Computational and mathematical methods in portfolio insurance - A MATLAB-based approach

Portfolio insurance is based on the principal of risk transfer i.e., one person’s protection is another person’s liability. The cost of portfolio insurance is the mechanism to equilibrate its demand with supply. In the theory of finance minimum-cost portfolio insurance has been characterized as a very important investment strategy. In this chapter, we discuss the investment strategy called minimum-cost portfolio insurance as a solution of a cost minimization problem and we propose computational methods that translate the economics problem into the language of computing. This strategy not only enables an investor to avoid losses but also allows him/her to capture the gains at the minimum cost. In general, it is well known that the minimum-cost insured portfolio depends on security prices. The cases where it is priceindependent (i.e., it does not depend on arbitrage-free security prices) are very important not only because the insured portfolio can be selected without knowledge of current security prices but also because we can present it in a simple form. Market structures in which minimum-cost portfolio insurance is price-independent relies on the theory of vector lattices (Riesz spaces). In particular, we focus our study in two very important classes of subspaces of a vector lattice, namely vector sublattices and lattice-subspaces. Vector lattices have been used by Brown & Ross (1991) and by Green & Jarrow (1987) in the framework of options markets. Also, Ross (1976) gave a characterization of complete markets by observing that derivative markets are complete if and only if the asset span is a vector sublattice of Rk. Completeness of derivative markets is a sufficient but not necessary condition for the minimum-cost portfolio insurance to be price-independent. Let us denote by X the subspace of payoffs of all portfolios of securities; then in Aliprantis et al. (2000) it is proved that the minimum-cost insured portfolio exists and is price-independent for every portfolio and at every floor if and only if X is a lattice-subspace of Rk. An equivalent necessary and sufficient condition so that X is a latticesubspace is the existence of a positive basis for X, that is a basis of limited liability payoffs such that every marketed limited liability payoff has a unique representation as a nonnegative linear combination of basis payoffs. The notion of a positive basis for X is a generalization to incomplete markets of a basis of Arrow securities for complete markets. From the previous discussion, it is evident that the mathematical theory of lattice-subspaces has been used in order to provide a characterization of market structures in which the cost minimizing portfolio is price-independent. In general, the theory of lattice-subspaces has been extensively used in !"