Abstract
We investigate default-free bond markets and relax assumptions on the numéraire, which is typically chosen to be the bank account. Considering numéraires different from the bank account allows us to study bond markets where the bank account process is not a valid numéraire or does not exist at all. We argue that this feature is not the exception, but rather the rule in bond markets when starting with, e.g., terminal bonds as numéraires. Our setting are general càdlàg processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price processes 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 numéraire, 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. The theory is illustrated by several examples.
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Notes
- 1.
We write \(\cdot ^{-\kappa } f(\cdot )\) short for the function \(u \mapsto u^{-\kappa } f(s)\).
- 2.
See Theorem 1.3.15 in [25].
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Acknowledgements
The authors thank ETH Foundation for its support of this research project. The first and second author thank the Forschungsinstitut Mathematik at ETH Zürich for its generous hospitality.
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Klein, I., Schmidt, T., Teichmann, J. (2016). No Arbitrage Theory for Bond Markets. In: Kallsen, J., Papapantoleon, A. (eds) Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-45875-5_17
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