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Non-Arbitrage up to Random Horizon for Semimartingale Models

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  • Anna Aksamit
  • Tahir Choulli
  • Jun Deng
  • Monique Jeanblanc
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    Abstract

    This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.

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    File URL: http://arxiv.org/pdf/1310.1142
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1310.1142.

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    Date of creation: Oct 2013
    Date of revision: Feb 2014
    Handle: RePEc:arx:papers:1310.1142

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    1. Ashkan Nikeghbali & Eckhard Platen, 2013. "A reading guide for last passage times with financial applications in view," Finance and Stochastics, Springer, Springer, vol. 17(3), pages 615-640, July.
    2. Freddy Delbaen & Walter Schachermayer, 1998. "A Simple Counterexample to Several Problems in the Theory of Asset Pricing," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 8(1), pages 1-11.
    3. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, Elsevier, vol. 20(3), pages 381-408, June.
    4. Tahir Choulli & Christophe Stricker & Jia Li, 2007. "Minimal Hellinger martingale measures of order q," Finance and Stochastics, Springer, Springer, vol. 11(3), pages 399-427, July.
    5. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
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