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Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads

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  • Fulvio Ortu

Abstract

In a general, finite-dimensional securities market model with bid-ask spreads, we characterize absence of arbitrage opportunities both by linear programming and in terms of martingales. We first show that absence of arbitrage is equivalent to the existence of solutions to the linear programming problems that compute the minimum costs of super-replicating the feasible future cashflows. Via duality, we show that absence of arbitrage is also equivalent to the existence of underlying frictionless (UF) state-prices. We then show how to transform the UF state-prices into state-price densities, and use them to characterize absence of arbitrage opportunities in terms of existence of a securities market with zero bid-ask spreads whose price process lies inside the bid-ask spread. Finally, we argue that our results extend those of Naik (1995) and Jouini and Kallal (1995) to the case of intermediate dividend payments and positive bid-ask spreads on all assets. Copyright Springer-Verlag Italia 2001

Suggested Citation

  • Fulvio Ortu, 2001. "Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 24(2), pages 79-105, November.
    • Handle: RePEc:spr:decfin:v:24:y:2001:i:2:p:79-105
    DOI: 10.1007/s102030170001
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    Citations

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    Cited by:

    1. Martin Brown & Tomasz Zastawniak, 2019. "Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs," Papers 1905.01859, arXiv.org, revised May 2019.
    2. Alet Roux, 2007. "The fundamental theorem of asset pricing under proportional transaction costs," Papers 0710.2758, arXiv.org.
    3. M. Pınar & A. Camcı, 2012. "An Integer Programming Model for Pricing American Contingent Claims under Transaction Costs," Computational Economics, Springer;Society for Computational Economics, vol. 39(1), pages 1-12, January.
    4. Lionel Martellini, 2000. "Efficient Option Replication in the Presence of Transactions Costs," Review of Derivatives Research, Springer, vol. 4(2), pages 107-131, May.
    5. Roux, Alet, 2011. "The fundamental theorem of asset pricing in the presence of bid-ask and interest rate spreads," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 159-163, March.
    6. Tomasz Zastawniak, 2024. "Fundamental Theorem of Asset Pricing under fixed and proportional costs in multi-asset setting and finite probability space," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 137-149, June.
    7. Tokarz, Krzysztof & Zastawniak, Tomasz, 2006. "American contingent claims under small proportional transaction costs," Journal of Mathematical Economics, Elsevier, vol. 43(1), pages 65-85, December.
    8. Martin Brown & Tomasz Zastawniak, 2020. "Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs," Annals of Finance, Springer, vol. 16(3), pages 423-433, September.

    More about this item

    Keywords

    Mathematics Subject Classification (2000): 91B28; 91B70; Journal of Economic Literature Classification: C61; C63; G10; G12;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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