Mathematical Methods of Operations Research

, Volume 85, Issue 2, pp 155–177 | Cite as

Pricing and clearing combinatorial markets with singleton and swap orders

Efficient algorithms for the futures opening auction problem
  • Johannes C. MüllerEmail author
  • Sebastian Pokutta
  • Alexander Martin
  • Susanne Pape
  • Andrea Peter
  • Thomas Winter
Original Article


In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.


Equilibrium problems Hierarchical objectives Linear programming Network flows Combinatorial auctions Futures exchanges 

Mathematics Subject Classification

90C33 90C29 90C05 90C35 91B26 



We want to thank M. Rudel for providing us insight into his research results and for providing us with his test data that enabled us to quickly verify our model. We also would like to thank H. Schäfer for supporting our work and D. Weninger and A. Jüttner for the helpful discussions. We thank the DFG for their support within projects A05 and B07 in CRC TRR 154. Last but not least, we would like to thank the reviewers for their valuable comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Johannes C. Müller
    • 1
    Email author
  • Sebastian Pokutta
    • 2
  • Alexander Martin
    • 1
  • Susanne Pape
    • 1
  • Andrea Peter
    • 1
  • Thomas Winter
    • 3
  1. 1.Department of MathematicsFAU Erlangen-NürnbergErlangenGermany
  2. 2.ISyE, Georgia Institute of TechnologyAtlantaUSA
  3. 3.Eurex Frankfurt AGFrankfurtGermany

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