Combinatorics of Option Spreads: The Margining Aspect

Abstract

In December 2005, the U.S. Securities and Exchange Commission approved margin rules for complex option spreads with 5, 6, 7, 8, 9, 10 and 12 legs. Only option spreads with 2, 3 or 4 legs were recognized before. Taking advantage of option spreads with a large number of legs substantially reduces margin requirements and, at the same time, adequately estimates risk for margin accounts with positions in options. In this paper we present combinatorial models for known and newly discovered option spreads with up to 134 legs. We propose their full characterization in terms of matchings, alternating cycles and chains in graphs with bicolored edges. We show that the combinatorial analysis of option spreads reveals powerful hedging mechanisms in the structure of margin accounts, and that the problem of minimizing the margin requirement for a portfolio of option spreads can be solved in polynomial time using network flow algorithms. We also give recommendations on how to create more efficient margin rules for options.