Eur. Phys. J. B 27, 263-272 (2002)
https://doi.org/10.1140/epjb/e20020143

Portfolio optimization with short-selling and spin-glass

¹ Istituto Nazionale di Fisica della Materia, Largo E. Fermi 2, 50125 Firenze, Italy
² Dipartimento di Matematica dell' Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
³ Dipartimento di Economia dell' Università di Parma, via Kennedy 6, 43100 Parma, Italy

Corresponding author: ^a augusto.schianchi@ipruniv.cce.unipr.it

Received: 31 December 2001
Published online: 15 May 2002

Abstract

In this paper, we solve a general problem of optimizing a portfolio in a futures markets framework, extending the previous work of Galluccio et al. [Physica A 259, 449 (1998)]. We allow for long buying/short selling of a relatively large number of assets, assuming a fixed level of margin requirement. Because of non-linearity in the constraint, we derive a multiple equilibrium solution, in a size exponential respect to the number of assets. That means that we can not obtain the unique efficiency frontier, but many of them and each one is related to different levels of risk. Such a problem is analogous to that of finding the ground state in long-ranged Ising spin glass with external field. In order to get the best portfolio (i.e. that is along the best efficiency frontier), we have to implement a two-step procedure, performing the exhaustive enumeration of all local minima. We develop a concrete application, where the different part of the proposed solution are computed.