A heuristic process on the existence of positive bases with applications to minimum-cost portfolio insurance in C[a, b]

Abstract

In this work we propose an algorithmic process that finds the minimum-cost insured portfolio in the case where the space of marketed securities is a subspace of C[a, b]. This process uses, effectively, the theory of positive bases in Riesz spaces and does not require the presence of linear programming methods. The key for finding the minimum-cost insured portfolio is the existence of a positive basis. Until know, we could check, under a rather complicated procedure, the existence of a positive basis in a prescribed interval [a, b]. In this paper we propose a heuristic method for computing appropriate intervals [a, b], so that the existence of a positive basis is guaranteed. All the proposed algorithmic processes are followed by appropriate Matlab code.

Introduction

It is well known that the theory of Riesz spaces (or vector lattices) has been extensively used in the last years to solve different problems in Mathematical Economics. For example, a lot of work has been done in the areas of incomplete markets (see [2], [4], [15], [16]) and especially in portfolio insurance (see [2], [3], [5], [6], [7], [8], [14]), in equilibrium theory (see [1], [4], [16], [17], [18], [19]) and option replication (see [9], [10], [11], [12], [13]). Finite-dimensional Riesz spaces and their applications in economics are important since it is well known that many economic models are finite, such as, for example the Arrow–Debreu model.

Lattice-subspaces, is a class of subspaces, of a Riesz space, with particular interest in the study of the problem of finding the minimum-cost insured portfolio. Let us denote by C[a, b], the space of all continuous real functions on the interval [a, b]. In the present manuscript we present the basic elements from the theory of lattice-subspaces in C[a, b], as well as elements from the theory of finance in order to construct a powerful and efficient package that finds the minimum-cost insured portfolio in the case where the marketed securities are time variant and especially, when they are elements of C[a, b].

In the literature, minimum-cost portfolio insurance has being characterized as a very important investment strategy. This strategy allows the investor to avoid losses but also allows him/her to catch the gains at the minimum cost. It has been shown in [3] that even if the derivative market is complete this is not a necessary condition for the minimum-cost portfolio insurance to be price-independent. Appropriate use of the theory of Riesz spaces can give characterizations of market structures in which the cost minimizing portfolio is price-independent (see [2], [3], [5], [6], [7], [8], [14]). It is a challenging task to program an algorithmic procedure for solving the minimum-cost portfolio insurance problem in C[a, b] since even for a small number of securities, it is an extremely demanding exercise to solve the problem by hand. In fact, the joint amount of tests, calculations and further considerations required to reach the goal may well render the manual solution process a prohibiting task. For the exhibition of our proposed computational method effectiveness, we have simplified the procedure to the extent that the interested user can reach a fast computational solution using a reduced amount of computational resources. An efficient method for solving the problem of finding the minimum cost insured portfolio in C[a, b] is described in [6]. The method in [6] is complicated, as expected in the case of C[a, b], and uses several different Matlab functions to reach to a solution. The computational method presented in this article uses a different, more efficient, compact and simplified approach that leads to only one Matlab function for the solution of the minimization problem. The key for finding the minimum-cost insured portfolio is the existence of a positive basis. In this paper we shall propose a heuristic method, supported by an algorithm and an easy to use Matlab function, in order to find (if any) appropriate intervals of real numbers, [a, b], such that the subspace $X=Span\{x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right)\},$ where $x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right)$ are linearly independent, positive functions of C[a, b], forms a lattice-subspace of C[a, b]. Equivalently, we will find appropriate intervals [a, b], so that X has a positive basis. Main results of the present paper are summarized as follows. Section 1 describes the mathematical problem of finding positive bases. In Section 2 we describe the mathematical problem. Section 3 describes the problem of finding the minimum-cost portfolio insurance. Section 4 proposes a new algorithm for finding the minimum-cost portfolio insurance in C[a, b]. Section 5 presents a heuristic method on the existence of positive bases followed by several examples. Last sections are, section 6 with concluding remarks and the appendix, Section 7, with all relevant Matlab codes.