Money is a fascinating subject to the average person on the street. Whether one has a little or a lot, nearly everyone spends a good portion of his or her time trying to figure how to make more, spend less or stretch the money just a little further. The seemingly insatiable demand for books on money and investing suggests that those who know little about it seem in awe of those who know - or profess to know - a lot. Yet finance - the study of how money is acquired and how it is invested - is a relatively young field, having emerged out of the shadows of economics in the post-World War II era. In this short period of time, finance has evolved into a critically important discipline, as evidenced by the influence it has had on so many people and institutions in society.

From the days when bankers were called moneychangers, finance has proven to be a quantitative discipline, borrowing frequently from mathematics. In recent years, this evolution has begun to embrace both physics and mathematics. This marriage of Wall Street and Los Alamos has been a propitious one for both parties. For scientists with advanced degrees, it has created new employment opportunities. For investment bankers, financial managers and commercial bankers, the mathematical revolution in finance has spawned opportunities to expand product lines and solve complex financial problems in increasingly globalized markets.

The scientific character of finance is probably due to its preoccupation with the concept of risk. From ancient times, when sailing the high seas meant subjecting one’s life and fortune to the uncertainties of weather, hostile enemies and mutinies, people have sought to analyze, understand and control the various risks they face. Fortunately, the development of probability theory, attributed by Peter Bernstein to the Italian scholar Girolamo Cardano and later refined by the French mathematicians Blaise Pascal and Pierre de Fermat, has given the process of risk management a set of scientific and mathematical tools with which risk could be studied, understood and - if not reduced - at least, faced with greater confidence and an awareness of the possible losses one might incur.

Finance as an academic subject focuses on the risks and rewards of investing monetary resources. As a scholarly discipline, finance, over the last half-century, has taken a well-defined body of economic theories of how people behave when faced with risk and translated them into models of the way in which individuals make decisions about how to obtain funds - called the financing decision - and how to invest funds - called the investment decision. But just as scientists have developed a large body of formal theories, so too is the need for experiments to validate those theories under the most demanding and scientifically rigorous standards. Finance is especially well-suited for theory validation through the process of empirical testing. The financial markets have, for nearly a century, generated vast quantities of recorded data on prices, interest rates and exchange rates. With advances in information technology and the development of increasingly powerful statistical techniques, finance has become an empirical discipline, demanding replicable experiments, increasing mathematical accuracy, and unbiased assessment. This evolution of finance has not been confined to the hallowed halls of academe. Wall Street’s demand for finance scholars as employees and consultants has brought the classroom to the real world and the real world to the classroom. Theories and tests of financial models are as likely to be undertaken in major brokerage houses as in universities.

The importance of finance research has not gone unnoticed by those who recognize and reward the great discoveries of today. Since 1985 the Nobel prize in economic science has been won or shared by William Sharpe, Franco Modigliani, Merton Miller and Harry Markowitz for their pioneering research on how investors and firms interact in financial markets to produce prices. Finance as an academic discipline can trace its origins to the early work of these Nobel Laureates and other economists, whose ideas can be seen today in innovations such as index funds and risk management.

One of the primary tasks of finance is valuation. Valuation is the process, and sometimes the art, of estimating what something in the future is worth today. The principles of valuation are not new. For example, Alfred Marshall in his book Principles of Economics in 1890, discussed the concept of a future benefit being discounted to an equivalent present value.

The interest rate or exchange rate of values through time is generally referred to as the compound rate when translating values forward in time and the discount rate when translating back in time. This exchange rate comprises two elements. The first reflects timing. A dollar today is not worth a dollar tomorrow, one year from today, or ten years from today. Each of these dollars has a different "time value" that reflects the opportunity cost of funds. The second element is uncertainty. The more uncertain are future cash flows, the less valuable are they in the present.

The exchange rate should also consider the interest paid on interest. Suppose that the interest rate is 5% on a loan of $1. If the loan is for two periods, then the issue is whether to charge interest on just the original amount of the loan, the $1, or additionally charge interest on any unpaid or accumulated interest. We refer to the former as simple interest and the latter as compound interest. Most financial transactions and analyses use compound interest. The relationship between the amount of a loan, its present value (PV), and what must be paid in the future, its future value (FV), in the case of compound interest is:

where r is the interest rate per period and t is the number of periods. The term (1 + r)t is the compound interest factor. If we look at the future value for a range of compound periods, we see that compound interest increases the future value at a constant rate, r. As an example of the power of compounding, $100 invested for 100 periods at 5 percent interest is worth $13,150 using compound interest, but only $600 using simple interest.

The more common application of the time value of money mathematics is translating future values into present values. The relationship between the present value and the future value is an algebraic rearrangement of the first equation, with the rate of translation from future value to present value being the discount factor, .

Valuation mathematics have been applied to price stocks and bonds. The idea that the value of a stock is the present value of all future dividends can be traced to the work of John Burr Williams in 1938. Myron Gordon took this idea further and developed a model in which dividends grow at a constant rate in the future, which is an accurate depiction of many companies’ dividend patterns. The dividend valuation model, sometimes referred to as the Gordon model, states that the value of a stock today, P, is the ratio of next period's dividend, D1, divided by the difference between the required rate of return on the stock, k, and the expected growth rate of dividends, g:

FUTURE VALUE WITH COMPOUND INTEREST of $100 invested at 5 percent interest for up to one-hundred periods in the future, using both a linear scale and a logarithmic scale for future value.

The ratio of D₁ to k-g is the limit of the present value of a perpetually growing periodic future cash flow. The required rate of return is the return that investors demand on the stock and is called the cost of capital. The Gordon model and its many variants with different patterns of dividend growth are widely used by many financial analysts in stock valuation.

Bond valuation is an application of the valuation math where the future cash flows to investors comprise the periodic coupon interest payments and the repayment of the bond’s principal at the maturity date of the bond.

In a historical context the returns earned on securities do tend to reflect the associated risks. Small capitalization stocks, as measured by total market value of equity, have earned higher returns than large capitalization stocks, which have earned greater returns than corporate bonds and government bonds. For example, an investment of $100 from 1926 to 1995, provided an investor in small capitalization stocks with $382,828, an investor in large capitalization stocks $111,406, and an investor in long-term U.S. government bonds with $3,405. This rank ordering in earned returns reflects the variability of the returns on these securities: small capitalization stocks have greater year-to-year variability than large capitalization stocks, and so on.

Looking at the holdings of investors, we usually observe that they invest in many different types of assets, such as stocks, bonds, and real estate, and within an asset class they hold a broad array of assets. For example, investors may hold shares of stock in many different U.S. and non-U.S. companies. The reason why investors do this is to diversify. The idea behind diversification is that the future returns on different assets are not perfectly correlated with one another so that by holding different assets, the variability of the total investment -- the portfolio -- is less than the variability of the individual assets. Consider two stocks. D and E. The returns investors expect and the risks, as measured by the standard deviation, are

Suppose that you invest 40% or your funds in D and 60% in E. The risk of your portfolio, measured in terms of the standard deviation, depends on the correlation between the returns on D and E. If the returns of D and E are perfectly negatively correlated, a correlation of -1.0, the portfolio standard deviation is minimized at 7%; if the returns of D and E are perfectly positively correlated, a correlation of 1.0, the portfolio standard deviation is 23%, the simple weighted average of the two securities’ standard deviations.

We can extend the calculations of portfolio risk beyond two assets, but the basic idea remains the same: as long as the assets’ returns are not perfectly positively correlated with one another, adding assets to the portfolio will reduce the risk of the portfolio. This is the principle of diversification. When we apply this to stocks, this works to a point -- there is a minimum risk for all stock portfolios, which we refer to as market risk. This results because stocks generally are positively correlated with one another, suggesting the existence of a common factor driving some of the performance of all stocks. If we take a stock-only portfolio, however, and add another asset class like long-term corporate bonds, we can reduce the risk further. This process is called asset allocation, which is the distribution of an investor's portfolio between the different asset classes including stocks, corporate bonds, government bonds, real estate, and cash.

The analysis of how assets are priced has occupied much attention in the past three decades. We generally believe that investors are risk averse, meaning that they do not like risk and if they assume risk they demand to be compensated for it. We also believe that investors prefer more return to less return. Given these two beliefs, the investment decision focuses almost exclusively on return and risk. When we select among investments, we are making a decision today based on our expectations of the future benefits of the investment. As noted above, we measure return by the expected return and risk by the standard deviation of the return.

PORTFOLIO STANDARD DEVIATIONS for different values of the correlation between the returns on assets E and D. The greater the correlation, the greater the portfolio standard deviation.

THE CONCEPT OF DIVERSIFICATION is based on the idea that as more stocks are added to a portfolio, the portfolio risk, as measured by the standard deviation of the portfolio's expected returns, declines -- to a point.

Consider three assets: A, B, and C:

A rational investor would prefer A to B, C to A, and C to B. Given the ability to form portfolios, we can create an infinite number of possible combinations of A, B and C. From among these portfolios is a set of preferred assets that form what is referred to as the efficient frontier: the combination of assets that dominate all other combinations of assets by offering a higher expected return for the same risk. The efficient frontier is found by the optimization of a quadratic function subject to linear constraints.

If we combine the efficient frontier with the idea that investors can expand their return-risk opportunities through borrowing and lending, the set of optimal portfolios is even better than the efficient frontier in terms of risk and return. This better set of portfolios is referred to as the Capital Market Line.

There is one more element that completes the picture: just which one of the optimal portfolios does an investor choose? The preferences of individuals is derived from a branch of economics called utility theory. Irving Fisher brought mathematics and economics together in his dissertation Mathematical Investigations in the Theory of Value and Prices (1892), developing the concept of combinations of different

THE EFFICIENT FRONTIER is the set of portfolios that dominate all others in terms of expected return and risk (as measured by standard deviation in these figures). If borrowing or lending opportunities are available, the set of optimal portfolios is expanded to consist of those along the Capital Market Line.

factors that have equivalent satisfaction, quantified as "utils". The mathematical specification of this concept is called a utility function. Combining the idea of investor utility functions with portfolios, we can determine the optimal portfolio by choosing the portfolio that is tangent to the highest utility curve.. Though the utility function is an abstract concept, the basic idea is sound: individuals will seek out the best portfolio in terms of return and risk that they feel suits them. The implication for asset pricing is that, given optimal portfolio seeking behavior on the part of investors, the price of an asset will reflect the risk of the asset, with riskier assets carrying lower prices and higher expected returns than safer assets.

INDIVIDUAL INVESTOR UTILITY FUNCTIONS, when combined with the efficient frontier, dictate the optimal portfolio for each investor. In this graph, a sample of the utility functions of investor A (denoted A1, A2, and A3) and investor B (B1, B2, and B3) is plotted . A utility function represent the individual’s return-risk combinations perceived to produce equivalent "utility" or satisfaction. Different levels of satisfaction are represented by the different utility functions for the investor. In the case of investor A, A3 represents greater utility than A2. However, for the different return-risk combinations represented along function A3, the investor achieves equivalent utility.

An alternative to the CAPM is the Arbitrage Pricing Theory or APT, developed by Stephen Ross, which is a more general model of asset pricing. The APT is based on the idea of arbitrage, a concept we’ll discuss in more detail later, but which generally means that if prices are out of line, there will always be some investors ready to take advantage of this misalignment, forcing prices back in line. The APT is more general than the CAPM; the CAPM states that returns on stocks are a function of the returns on one factor, the market, whereas the APT states that returns on stocks are a function of potentially many more factors.

Empirical tests of the CAPM tend to support the model but as Richard Roll has demonstrated, any test of the CAPM is simply a test of whether the market portfolio is efficient. He showed that the efficiency of any portfolio is highly sensitive to the inclusion or exclusion of assets. Since the market portfolio must include all risky assets, any proxy will surely omit some assets. Consequently, the CAPM is essentially untestable. The APT does not impose such stringent requirements, but it gives no guidance as to the number or identity of the factors that affect asset returns. Thus, asset pricing theory, while providing considerable insights into the risk-return relationship, cannot truly be subjected to unbiased, scientific testing.

A great deal of time and effort over the past three decades has been spent grappling with the concept referred to as market efficiency. A market is efficient if the prices of assets quickly reflect all available information. The concept of market efficiency is based on several assumptions. First, there are many informed profit-maximizing market participants whom we call investors. Second, information arrives in the market in a random fashion. Third, prices adjust to reflect the effect of new information. Combined with the concepts of asset pricing, the price and expected return should immediately reflect all relevant information. Thus, we would observe no mispriced securities.

The seminal work in market efficiency was published by Eugene Fama in 1970, who classified market efficiency three ways according to the type of information : past prices, publicly-available information, and private information. The basic idea behind any test for market efficiency is that if investors can use the information to earn abnormal profits, the market is not efficient. Abnormal profits are returns in excess of the return expected based on the risk of the stock and transactions costs. If an investor can consistently earn positive abnormal profits, we casually refer to this as "beating the market".

If asset values reflect all past prices and other market-generated information such as the number of odd-lot transactions, we refer to this market as weak-form efficient. This means that past prices cannot be used to predict future prices. The implication of this for security markets is that studying charts of stock prices will not help investors earn abnormal profits.

If asset values reflect not only past prices but all publicly-available information, we refer to this market as semi-strong form efficient. The implication of this for security markets is that trading on the basis of information available to investors does not help them earn abnormal returns. This means that performing financial analyses of companies will not help investors beat the market.

If asset values reflect not only all publicly-available information but all private or "insider" information, we refer to the market as strong-form efficient. The implication of this is that insiders cannot beat the market using any information whatsoever.

Many studies have been performed using monthly, daily, and even trade-to-trade prices of stocks in both the U.S. and elsewhere to see whether security markets are efficient. Researchers use regression and time-series analysis, among other statistical tools to test market efficiency. The challenges facing researchers in examining market efficiency include identifying the point in time at which investors have access to information, quantifying security risk, and analyzing security returns. As is the case with most theories in economic and finance, empirical evidence on efficient markets is mixed.

Researchers examine weak-form efficiency using statistical tests of the independence of returns over time and tests of technical trading rules. The evidence tends to support the idea that U.S. stock markets are at least weak-form efficient, though recent results suggest that there may be some calendar-based patterns of stock prices that suggest otherwise.

Researchers examine whether the market is semi-strong efficient by using either event studies or trading rules. The event study approach focuses on the period that information is received by the market -- the event date -- and tests whether there are abnormal security returns before, after, or during the information’s arrival. The tests are basically tests of the statistical significance of forecast errors, a period before or after the event period as a baseline for developing the forecasted returns. Often we look at a plot of the cumulative sum of the abnormal returns called the cumulative abnormal return or CAR.

We expect abnormal returns to occur at the time the information hits the market. We do not expect abnormal returns prior to the public release of the information, except in cases where investors anticipate the information or insiders are trading. We do not expect abnormal returns subsequent to the release and if we do observe them, we would conclude that the market has not responded efficiently to the information. In the case of the earnings announcements shown in the graph, we see that the abnormal returns continue for some period following the announcement, suggesting that the information is not impounded in the stock price in an efficient manner.

The trading rule approach simulates the purchase or sale of stock at the time of the information arrival and tests whether abnormal profits can be earned following the information’s arrival. Most evidence suggests that the markets are semi-strong efficient, with stock prices reflecting information within fifteen minutes, though there is some evidence to the contrary. For example, researchers have found that buying stocks following the announcement of earnings that are better than expected provides abnormal profits.

There is limited evidence on whether markets are strong-form efficient, though the suspicion confirmed by well-publicized insider-trading cases is that there are profits to be made, though most likely illegally, if investors trade using inside information. This implies that the markets are not strong-form efficient because if the markets were efficient in this sense, it would mean there would be no benefit from trading on inside information. If we broaden the definition of insiders to include those with presumably financial analysis skills, such as the mutual fund managers, the evidence is supportive of strong-form efficiency. Thus, the overall conclusion is that the evidence is mixed, but for an average investor without special, and perhaps illegal, information, the market is very tough to beat.

CUMULATIVE ABNORMAL RETURNS up to sixty trading days following the announcement of greater-than-expected earnings.

Option pricing is the area of finance where science has probably played the most significant role. An option is a contract between two parties, the buyer and writer, that grants the buyer the right to buy or sell an asset from the writer at a fixed price for a predetermined period of time. To obtain that right, the buyer pays the seller a sum of money, called the option price or premium. An option to buy an asset is a call and an option to sell an asset is a put. The fixed price at which the option buyer can purchase or sell the asset is called the strike price or exercise price.

Deriving a formula for the price of the option was a challenging task. Though options had existed in this country since the mid 1800’s, it was not until the early 1970s that the pricing problem was solved. Let us take a look at how the financial science of option pricing evolved.

At the turn of the century at the Sorbonne in Paris, the doctoral student Louis Bachelier was studying for his degree under the tutelage of the great mathematician Henri Poincaré. Bachelier chose as his dissertation topic the movements of commodity prices and their associated option prices. Poincaré was not impressed with such an applied topic and, though approving the research, he gave Bachelier a mark of less than the highest distinction, thereby condemning the young scholar to an academic career at one of France’s lessor-known institutions. Sadly, little if anything was heard from Bachelier for the rest of his career.

At the same time, however, young Albert Einstein was developing his work on relativity. In the course of that research Einstein wrote papers on the mathematics of the physical phenomenon of Brownian motion, a statistical model of the collision of particles suspended in liquid that had first been observed by the Scottish physicist Robert Brown around 1827. The equations of Brownian motion were subsequently refined by the American mathematician Norbert Wiener in the 1920s. The equation of Brownian motion that played so important a role in option pricing was

The variable X(t) evolves through time according to a Brownian motion/Wiener process, with an expected movement of m [X(t),t] and volatility of s [X(t),t], both of which may change with the level of X(t), which is driven by the variable Z(t). Z(t) is a random process characterized by a normal, bell-shaped distribution with zero expected change and a volatility or diffusion proportional to D t. The variable X(t) is a generalized Brownian motion, whereby a non-zero expected change and volatility coefficient make the equation widely applicable to a variety of physical and financial phenomena. Equations such as this belong to the family of processes called stochastic differential equations, which are characterized by extremely rapid oscillations that decrease in magnitude as the time interval D t shrinks. Ordinary derivatives, such as dX(t)/dt, and their corresponding Riemann-Stieltjes integrals do not exist, necessitating the development of a whole new branch of mathematics called stochastic calculus.

A BROWNIAN MOTION/WEINER PROCESS showing the evolution of the variable X(t) as it moves through time, starting with a value of 100.

The new mathematics was supplied in the period immediately following World War II by the Japanese mathematician named Kiyoshi Itô. Probably his most influential work was the derivation of a lemma, which is stated as

which describes the dynamics of a random variable F, driven by a Brownian motion X(t) as it evolves through time. Itô’s Lemma, as it has since been called, is simply a Taylor series expansion of the stochastic function F[X(t),t], giving the total differential, with the added recognition that in stochastic processes of the Wiener type, differentials such as dX² do not disappear as they do in ordinary calculus. Just as Taylor’s theorem leads to the total differential of ordinary calculus and is known as the fundamental theorem of calculus, Itô’s Lemma has become known as the fundamental theorem of stochastic calculus.

While these ideas may seem far removed from modeling the world of finance, they are in fact ideally suited. Empirical tests, as noted earlier, had supported the idea of the randomness of financial markets. The marriage of these mathematical theories of physical processes and the movements of financial market prices is often credited to M. F. M. Osborne, a Naval physicist. Claiming that because he was a scientist with no economics training, he had impartially observed financial market prices and noticed that their behavior was remarkably akin to the well-known Brownian movement.

Osborne’s work provided much of the impetus for the formal theories of weak-form market efficiency and their empirical testing that we described earlier. The mathematical techniques for analyzing Brownian motion soon reached the halls of business schools and economics departments. One of the more interesting problems to which they seemed applicable was that of option pricing. A number of option pricing models had been developed by various researchers. Yet, all of the models either required knowledge of expected changes in stock prices or difficult to observe measures such as how investors feel about risk.

It was not until two young MIT professors named Fischer Black and Myron Scholes applied the aforementioned concept of arbitrage to the problem that a tractable model was discovered. Consider the basic fact that two financial instruments or combinations of financial instruments - portfolios - that produce the same returns in every situation logically must sell for the same price, a result is known as the Law of One Price. Arbitrage is a transaction executed when the Law of One Price is violated. The arbitrageur purchases the lower priced security or portfolio, sells the higher priced security or portfolio and profits at no risk from the difference in the prices. If markets are efficient, the law of one price will rarely be violated, which is a powerful result that plays a major role in option pricing models. It requires that we assume nothing more about how people will behave under risk than the simple fact that if given an opportunity to earn arbitrage profits, they will take action to do so. The combined transactions of all individuals and institutions engaging in arbitrage would force the lower priced instrument or portfolio up and the higher priced instrument or portfolio down until their prices converge.

Black and Scholes applied this principle to the strategy of purchasing a share of stock and selling a call option on the stock. Under the assumption that stock prices evolve according to Brownian motion, the instantaneous change in the option price is completely determined by infinitesimal movements in the stock price and increments in time. That being the case, the local risk in owning the stock can be eliminated by selling the option and continuously revising the ratio of options to stock, leading to a globally riskless position. The resulting combination should, therefore, earn a return equivalent to the return offered by riskless bonds, a rate easily observed in the market. From there Black and Scholes backed out the option price by obtaining the following parabolic partial differential equation:

with the solution subject to the boundary condition that at the expiration, t = T, the option is worth the exercise value, or c[S(T),T] = (S(T) - K)⁺. Although Black held a Ph.D. in applied mathematics from Harvard, he was not a specialist in differential equations. He derived the solution by one of what we now know are several alternative approaches, took the partial derivatives of the solution, inserted them back into the above PDE and verified that he had the right answer. Later it was pointed out to him that the above equation could be transformed into the heat transfer equation of thermodynamics, whose solution was already well-known.

The option pricing formula is

where S(t) is the current stock price, K is the exercise price, r_f is the risk-free rate, s is the volatility or standard deviation of the stock and T is the expiration of the option, meaning that T-t is the time to expiration in years. N(d₁) and N(d₂) are probabilities from the normal, bell-shaped curve.

THE BLACK-SCHOLES MODEL as the solution to a parabolic partial differential equation in S(t) and t.

Although the model is generally referred to as the Black-Scholes model, a young finance scholar at MIT named Robert C. Merton, working independently but in contact with Black and Scholes, derived the solution at around the same time. Merton, however, held up publication of his article in deference to Black and Scholes, who he felt deserved the primary recognition. After some difficulty convincing journal editors of the paper’s merits, the article eventually appeared in the May-June 1973 issue of The Journal of Political Economy, timed coincidentally with the opening of the Chicago Board Options Exchange, the nation’s first organized facility for trading options.

The Black-Scholes model found its way into finance textbooks over the next decade and by the middle of the 1980s, Black was busy putting his theories to work at the Wall Street firm of Goldman Sachs. Many others followed Black from academe to the real world. As the next generation of business school graduates reached the working world filled with knowledge of the model, a new revolution was created. Large financial institutions began offering options to corporations and investment funds. This new application of mathematical knowledge to the financial markets spawned a host of new products, such as swaps, structured notes and asset-backed securities, which integrated well with the array of forward and futures contracts that had been trading from many years. These instruments collectively came to be known as derivatives, their values being derived from the values of stocks, bonds, currencies and commodities.

Increasingly these derivative instruments became more complex, owing not to the ingenious scheming of high-tech financiers, but more to a recognition that risk in modern fast-moving and global financial markets is complex. The new instruments began playing a vital role in the buying and selling of risk. Firms needing to reduce risk could sell it to firms willing to bear it. The theories of Black and Scholes laid the groundwork for the development of most of this revolution, which came to be called financial engineering. For scientists it became a bonanza. Their services were now needed in places other than the laboratories of industrial corporations and universities. Mathematicians, physicists, systems engineers and computer scientists are now widely employed in the world’s financial institutions and increasingly by the finance divisions of corporations who buy these risk management products to control their interest rate and currency risks.

The outlook for further developments in finance is bright. There are now an extraordinary number of brilliant people working on interesting theoretical and applied problems in both academia and on Wall Street. The cross-fertilization of science and finance has without doubt been a beneficial partnership for both sides.

Asset-backed security: A financial obligation issued by one party entitling the holder to a claim on the returns produced by a combination of other assets, which are normally mortgages, credit card receivables or bonds.

Derivative: A contract between two parties in which one party ultimately pays the other party a sum of money according to how an underlying asset or other derivative performs in a market. Derivatives are said to "derive" their values from the values of other assets or derivatives. Types of derivatives include options, forwards, futures, swaps and sometimes asset-backed securities.

Forward contract: A contract in which one party, the buyer, agrees to buy an asset or other derivative from the other party, the seller, at a later date at a price agreed on in advance. The arrangement is a private transaction and is subject to the possibility that one of the parties could default.

Futures contract: A contract in which one party, the buyer, agrees to buy an asset or other derivative from the other party, the seller, at a later date at a price agreed on in advance, but which trades on an organized futures exchange and is subject to a daily settling of profits and losses. The arrangement is subject to the regulations of the exchange and normally a federal regulatory agency.

Option: A contract in which one party, the buyer, pays another party, the seller, a sum of money, called the premium, and receives the right to buy or sell an asset or other derivative from or to the seller at a price agreed on in advance. The right to buy is referred to as a call and the right to sell is referred to as a put. Options can be private transactions, subject to the possibility that the seller will default, or they can be transacted on organized options exchanges, where they are guaranteed against default and subject to exchange and federal regulatory rules.

Structured note: A debt security normally issued with a maturity of one to ten years in which the interest rate on the security adjusts according to movements in a market interest rate. The interest rate on the note may move opposite to the market interest rate and/or may adjust as a multiple of the move in the market interest rate. A structured note may also contain option-like features and/or may pay interest only if rates stay within or outside of a particular range. A variety of other features that adjust the way in which interest is paid are often found in structured notes.

Swap: A contract between two parties in which each agrees to make a series of payments to the other at predetermined dates where at least one of the two sets of payments is not determined until a later date. The parties are said to exchange or "swap" cash flows. The arrangement is a private transaction and is subject to the possibility that one of the two parties could default.

DON M. CHANCE is First Union Professor of Financial Risk Management at Virginia Tech, where he has taught since 1980. He holds a Ph.D. in finance from Louisiana State University and is a Chartered Financial Analyst. His research, teaching and consulting are concentrated in the area of financial derivatives and risk management. He is widely published in the academic and professional literature and is frequently quoted in the media.

PAMELA P. PETERSON is a Professor of Finance at Florida State University, where she has taught since 1981. She holds a Ph.D. from the University of North Carolina at Chapel Hill with an area of concentration of finance. She is also a Chartered Financial Analyst. Her research and teaching are concentrated primarily in corporate finance and empirical methods. Professor Peterson has numerous published articles and has served as an associate editor for several academic journals and is currently editor of Contemporary Finance Digest.