A hypothetical company, XYZ Inc., manufactures circular metal disks for sale to ninjas. In the year 2010, ninja activity is on the rise, and XYZ wants to borrow money to expand production while hiring new employees. The best way to get the best people, in turns out, involves the offer of stock options as part of the compensation package. This situation is rife with opportunities for the use of calculus.

## Manufacturing Tolerance

In order to manufacture a circular metal disk with an area of X square inches, it will be critical to know what radius will produce such a disk. That is a straightforward arithmetical calculation, given pi. Going further, though, suppose the machinist is allowed a predetermined error tolerance in the area of this disk. It might be critical to know how to derive the error tolerance in the radius as a function of the error tolerance in the area. This involves two of the key conceptions of calculus: function and limit.

## Borrowing and Hedging

XYZ needs to borrow money to expand operations. There are many ways of doing this, and all of them have drawbacks. For example, a company's chief financial officer might be nervous if too much of the firm's borrowing is at variable rates of interest. He is concerned there might be a sharp upward move in rates. How could he reduce this risk? There are many financial products that will allow him to do so. The specific product he uses -- an option, a future, an interest-rate swap, a swaption -- depends upon the specifics of the company's existing debt and its plans. Valuing these products, ensuring that the company is buying what it needs and isn't paying too much for it, will require calculus.

## Employee Compensation Packages

The human resources department reports that XYZ needs to hire new talent and can best do so by offering a package that includes stock options. A stock option is an instrument that gives its recipient the right, but not the obligation, to purchase XYZ's stock at a given price on or before a given date. Generally Accepted Accounting Principles require that the stock options be treated on the company's books as an expense. How much of an expense is a question of valuation, and that in turn can become an opportunity for the application of calculus.

## Stock Options

Suppose that on June 1, 2010, XYZ's stock sold at \$50 a share. It issued stock options to its new employees authorizing them to buy XYZ stock on or before Jan. 1, 2010, at \$40 a share. Though the value of the option carries a speculative element when issued, it is surely not zero. So ... how is it to be valued? This is conventionally done through a formula within the field of "stochastic calculus" known as the Black-Scholes equation. Dan Oglevee, of Ohio State University, has explained that this equation is "independent of all variables affected by risk preference," a form of neutrality that is crucial to its appeal.