An Introduction to Financial Markets

Chapter Fifteen Optimization Model Building

In this chapter, we describe a range of optimization models aimed at financial applications. Emphasis will be almost exclusively on model building, rather than model solving, which is deferred to the next chapter. To this aim, it is important to have a broad and clear picture of the types of model formulations that may be solved by available software tools. Intuition suggests that a large problem is a harder nut to crack than a small one and that a linear problem is easier than a nonlinear one. Actually, this is not necessarily true, and we shall learn that the main problem feature, drawing the line between relatively easy and difficult problems, is convexity.

We begin, in Section 15.1, with a classification of optimization problems, revolving around the concepts of convex sets and convex functions. The impor tant class of linear programming (LP) models is the topic of Section 15.2. We have already appreciated the role of LP models in the mathematics of arbitrage in Section 2.4. Here, we see how it may be an essential tool in solving possibly large-scale problems. The next level of the hierarchy is convex quadratic programming (QP), which is dealt with in Section 15.3. Convex QP models are the foundation of the mean–variance portfolio theory illustrated in Chapter 8. In Section 15.4, we take a detour into the realm of hard, nonconvex problems. While LPs and convex QPs can be solved quite efficiently, here we add an integrality restriction ...