Integer programs for margining option portfolios by option spreads with more than four legs

Margining is a crucial brokerage operation. In application to option portfolios it becomes exceptionally challenging because margin offsets with options require solving a highly intractable integer program. All these offsets are based on option spreads with a maximum of four legs. Although option spreads with more than four legs can be traced in regulatory literature of 2003, they have not yet been studied and used. Their usage in margin calculations would substantially increase the size of the program and therefore make it practically unsolvable. On the other hand, option spreads with more than four legs would allow the brokers to substantially increase the accuracy of margin calculations for option portfolios. In this paper we develop a theoretical framework for option spreads with any number of legs. We show that these spreads can be naturally described by homomorphisms of free abelian groups associated with option portfolios and option spreads with up to four legs. Using this observation we propose alternative integer programs that use option spreads with any number of legs and whose size does not depend on the number of legs. These programs can be solved in reasonable time and substantially increase the accuracy of margin calculations for option portfolios.