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When roll-overs do not qualify as num\'eraire: bond markets beyond short rate paradigms

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  • Irene Klein
  • Thorsten Schmidt
  • Josef Teichmann
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    Abstract

    We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not a valid num\'eraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bonds as num\'eraires. Our setting are general c\`adl\`ag processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price process to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as num\'eraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds. Additionally we provide an alternative definition of the concept of a num\'eraire, leading to a possibly interesting connection to bubbles. If we can construct a bank account process through roll-overs, we can relate the impossibility of taking the bank account as num\'eraire to liquidity effects. Here we enter endogenously the arena of multiple yield curves. The theory is illustrated by several examples.

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    Paper provided by arXiv.org in its series Papers with number 1310.0032.

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    Date of creation: Sep 2013
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    Handle: RePEc:arx:papers:1310.0032

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    1. Ivar Ekeland & Erik Taflin, 2003. "A theory of bond portfolios," Papers math/0301278, arXiv.org, revised May 2005.
    2. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 16(1), pages 131-151.
    3. Alan Brace & Dariusz G´┐Żatarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(2), pages 127-155.
    4. Ekeland, Ivar & Taflin, Erik, 2005. "A theory of bond portfolios," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/6041, Paris Dauphine University.
    5. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, Springer, vol. 9(4), pages 477-492, October.
    6. W. Schachermayer, 1994. "Martingale Measures For Discrete-Time Processes With Infinite Horizon," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 4(1), pages 25-55.
    7. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, Springer, vol. 2(2), pages 143-172.
    8. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, Springer, vol. 1(2), pages 149-177, November.
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