Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 7

Optimization

In mathematics and statistics, optimization means the selection of a best element (with regard to some criteria) from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a function by choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. In finance, optimization is widely used in asset allocation, bond portfolio management, and derivative pricing. Using optimization:

• One can determine the mean-variance efficient frontier by maximizing expected return of a portfolio subject to a risk constraint. Alternatively, one can minimize the portfolio’s risk subject to an expected return constraint.
• One can find the optimal portfolio weights by maximizing expected utility of a risk-averse investor defined as the expected return of a portfolio minus the product of the risk aversion parameter and the portfolio’s variance.
• One can construct an immunized portfolio, which means a portfolio created so as to have an assured return for a specific time horizon irrespective of interest rate changes, one can deal with cash flow matching, also referred to as a dedicated portfolio strategy or the problem of matching a predetermined set of liabilities with an investment portfolio that produces a deterministic stream of cash flows.
• One can compute the optimal ...