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A NEW PERSPECTIVE ON THE FUNDAMENTAL THEOREM OF ASSET PRICING FOR LARGE FINANCIAL MARKETS

Cuchiero C

Klein I

Teichmann J

Summary

In the context of large financial markets we formulate the notion of no asymptotic free lunch with vanishing risk (NAFLVR), under which we can prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (even uncountably) infinite number of assets, as it is, for instance, the case in bond markets. We work in the general setting of admissible portfolio wealth processes as laid down by Kabanov in [Statistics and Control of Stochastic Processes (Moscow, 1995/1996), World Sci. Publ., River Edge, NJ, 1997, pp. 191-203] under a substantially relaxed concatenation property and adapt the FTAP proof variant obtained in [C. Cuchiero and J. Teichmann, Finance Stoch., 19 (2015), pp. 743-761] for the classical small market situation to large financial markets. In the case of countably many assets, our setting includes the large financial market model considered by De Donno, Guasoni, and Pratelli [Stochastic Process. Appl., 115 (2005), pp. 2006-2022] and its abstract integration theory. The notion of (NAFLVR) turns out to be an economically meaningful "no arbitrage" condition (in particular, not involving weak-*-closures), and (NAFLVR) is equivalent to the existence of a separating measure. Furthermore, we show by means of a counterexample that the existence of an equivalent separating measure does not lead to an equivalent sigma-martingale measure, even in a countable large financial market situation.

Keywords

fundamental theorem of asset pricing; large financial markets; Emery topology; (NFLVR) condition; (P-UT) property

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