A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing

Abstract

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.

Keywords::

• arbitrage
• fundamental theorem of asset pricing
• Itô-process
• complete market
• equivalent local martingale measure
• martingale deflator

AMS Subject Classification::

• 60G44
• 60H05
• 91B70
• 91G10

Acknowledgements

The author is thankful to two anonymous referees for valuable comments that helped to improve the paper. Part of this research was carried out while the author was a post-doctoral researcher at INRIA Paris-Rocquencourt, MATHRISK project.

Notes

^1. The assumption that has P-a.s. full rank for every can be relaxed by only assuming that P-a.s. for every (which corresponds to exclude pathological arbitrage possibilities known as increasing profits; see [6], Proposition 4.3.4), with , and by replacing the matrix with the Moore–Penrose pseudoinverse of in (5).

^2. Models based on three-dimensional Bessel processes are classical examples of financial markets that allow for arbitrage opportunities, see for instance [2], Example 6.8 in [4] and the example on p. 59 of [6].

Additional information

Funding

This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme under [grant agreement PIEF-GA-2012-332345].