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Algebraic Properties of Arbitrage: An Application to Additivity of Discount Functions

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  • Salvador Cruz Rambaud

    (Departamento de Economía y Empresa, Universidad de Almería (Spain), 04120 Almería, Spain
    La Cañada de San Urbano, s/n (04120) Almería, Spain.)

Abstract

Background: This paper aims to characterize the absence of arbitrage in the context of the Arbitrage Theory proposed by Kreps (1981) and Clark (2000) which involves a certain number of well-known financial markets. More specifically, the framework of this model is a linear (topological) space X in which a (convex) cone C defines a vector ordering. There exist markets for only some of the contingent claims of X which assign a price p i to the marketed claim m i . The main purpose of this paper is to provide some novel algebraic characterizations of the no arbitrage condition and specifically to derive the decomposability of discount functions with this approach. Methods: Traditionally, this topic has been focused from a topological or probabilistic point of view. However, in this manuscript the treatment of this topic has been by using purely algebraic tools. Results: We have characterized the absence of arbitrage by only using algebraic concepts, properties and structures. Thus, we have divided these characterizations into those concerning the preference relation and those involving the cone. Conclusion: This paper has provided some novel algebraic properties of the absence of arbitrage by assuming the most general setting. The additivity of discount functions has been derived as a particular case of the general theory.

Suggested Citation

  • Salvador Cruz Rambaud, 2019. "Algebraic Properties of Arbitrage: An Application to Additivity of Discount Functions," Mathematics, MDPI, vol. 7(9), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:868-:d:268775
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    References listed on IDEAS

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    1. Roll, Richard & Ross, Stephen A, 1980. "An Empirical Investigation of the Arbitrage Pricing Theory," Journal of Finance, American Finance Association, vol. 35(5), pages 1073-1103, December.
    2. Fama, Eugene F, 1991. "Efficient Capital Markets: II," Journal of Finance, American Finance Association, vol. 46(5), pages 1575-1617, December.
    3. Clark, Stephen A., 2000. "Arbitrage approximation theory," Journal of Mathematical Economics, Elsevier, vol. 33(2), pages 167-181, March.
    4. Ross, Stephen A, 1978. "A Simple Approach to the Valuation of Risky Streams," The Journal of Business, University of Chicago Press, vol. 51(3), pages 453-475, July.
    5. LeRoy,Stephen F. & Werner,Jan, 2014. "Principles of Financial Economics," Cambridge Books, Cambridge University Press, number 9781107024120, October.
    6. Clark, Stephen A., 1993. "The valuation problem in arbitrage price theory," Journal of Mathematical Economics, Elsevier, vol. 22(5), pages 463-478.
    7. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2009, January-A.
    8. Constantinos Kardaras, 2012. "Market viability via absence of arbitrage of the first kind," Finance and Stochastics, Springer, vol. 16(4), pages 651-667, October.
    9. Salvador Cruz Rambaud, 1997. "Some new ideas in the concept of financial law," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 20(1), pages 23-43, June.
    10. Freddy Delbaen & Walter Schachermayer, 1994. "Arbitrage And Free Lunch With Bounded Risk For Unbounded Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 343-348, October.
    11. Claudio Fontana, 2015. "Weak And Strong No-Arbitrage Conditions For Continuous Financial Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-34.
    12. Salvador Cruz Rambaud, 1996. "Financial laws with algebraic automata theory," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 6, pages 65-102, Diciembre.
    13. Harvey, Charles M., 1994. "The reasonableness of non-constant discounting," Journal of Public Economics, Elsevier, vol. 53(1), pages 31-51, January.
    14. Stephen A. Clark, 2003. "An Infinite-Dimensional LP Duality Theorem," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 233-245, May.
    15. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, July-Dece.
    16. Fama, Eugene F., 1998. "Market efficiency, long-term returns, and behavioral finance," Journal of Financial Economics, Elsevier, vol. 49(3), pages 283-306, September.
    17. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
    18. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    19. Jouini,E. & Cvitanic,J. & Musiela,Marek (ed.), 2001. "Handbooks in Mathematical Finance," Cambridge Books, Cambridge University Press, number 9780521792370, October.
    20. Werner, Jan, 1987. "Arbitrage and the Existence of Competitive Equilibrium," Econometrica, Econometric Society, vol. 55(6), pages 1403-1418, November.
    21. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    22. Nikolaos A. Kyriazis, 2019. "A Survey on Efficiency and Profitable Trading Opportunities in Cryptocurrency Markets," JRFM, MDPI, vol. 12(2), pages 1-17, April.
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