Financial Derivatives and Partial Differential Equations

1. ASSETS AND DERIVATIVES. Assets of all sorts are traded in financial markets: stocks and stock indices, foreign currencies, loan contracts with various interest rates, energy in many forms, agricultural products, precious metals, etc. The prices of these assets fluctuate, sometimes wildly. As an example, Figure 1 shows the price of IBM stock within a single day. The picture would look more or less the same across a month, a year, or a decade, though the axis scales would be different. If you could anticipate the price fluctutations to any significant extent, then you could clearly make a great amount of money very quickly. The fact that many people are trying to do exactly that makes the fluctuations essentially unpredictable for practical purposes. A fundamental principle of finance, the efficient market hypothesis [9] asserts that all information available to anyone anywhere is instantly expressed in the current price, as market participants race to be the first to profit from new information. Thus successive price changes may be considered to be uncorrelated random variables, since they depend on as-yet unrevealed information. This principle is the subject of intensive analytical testing and some controversy [7], but is an excellent approximation for our purposes. Although the directions of the price motions are completely unpredictable, statistics can tell us a lot about their expected size. Figure 2 shows the distribution of percentage changes in IBM stock price across half hour time intervals. We can identify a typical size of the fluctuations, about half of one percent in this example. Since the fluctuations are uncorrelated and have mean near zero, this typical size is the single most important statistical quantity that we can extract from the price history. We may additionally ask about the form of this distribution, for example, whether or not it is a Gaussian. Again, this is the subject of active research [10]. ∗To appear in American Mathematical Monthly