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Positivity

, Volume 13, Issue 1, pp 201–224 | Cite as

Geometry of polar wedges in Riesz spaces and super-replication prices in incomplete financial markets

  • Frank Oertel
  • Mark P. Owen
Article
  • 53 Downloads

Abstract

This paper is devoted to a further generalisation of the main results in [5] including the representation of the weak super-replication price (cf. equation (1.6)). In addition to the already established weakening of the technical assumptions in [5] (cf. [24] and [25]), the main results in [5] can be still generalised by considering the geometric structure of the underlying problem (based on the properties of Riesz spaces and polar wedges therein). In Section 5 we show under which geometric conditions of the relevant sets the results still hold (cf. Theorem 5.3 and Corollary 5.5). In particular, we can completely remove the restrictive admissibility assumption and carry forward equation (1.4) to a larger class of wedges \(K \subseteq L^{0}\) (cf. Corollary 5.5).

Keywords

Super-replication Incomplete markets Contingent claims Duality theory Weak topologies Riesz spaces 

Mathematics Subject Classification (2000)

1B16 46N10 60G44 

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References

  1. 1.
    C.D. Aliprantis, O. Burkinshaw, Positive Operators. Springer, Netherlands (2006).Google Scholar
  2. 2.
    C.D. Aliprantis, R. Tourky, Cones and Duality. AMS, Graduate Studies in Mathematics, vol. 84 (2007).Google Scholar
  3. 3.
    P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent measures of risk. Math. Finance, 9(3), 203–228 (1999).Google Scholar
  4. 4.
    M. Baxter, A. Rennie. Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press, Cambridge (1996).Google Scholar
  5. 5.
    S. Biagini, M. Frittelli, On the super-replication price of unbounded claims. Ann. Appl. Prob., 14(4), 1970–1991 (2004).Google Scholar
  6. 6.
    S. Biagini, M. Frittelli, Utility maximization in incomplete markets for unbounded processes. Finance Stochast., 9, 493–517 (2005).Google Scholar
  7. 7.
    S. Biagini, M. Frittelli, A unified framework for utility maximization problems: an Orlicz space approach. Ann. Appl. Prob., 18(3), 929–966 (2008).Google Scholar
  8. 8.
    J. Borwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization. Springer, CMS Books in Mathematics, 2nd edn (2006).Google Scholar
  9. 9.
    F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann., 300, 463–520 (1994).Google Scholar
  10. 10.
    F. Delbaen, W. Schachermayer, The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep., 53, 213–226 (1995).Google Scholar
  11. 11.
    F. Delbaen, W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann., 312, 215–250 (1998).Google Scholar
  12. 12.
    F. Delbaen, W. Schachermayer, Applications to mathematical finance. In: Handbook of the geometry of Banach spaces, vol. 1, North-Holland, Amsterdam, pp. 369–391 (2001).Google Scholar
  13. 13.
    F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage. Springer Finance (2006).Google Scholar
  14. 14.
    R.J. Elliott, P.E. Kopp, Mathematics of Financial Markets. Springer Finance, 2nd edn (2005).Google Scholar
  15. 15.
    M. Frittelli, Optimal solutions to utility maximization and to the dual problem. Preprint (2000).Google Scholar
  16. 16.
    M. Frittelli, E. Rosazza Gianin, Equivalent formulations of reasonable asymptotic elasticity. Preprint (2002).Google Scholar
  17. 17.
    H.G. Heuser, Functional Analysis. John Wiley & Sons, New York (1982).Google Scholar
  18. 18.
    J. Hugonnier, D. Kramkov, Optimal investment with random endowments in incomplete markets. Ann. Appl. Prob., 14, 845–864 (2004).Google Scholar
  19. 19.
    H. Jarchow, Locally convex spaces. B. G. Teubner, Stuttgart (1981).Google Scholar
  20. 20.
    F.C. Klebaner, Introduction to Stochastic Calculus with Applications. Imperial College Press (1998).Google Scholar
  21. 21.
    D. Kramkov, W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob., 9, 904–950 (1999).Google Scholar
  22. 22.
    W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces I. North-Holland, Amsterdam (1971).Google Scholar
  23. 23.
    P. Meyer-Nieberg, Banach Lattices. Springer, Berlin (1991).Google Scholar
  24. 24.
    M.P. Owen, On utility based super-replication prices. Preprint (2003).Google Scholar
  25. 25.
    M.P. Owen, F. Oertel, On utility-based super-replication prices of contingent claims with unbounded payoffs. J. Appl. Prob., 44(4), 880–888 (2007).Google Scholar
  26. 26.
    M.P. Owen, G. Žitković, Optimal investment with an unbounded random endowment and utility-based pricing methods. To appear in Math. Finance (2008).Google Scholar
  27. 27.
    P. Protter. Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2003).Google Scholar
  28. 28.
    R.T. Rockafellar, Convex Analysis. Princeton University Press (1972).Google Scholar
  29. 29.
    S. Roman, Introduction to the Mathematics of Finance. Springer Undergraduate Texts in Mathematics (2004).Google Scholar
  30. 30.
    W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Prob., 11, 694–734 (2001).Google Scholar
  31. 31.
    W. Schachermayer, Utility maximisation in incomplete markets. Stochastic Methods in Finance. Lectures given at the CIME–EMS Summer School in Bressanone/Brixen, Italy, July 6–12, 2003, M. Frittelli, W. Runggaldier, (eds). Springer Lecture Notes in Mathematics 1856, 225–288 (2004).Google Scholar
  32. 32.
    H.H. Schaefer, Banach Lattices and Positive Operators. Springer (1974).Google Scholar
  33. 33.
    Y.-C. Wong, Some Topics in Functional Analyis and Operator Theory. Science Press (1993).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Frank Oertel
    • 1
  • Mark P. Owen
    • 2
  1. 1.Department of MathematicsUniversity College CorkCorkIreland
  2. 2.Department of Actuarial Mathematics and StatisticsMaxwell Institute for Mathematical Sciences and Heriot-Watt UniversityEdinburghScotland

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