Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 8

Difference Equations

In mathematics and statistics, a difference equation refers to a recurrence relation, or an equation that recursively defines a sequence, once one or more initial terms are given. Each further term of the sequence is defined as a function of the preceding terms. Difference equations are extensively used in financial economics. For example, in a theoretical asset pricing framework, one might develop a model of various broad sectors of the economy in which some agents’ actions depend on lagged variables. The model would then be solved for current values of key variables (interest rate, output growth, etc.) in terms of exogenous variables and lagged endogenous variables. Difference equations are also useful in financial econometrics. For example, time-series forecasting uses a recursive model to predict future values of financial and economic indicators based on the previously observed values of these variables. In finance, difference equations are also used to model persistence structure of asset returns and asset return volatility. Using difference equations:

• One can determine the serial correlation structure of asset returns in the context of a dynamic econometric model.
• One can examine the stochastic behavior of asset prices expressed as a linear difference equation with random disturbances added.
• One can understand the joint dynamics and dependencies of two persistent financial and macroeconomic variables.
• One can provide a better understanding ...