An Elementary Introduction to Mathematical Finance

Introduction

In this chapter we consider some optimization problems involving onetime investments not necessarily tied to the movement of a publicly traded security. Section 11.2 introduces a deterministic optimization problem where the objective is to determine an efficient algorithm for finding the optimal investment strategy when a fixed amount of money is to be invested in integral amounts among n projects, each having its own return function. Section 11.2.1 presents a dynamic programming algorithm that can always be used to solve the preceding problem; Section 11.2.2 gives a more efficient algorithm that can be employed when all the project return functions are concave; and Section 11.2.3 analyzes the special case, known as the knapsack problem, where project investments are made by purchasing integral numbers of shares, with each project return being a linear function of the number of shares purchased. Models in which probability is a key factor are considered in Section 11.3. Section 11.3.1 is concerned with a gambling model having an unknown win probability, and Section 11.3.2 examines a sequential investment allocation model where the number of investment opportunities is a random quantity.

A Deterministic Optimization Model

Suppose that you have m dollars to invest among n projects and that investing x in project i yields a (present value) return of fi(x), i = 1, …, n. The problem is to determine the integer amounts to invest in each project so as to maximize the sum of the returns.