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M6 - On Minimal Market Models and Minimal Martingale Measures

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Abstract

The well-known absence-of-arbitrage condition NFLVR from the fundamental theorem of asset pricing splits into two conditions, called NA and NUPBR. We give a literature overview of several equivalent reformulations of NUPBR; these include existence of a growth-optimal portfolio, existence of the numeraire portfolio, and for continuous asset prices the structure condition (SC). As a consequence, the minimal market model of E. Platen is seen to be directly linked to the minimal martingale measure. We then show that reciprocals of stochastic exponentials of continuous local martingales are time changes of a squared Bessel process of dimension 4. This directly gives a very specific probabilistic structure for minimal market models.

Suggested Citation

  • Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series 280, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:280
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-03/QFR-rp280.pdf
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    References listed on IDEAS

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    1. Eva Strasser, 2005. "Characterization of arbitrage-free markets," Papers math/0503473, arXiv.org.
    2. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 263-286, July.
    3. Michel Chatelain & Christophe Stricker, 1994. "ON COMPONENTWISE and VECTOR STOCHASTIC INTEGRATION," Mathematical Finance, Wiley Blackwell, vol. 4(1), pages 57-65.
    4. 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.
    5. Eckhard Platen, 2004. "Diversified Portfolios with Jumps in a Benchmark Framework," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 1-22, March.
    6. repec:wsi:ijtafx:v:07:y:2004:i:04:n:s0219024904002499 is not listed on IDEAS
    7. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    8. Eckhard Platen, 2004. "Modeling The Volatility And Expected Value Of A Diversified World Index," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(04), pages 511-529.
    9. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Citations

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    Cited by:

    1. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2015. "Non-Arbitrage Under Additional Information for Thin Semimartingale Models," Papers 1505.00997, arXiv.org.
    2. Christa Cuchiero, 2017. "Polynomial processes in stochastic portfolio theory," Papers 1705.03647, arXiv.org.
    3. Tahir CHOULLI & Martin SCHWEIZER, 2015. "Locally Phi-Integrable Sigma-Martingale Densities for General Semimartingales," Swiss Finance Institute Research Paper Series 15-15, Swiss Finance Institute.
    4. Francesca Biagini & Alessandra Cretarola & Eckhard Platen, 2012. "Local Risk-Minimization under the Benchmark Approach," Research Paper Series 319, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series 296, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Tahir Choulli & Jun Deng & Junfeng Ma, 2012. "How Non-Arbitrage, Viability and Num\'eraire Portfolio are Related," Papers 1211.4598, arXiv.org, revised Jun 2014.
    7. Claudio Fontana & Wolfgang J. Runggaldier, 2012. "Diffusion-based models for financial markets without martingale measures," Papers 1209.4449, arXiv.org, revised Feb 2013.
    8. Ceci, Claudia & Colaneri, Katia & Cretarola, Alessandra, 2014. "A benchmark approach to risk-minimization under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 129-146.
    9. Gabriel Frahm, 2016. "Pricing And Valuation Under The Real-World Measure," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-39, February.
    10. Baldeaux Jan & Ignatieva Katja & Platen Eckhard, 2014. "A tractable model for indices approximating the growth optimal portfolio," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(1), pages 1-21, February.
    11. Claudio Fontana, 2013. "Weak and strong no-arbitrage conditions for continuous financial markets," Papers 1302.7192, arXiv.org, revised May 2014.
    12. Gabriel Frahm, 2013. "Pricing and Valuation under the Real-World Measure," Papers 1304.3824, arXiv.org, revised Jan 2016.
    13. Christa Cuchiero & Walter Schachermayer & Ting-Kam Leonard Wong, 2016. "Cover's universal portfolio, stochastic portfolio theory and the numeraire portfolio," Papers 1611.09631, arXiv.org.
    14. Martin HERDEGEN & Martin SCHWEIZER, 2016. "Economically Consistent Valuations and Put-Call Parity," Swiss Finance Institute Research Paper Series 16-02, Swiss Finance Institute.
    15. Koichiro Takaoka & Martin Schweizer, 2014. "A note on the condition of no unbounded profit with bounded risk," Finance and Stochastics, Springer, vol. 18(2), pages 393-405, April.
    16. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2012. "On arbitrages arising from honest times," Papers 1207.1759, arXiv.org, revised Jul 2013.
    17. Biagini, Francesca & Zhang, Yinglin, 2016. "Polynomial diffusion models for life insurance liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 114-129.
    18. Jacopo Mancin & Wolfgang J. Runggaldier, 2015. "On the Existence of Martingale Measures in Jump Diffusion Market Models," Papers 1511.08349, arXiv.org.
    19. repec:eee:insuma:v:76:y:2017:i:c:p:149-163 is not listed on IDEAS
    20. Claudio Fontana, 2013. "No-arbitrage conditions and absolutely continuous changes of measure," Papers 1312.4296, arXiv.org, revised Mar 2014.
    21. Tahir Choulli & Jun Deng & Junfeng Ma, 2015. "How non-arbitrage, viability and numéraire portfolio are related," Finance and Stochastics, Springer, vol. 19(4), pages 719-741, October.
    22. Francesca Biagini & Yinglin Zhang, 2016. "Polynomial Diffusion Models for Life Insurance Liabilities," Papers 1602.07910, arXiv.org, revised Sep 2016.
    23. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    24. Oleksii Mostovyi & Mihai S^irbu, 2017. "Sensitivity analysis of the utility maximization problem with respect to model perturbations," Papers 1705.08291, arXiv.org.

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