Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 9

Differential Equations

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in financial economics. Differential equations arise in many areas of financial economics, that is, whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in time or some other variable (expressed as derivatives) is known or postulated. Differential equations are mathematically studied from several different perspectives, mostly concerning their solutions—the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution can be numerically approximated using computer algorithms. Using differential equations:

• One can come up with a closed-form solution for the prices of options, as in the case of the Black-Scholes model.
• One can introduce the key idea behind the Black-Scholes model to perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently eliminate risk.
• One can compute the quantities (popularly referred to as the “Greeks”) representing ...