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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
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.

Suggested Citation

  • Irene Klein & Thorsten Schmidt & Josef Teichmann, 2013. "When roll-overs do not qualify as num\'eraire: bond markets beyond short rate paradigms," Papers 1310.0032, arXiv.org.
    • Handle: RePEc:arx:papers:1310.0032
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    References listed on IDEAS

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    Cited by:

    1. Harms, Philipp & Stefanovits, David, 2019. "Affine representations of fractional processes with applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1185-1228.
    2. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2016. "A general HJM framework for multiple yield curve modelling," Finance and Stochastics, Springer, vol. 20(2), pages 267-320, April.
    3. Christa Cuchiero & Irene Klein & Josef Teichmann, 2014. "A new perspective on the fundamental theorem of asset pricing for large financial markets," Papers 1412.7562, arXiv.org, revised Oct 2023.
    4. Zdzisław Brzeźniak & Tayfun Kok, 2018. "Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations," Finance and Stochastics, Springer, vol. 22(4), pages 959-1006, October.
    5. Claudio Fontana & Thorsten Schmidt, 2016. "General dynamic term structures under default risk," Papers 1603.03198, arXiv.org, revised Nov 2017.
    6. Frank Gehmlich & Thorsten Schmidt, 2014. "Dynamic Defaultable Term Structure Modelling beyond the Intensity Paradigm," Papers 1411.4851, arXiv.org, revised Jul 2015.
    7. Thomas Krabichler & Josef Teichmann, 2020. "A constraint-based notion of illiquidity," Papers 2004.12394, arXiv.org.

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