes created a model whereby much of the uncertainty surrounding the potential return on hedging investments may be mitigated. This paper will examine the "Black-Scholes" method for pricing, its fundamentals and its applications in the 21st century economy.

Keywords Call Options Profit; Derivative; Hedge Fund; Hedging; In the Money Call; Option; Out of the Money Call

Overview

Benjamin Franklin once stated, "I conceive that the great part of the miseries of mankind is brought upon them by false estimates that they have made of the value of things" (http://www.valuequotes.net/). Indeed, one of the most challenging aspects of life is assigning not just worth, but also value to a given subject.

Value is one of the most vexing of concepts in the world of international commerce and investment. Affixing a price to a given product, good or service is a relatively simple action based on current conditions surrounding the production of that item. Value, however, is more long-term; subject to the conditions not just of the present but of the short-term and longer-term future. Determining the value of an investment may, therefore, assist in generating a higher yield to the investor.

In the 21st century global economy, there exists a myriad of markets in which entrepreneurs may invest their assets in the hope of a return on that investment. Most investors, fearful of losing their money in a single market that experiences sudden collapse, diversify their portfolios across a broad spectrum of markets. Others, also mindful of the risks involved with investments (particularly in a volatile economy), seek to "insure" themselves from loss by hedging on their investments.

Hedging is an increasingly popular practice, offering investors an opportunity to mitigate potential losses by counterbalancing with futures contracts. However, the predictability of the potential returns on such contracts has long proven nebulous, due to the uncertainty surrounding long-term worth and value. In 1973, however, financial analyst Fischer Black and finance professor Myron Scholes created a model whereby much of the uncertainty surrounding the potential return on hedging investments may be mitigated. This paper will examine the "Black-Scholes" method for pricing, its fundamentals and its applications in the 21st century economy.

Further Insights

A Brief Introduction to Hedging

In order to understand the applicability of the Black-Scholes method, one must first understand the practice of hedging. This practice is seen by many as a recent development in the world of finance, although scholars trace its origins as far back as the days of Aristotle, who told a tale of Thales, the philosopher. Thales haggled with the owners of olive press equipment to gain exclusive rights to use the presses during the harvest later that year. In order to gain this access, he put forth a considerable investment against the industry, which solidified for him a significant profit on his wager (Rappeport, 2007).

In 1949, journalist and academic Alfred Winslow Jones decided to abandon journalism and take up finance, forming A.W. Jones and Company with three friends and investing $100,000 of his own money. The fund would employ a wide mix of short- and long-term positions with considerable incentives for participating investors. One year later, Jones's fund (to which Fortune magazine's Carol Loomis would refer two decades later as the first "hedge fund") earned more than 17 percent than in the previous year, and over the following decade would outperform every other mutual fund by more than 87 percent (Rappeport, 2007).

Today, hedge funds comprise an estimated $1 trillion industry, using a wide range of tactics that creates considerable diversity among funds (McWhinney, 2009). The practice is often given the more familiar moniker, "insurance." Indeed, hedging involves an investor's understanding the risks associated with a long-term investment and, in order to mitigate any unforeseen downturns in the performance of the company in question, taking out short-term, high-yield investments in order to offset any negative performance.

Derivatives

Central to the practice of hedging are derivatives, which are simply contracts between two or more parties. The value of derivatives depends on conditions surrounding the parties' underlying assets (such as stocks, bonds and currencies). For the purposes of hedging, derivatives are manifested in two forms. The first is the futures contract. Investors and companies alike may use these arrangements to ensure that their prices are stable. A manufacturer, for example, may purchase a futures contract on the price of oil, which is absolutely essential to the operation of the company's machinery. This contract will set a price at which the oil is purchased. When political instability, terrorist incidents or simply high demand send the price of oil skyward, that company's futures contract will keep its oil prices stable. In fact, the company may come out ahead, saving considerably on expenses. Of course, during times of economic and political stability, or even during spring and non-holiday summer months, when the price of oil seeps below the set price of the futures contract, the company is still obligated to pay for oil at the set price, which means that it may have been better off not entering such a hedge arrangement ("A beginner's guide," 2009).

The second derivative in hedging is the option, which is a derivative contract in which an individual agrees to either buy (put) or sell (call) stock in a company within a fixed, short period of time and at a set price (known as the "strike price"). If the stock in question falls below strike price, the holder of the option will still see a profit from the call, gains that may be used to offset any short-term losses associated with long-term holdings.

The most challenging part of hedging has long been setting a price for options. Far too many variables exist in the marketplace, particularly in one as globally expansive and diverse as the 21st century economy, making price setting for options a daunting task. Doing so is, in essence, placing a long-term value on a given contract, but basing that contract on a large collection of potentially volatile factors

In 1973, however, Fischer Black and Myron Scholes crafted a theoretical framework that for many answered this extremely vexing problem using mathematical formulae and even physics equations (Bookstaber, 2003). In introducing the theory (which contemporary Robert Merton later coined the "Black-Scholes Method for Pricing") in their paper, "The Pricing of Options and Corporate Liabilities," Black and Scholes laid a whole new foundation for finance and business. Merton would expand this theory and help apply it into one of the most significant frameworks in the marketplace.

The Black-Scholes Method

In addressing the inequities between parties in an options environment, Black, Scholes and Merton largely drew from a growing field in calculus known as "stochastic" process. As opposed to existing process models that examined outcomes along a single linear path, stochastic process (also known as "random process") takes into account the fact that the variables involved may be too substantive to simply incorporate into one "reality." In doing so, stochastic process states that although a starting point is evident, there are a great many possible outcomes, some more probable than others. This mode of non-linear thinking proved extremely attractive for those attempting to predict the short-term and long-term health of their investments.

Investors are primarily concerned with the call options profit, which is the difference between how much the option will pay off and the cost of purchasing the call. This figure is the central target at which the Black-Scholes method focuses. In calculating the option price, Black-Scholes uses six aspects of the stock in question:

• The price of the underlying stock,
• The price to exercise the option,
• The volatility of the stock price,
• Time to expiration,
• The interest rate and
• The dividend rate (Tucker, 2007).

Stock volatility is one of the more dynamic forces in this formula. By seeking options, an investor is seeking to avoid dramatic losses due to unforeseen adversities in the marketplace. The option gives him or her insurance and stability, by providing significant protection from such losses. Therefore, when stock volatility increases, so too does the call option value.

Black, Scholes and Merton's work in this arena are theoretical frameworks, but remain significant nonetheless in their application in the financial arena. In fact, in honor of their model, Scholes and Merton were awarded the Nobel Prize for Economics in 1997 (Black had died in 1995, rendering him ineligible to receive the award, although he was given recognition as a contributor).

The Black-Scholes Method for pricing is of an academic nature, established in a theoretical context. Applications in financial circles are interpretations and modifications of its groundbreaking work. In light of its scholarly origins, Black-Scholes makes a number of assumptions that enable it to function effectively under certain conditions. For example, the method assumes that an option is exercised only at the time of expiration. Also, it assumes that the risk-free interest rates and underlying stock volatility will remain constant over a given period, and that there will be no sudden jumps or drops in stock prices. Furthermore, the method acts on the basis that the underlying stock itself does not pay dividends (Black & Scholes, 1973).

Issues

The assumptions this model makes do not necessarily render the theory irrelevant in "real world" applications. Rather, Black-Scholes was designed to connect the constants in this environment (such as risk-free interest rates or current underlying stock prices) while identifying the elements that contribute to volatility. In essence, Black-Scholes created an integral financial model for a financial environment that was truly in need of one. In fact, this method, by assuming a volatility-free set of conditions, was creating flexibility for its application in an arena that had innumerable factors that could impact the market. "Real world" variables that work to increase volatility or the complexity of financial transactions were not discounted; rather, the formula was left intact with the expectation of upgrade when such variables occurred.

Because it fills in a major gap in the determination of option prices, the Black-Scholes method has been modified and applied in a wide range of environments. In many cases, this application has proven useful in more effectively calculating this invaluable figure. In other applications, however, its application has fallen short.

Generally, criticism of the Black-Scholes Method for Pricing centers on its assumptions, which analysts assert create either an inflexible, unviable set of conditions in which it may truly prove effective, or simply fails to prove useful in "real world" applications. With this issue in mind, many financial analysts and academics are looking to build on the important groundwork laid by this theoretical framework.

In 2004, for example, the Financial Accounting Standards Board (FASB) issued a plan that required companies to treat employee stock options as expenses, thereby avoiding the free-handed approach to stock option distribution that contributed heavily to the recession of the early 1990s. While implementing new reporting standards for companies to identify the manner by which they offered employees such options, the FASB employed a variation on Black-Scholes to determine the value of such options. However, a number of issues arose in the process. For one, employee stock options were often offered with conditions that necessitated employees to exercise them within the usual 10-year timeframe for option exercise; Black-Scholes assumes that exercises would take place closer to that 10-year mark. Such assumptions, in the eyes of industry experts, made calculating option prices based on a number of variables that were extraneous to Black-Scholes's framework unworkable. Analyst Ron Rudkin commented on the effect such variables would have on the issuance of employee stock options (and employees' propensity to exercise them), calling into question the method's ability to arrive at a reasonable option value: "You can't do that with Black-Scholes … it just doesn't have the flexibility" (cited in Schneider, 2004, p. 45-46).

Other criticisms take to task the fact that Black-Scholes often overprices "in the money calls" (calls that fall below the strike price) and underprices "out of the money" calls (which are above the market price for the underlying stock). Those critics claim that the issue lies to a large degree in the formula's distribution methodologies, which fails to take into account negative stock prices. Such stock prices could not exist, but the presence of such a negative range, critics argue, helps paint a more accurate placement of overall stock prices (McMillan, 2003). These miscalculations may cause serious issues in the determination of option prices in some cases, particularly if the analyst employing the method uses its broad conclusions as the sole basis for price calculation.

Conclusions

A traditional Indian parable tells of a group of blind men who seek to understand the appearance of an elephant. Each man stood at one part of the elephant and, based on what he happened to encounter, offered his assessment of the animal. One, for example held the animal's tail, while another felt the animals legs while still another touched its side. As a result of their arrogance, each man had but part of the description accurate, but none of them took into consideration the aggregate of their accounts. The elephant, as a result, remained a mystery.

In the world of finance and commerce, the "elephant" is the ability to determine value worth pursuing. For decades, investors supplied their money into stock markets; a gamble they felt was worth the potential loss due to the volatility of the market and the fact that this volatility could in fact give them a greater return on their investments. Of course, when this volatility did impact the markets in a severely negative manner, the losses (both real and potential) led many investors to buttress their long-term investments.

In the late 1940s, this "insurance" came in the form of hedging and hedge funds. Put simply, such a practice allowed investors to offset perceived future losses in their assets by placing short-term investments to create profitable returns against those losses. As shown in this paper, hedging grew exponentially over the next half century into and industry worth more than a trillion dollars.

The severity of the Great Depression and the subsequent recessions of the 20th century led many financial analysts and scholars to research ways to more accurately predict and calculate transactions and investments. One of the most significant of the resulting frameworks was the Black-Scholes Method for pricing. Its creators tasked themselves with calculating another sizable "elephant" — setting prices for options, an essential element in hedging.

Generally, the theory offered by Black, Scholes and Merton was well-received by virtue of the fact that no one previously could develop such a formula out of a near-chaos of variables. As a theoretical framework, Black-Scholes did not address a number of key issues regarding the volatility of the market, an approach that advocates maintain allows for future modifications and improvements.

Critics have charged that strict adoption of Black-Scholes as a primary mechanism for calculating option prices is problematic and unreliable, due to the number of key assumptions it makes in theoretical constructs. Nevertheless, many assert that the method cannot simply be discarded. Rather, they argue, the work must be reinforced through additional modifications. In this regard, the relevance of this important model will continue to be examined and the model itself will ultimately become more effectively applied in the marketplace.

Terms & Concepts

Call Options Profit: The difference between how much an option will pay off and the cost of purchasing the call.

Derivative: Financial contract between two or more parties.

Hedging: Investment practice whereby long-term assets are supported by short-term investments in order to offset any negative asset performance.

In the Money Call: Call that falls below the strike price.

Option: A derivative contract in which an individual agrees to either buy or sell stock in a company within a fixed and short period of time and at a set price.

Out of the Money Call: Call that falls above market price.

Bibliography

Ankirchner, S., Dimitroff, G., Heyne, G., & Pigorsch, C. (2012). Futures cross-hedging with a stationary basis. Journal of Financial & Quantitative Analysis, 47, 1361-1395. Retrieved November 24, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=87363444&site=ehost-live

Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81 , 637. Retrieved July 9, 2009 from EBSCO online database, Business Source Complete http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=5053921&site=ehost-live

Bookstaber, R. (2007). A demon of our own design. Hoboken, NJ: John Wiley and Sons.

A beginner's guide to hedging. (2009). Investopedia. Retrieved July 9, 2009 from Investopedia.com http://www.investopedia.com/articles/basics/03/080103.asp.

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McMillan, L. G. (2003). Options as a strategic investment, 2nd Ed. New York: Penguin Books.

McWhinney, J. E. (2009). A brief history of the hedge fund. Investopedia. Retrieved July 8, 2009 from Investopedia.com. http://www.investopedia.com/articles/mutualfund/05/HedgeFundHist.asp?vi ewed=1

Rappeport, A. (2007, March 27). A short history of hedge funds. CFO. Retrieved July 8, 2009 from CFO.com http://www.cfo.com/article.cfm/8914091/c%5f8913455

Schneider, C. (2004). Forget Black-Scholes? CFO Magazine, 20, 45-50. Retrieved July 10, 2009 from EBSCO online database, Business Source Complete, http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=13087649&site=ehost-live

Tucker, J. (2007, June 17). Black-Scholes option pricing model. Retrieved July 7, 2009 from Suite101.com. http://options-investing.suite101.com/article.cfm/blackscholes%5fopti on‗pricing‗model.

Suggested Reading

Curtis, G. (2002). Modern portfolio theory and quantum mechanics. Journal of Wealth Management, 5, 7-13. Retrieved July 12, 2009 from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=8594085&site=ehost-live

Eisinger, J. (2005, October 12). Hedge-fund activism wins plaudits, but the focus is really on firms' cash. Wall Street Journal — Eastern Edition, 246, C1-C4.

Gould, M. (2008). Accounting for stock-based compensation. Accounting for Stock-Based Compensation — Research Starters Business. Retrieved July 12, 2009 from EBSCO Online Database Research Starters Business. http://search.ebscohost.com/login.aspx?direct=true&db=e6h&AN=28544266&site=ehost-live

Gould, M. (2008). Trading and markets. Trading and Markets — Research Starters Business. Retrieved July 12, 2009 from EBSCO Online Database Research Starters Business. http://search.ebscohost.com/login.aspx?direct=true&db=e6h&AN=29964573&site=ehost-live

Peltomäki, J. (2011). On derivatives use by equity-specialized hedge funds. Journal of Derivatives & Hedge Funds, 17, 42-62. Retrieved November 24, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=65928760&site=ehost-live

Sprague, C. (2008). Financial derivatives. Financial Derivatives — Research Starters Business, 1-13. Retrieved July 12, 2009 from EBSCO Online Database Research Starters — Business. http://search.ebscohost.com/login.aspx?direct=true&db=e6h&AN=31962704&site=ehost-live