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

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Abstract

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.

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Notes

  1. We do not discuss here cross-margining principles for offsets involving positions with different underlying instruments. These principles are the same for both approaches; cf. Lindsey and Pecora (1998).

  2. For example, the lowest and highest valuation points for a stock should be 15 % lower and higher than its current market price. These percentages follow SEC Rule 15c3–1a, section (b)(1)(i)(B), for net capital requirements of brokers or dealers.

  3. The U.S. Securities and Exchange Commission.

  4. The National Association of Securities Dealers.

  5. The New York Stock Exchange.

  6. Margin account equity is the total market value of all its positions, where long/short positions have positive/negative market values.

  7. The Options Clearing Corporation.

  8. The Chicago Board Options Exchange.

  9. In 2007, the NASD merged with the NYSE’s regulation committee to form the Financial Industry Regulatory Authority, or FINRA.

  10. Offsets of minimal quantity; see Sec. 2.1 for formal definition.

  11. A leg of an option spread or offset based on this spread is a position in options with the same exercise price and expiry date.

  12. For example, according to http://finance.google.com, as of 2 Aug. 2011, 5:50 p.m., exercise prices of options on the IBM stock listed in NYSE and expiring on 20 Aug. 2011 generated the uniform exercise domain \(\{85,90,\ldots ,270\}\) of dimension \(38\).

  13. The term debit/credit indicates that the spread is a result of a net debit/credit transaction.

  14. A negative quantity of a spread should be interpreted as a positive quantity of its negation.

  15. The regulatory definitions given in SEC Release 34-52738, the CBOE Regulatory Circular and NYSE Rule 431 imply only these seven types. The CBOE gave some of complex spreads the same names as those of their resulting main spreads. To avoid confusions, we do not use these names. We should also emphasize that this paper presents our mathematical interpretation of CBOE’s informal definitions of complex spreads in a text form. Our goal was to follow the idea given in the definitions as close as possible and, at the same time, avoid inconsistencies that we found in them. Any omission that someone may find in our mathematical interpretation of CBOE’s complex spreads will be our responsibility.

    Table 5 CBOE’s complex spreads, their components and resulting spreads
  16. Centipedes, as all other creatures, have even number of legs (one pair of legs per body segment), and this number can reach 200 and more. Centipedes usually do not bite humans but a few species, when provoked, can bite inflicting painful wounds. Millipedes are creatures with number of legs multiple of four (two pairs of legs per body segment). Some species have over 400 legs. Millipedes are not predators as centipedes—Wikipedia (terrestrial animals). One can observe that multi-leg spreads introduced in this section have similar properties if their legs are counted not as positions in options but as option contracts involved.

  17. Some researchers provide evidence that the distribution of trades of main spreads is not uniform in certain option markets. For example, options on Eurodollar futures in 1999–2000, see Chaput and Ederington (2003), and options on the FTSE 100 Index in 2001–2004, see Fahlenbrach and Sandås (2010), were more frequently traded in spreads with two legs than spreads with three or four legs. One could argue that a similar trend should take place for stock options on NASDAQ: AAPL in 2011 (the year of our dataset), however, the stock market in 2011 was very volatile. Therefore, volatility option strategies which use spreads with three or four legs may have been traded more frequently than spreads with two legs. Under the absence of any data on these trades, we believe that using a uniform distribution of main spreads in option portfolios is justified.

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Matsypura, D., Timkovsky, V. Integer programs for margining option portfolios by option spreads with more than four legs. Comput Manag Sci 10, 51–76 (2013). https://doi.org/10.1007/s10287-012-0159-x

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