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Introduction to a theory of value coherent with the no-arbitrage principle

Author

Listed:
  • Marco Frittelli

    (Department of Quantitative Methods in Economics, University of Milano - Bicocca, 20126 Milano, Italy Manuscript)

Abstract

This paper defines the value of a general claim based on agent's preferences and coherent with the No Arbitrage Principle. This Value is a non trivial extension of the certainty equivalent since it takes into consideration the possibility of partially hedging the risk carried by the claim. When the market is complete this Value is the unique no arbitrage price. When the risk may not even be partially covered, this Value is the certainty equivalent. Between these two cases just some of the risk may be hedged and the no arbitrage principle requires the price to lie in the "arbitrage interval". The Value we propose is exactly designed to satisfy this condition.

Suggested Citation

  • Marco Frittelli, 2000. "Introduction to a theory of value coherent with the no-arbitrage principle," Finance and Stochastics, Springer, vol. 4(3), pages 275-297.
    • Handle: RePEc:spr:finsto:v:4:y:2000:i:3:p:275-297
    Note: received: April 1998; final version received: June 1999
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    Citations

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    Cited by:

    1. Grzegorz Hara'nczyk & Wojciech S{l}omczy'nski & Tomasz Zastawniak, 2007. "Relative and Discrete Utility Maximising Entropy," Papers 0709.1281, arXiv.org.
    2. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    3. Lixin Wu & Min Dai, 2009. "Pricing jump risk with utility indifference," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 177-186.
    4. Siu, Tak Kuen, 2016. "A functional Itô’s calculus approach to convex risk measures with jump diffusion," European Journal of Operational Research, Elsevier, vol. 250(3), pages 874-883.
    5. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
    6. Mark Owen & Gordan Zitkovic, 2007. "Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing," Papers 0706.0478, arXiv.org, revised Sep 2007.
    7. Mark P. Owen & Gordan Žitković, 2009. "Optimal Investment With An Unbounded Random Endowment And Utility‐Based Pricing," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 129-159, January.
    8. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    9. Barrieu, Pauline & El Karoui, Nicole, 2005. "Inf-convolution of risk measures and optimal risk transfer," LSE Research Online Documents on Economics 2829, London School of Economics and Political Science, LSE Library.
    10. Cerreia-Vioglio, S. & Maccheroni, F. & Marinacci, M. & Montrucchio, L., 2011. "Uncertainty averse preferences," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1275-1330, July.
    11. Miklos Rasonyi & Lukasz Stettner, 2005. "On utility maximization in discrete-time financial market models," Papers math/0505243, arXiv.org.
    12. Alet Roux & Zhikang Xu, 2019. "Optimal investment and contingent claim valuation with exponential disutility under proportional transaction costs," Papers 1909.06260, arXiv.org, revised May 2021.
    13. Robert Elliott & Tak Siu, 2010. "On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy," Annals of Operations Research, Springer, vol. 176(1), pages 271-291, April.
    14. Alessandro Doldi & Marco Frittelli, 2020. "Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality," Papers 2005.12572, arXiv.org, revised Sep 2021.
    15. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
    16. Ma, Hanmin & Tian, Dejian, 2021. "Generalized entropic risk measures and related BSDEs," Statistics & Probability Letters, Elsevier, vol. 174(C).
    17. Alessandro Doldi & Marco Frittelli & Emanuela Rosazza Gianin, 2024. "On entropy martingale optimal transport theory," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 1-42, June.
    18. Nikolai Dokuchaev, 2015. "Optimal portfolio with unobservable market parameters and certainty equivalence principle," Papers 1502.02352, arXiv.org.
    19. Becherer, Dirk, 2003. "Rational hedging and valuation of integrated risks under constant absolute risk aversion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 1-28, August.
    20. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2004. "Dynamic monetary risk measures for bounded discrete-time processes," Papers math/0410453, arXiv.org.
    21. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    22. Elisa Pagani, 2015. "Certainty Equivalent: Many Meanings of a Mean," Working Papers 24/2015, University of Verona, Department of Economics.
    23. Dejian Tian, 2022. "Pricing principle via Tsallis relative entropy in incomplete market," Papers 2201.05316, arXiv.org, revised Oct 2022.
    24. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.

    More about this item

    Keywords

    Certainty Equivalent; Asset Pricing; No Arbitrage; Equivalent Martingale Measure; Incomplete Market.;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory

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