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M6 - On Minimal Market Models and Minimal Martingale Measures

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.

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File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp280.pdf
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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 280.

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Length: 21 pages
Date of creation: 01 Jun 2010
Date of revision:
Publication status: Published as: Hulley, H. and Schweizer, M., 2010, "M6 - On Minimal Market Models and Minimal Martingale Measures", In: Carl Chiarella and Alexander Novikov (eds) Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, 35-51.
Handle: RePEc:uts:rpaper:280
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  1. Eckhard Platen, 2003. "Modeling the Volatility and Expected Value of a Diversified World Index," Research Paper Series 103, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Eckhard Platen, 2004. "Diversified Portfolios with Jumps in a Benchmark Framework," Research Paper Series 129, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. Michel Chatelain & Christophe Stricker, 1994. "ON COMPONENTWISE and VECTOR STOCHASTIC INTEGRATION," Mathematical Finance, Wiley Blackwell, vol. 4(1), pages 57-65.
  4. 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.
  5. 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.
  6. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
  7. Eva Strasser, 2005. "Characterization of arbitrage-free markets," Papers math/0503473, arXiv.org.
  8. 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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