A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries. Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities. Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day. This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it. While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads). In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections. Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.
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