Abstract:
For a large financial market (which is a sequence of usual, “small” financial
markets), we introduce and study a concept of no asymptotic arbitrage (of the
first kind), which is invariant under discounting. We give two dual
characterizations of this property in terms of (1) martingale-like properties
for each small market plus (2) a contiguity property, along the sequence of
small markets, of suitably chosen “generalized martingale measures.” Our
results extend the work of Rokhlin, Klein, and Schachermayer and Kabanov and
Kramkov to a discounting-invariant framework. We also show how a market on
[0,∞) can be viewed as a large financial market and how no asymptotic
arbitrage, both classic and in our new sense, then relates to no-arbitrage
properties directly on [0,∞).
Keywords:
large financial markets, asymptotic arbitrage, discounting, no asymptotic arbitrage (NAA), no unbounded profit with bounded risk (NUPBR), asymptotic strong share maximality, dynamic share viability, asymptotic dynamic share viability, tradable discounter.
Citation:
D. A. Balint, M. Schweizer, “Large financial markets, discounting, and no asymptotic arbitrage”, Teor. Veroyatnost. i Primenen., 65:2 (2020), 237–280; Theory Probab. Appl., 65:2 (2020), 191–223
\Bibitem{BalSch20}
\by D.~A.~Balint, M.~Schweizer
\paper Large financial markets, discounting, and no asymptotic arbitrage
\jour Teor. Veroyatnost. i Primenen.
\yr 2020
\vol 65
\issue 2
\pages 237--280
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\crossref{https://doi.org/10.4213/tvp5353}
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\transl
\jour Theory Probab. Appl.
\yr 2020
\vol 65
\issue 2
\pages 191--223
\crossref{https://doi.org/10.1137/S0040585X97T98991X}
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Linking options:
https://www.mathnet.ru/eng/tvp5353
https://doi.org/10.4213/tvp5353
https://www.mathnet.ru/eng/tvp/v65/i2/p237
This publication is cited in the following 4 articles:
Oleksii Mostovyi, Pietro Siorpaes, “Pricing of contingent claims in large markets”, Finance Stoch, 2024
E. Platen, S. Tappe, “No arbitrage and multiplicative special semimartingales”, Adv. Appl. Probab., 55:3 (2023), 1033–1074
C. Fontana, S. Pavarana, W. J. Runggaldier, “A stochastic control perspective on term structure models with roll-over risk”, Finance Stoch., 27:4 (2023), 903–932
D. Á. Bálint, M. Schweizer, “Making no-arbitrage discounting-invariant: A new FTAP version beyond NFLVR and NUPBR”, Frontiers of Mathematical Finance, 1:2 (2022), 249