A quantitative and a dual version of the Halmos-Savage theorem with applications to mathematical finance

Klein, Irene; Schachermayer, Walter

doi:10.1214/aop/1039639366

Cited by 33 publications

(48 citation statements)

References 13 publications

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“…However, there is still the possibility of various approximations of an arbitrage profit by trading on the sequence of small markets, compare for example [9], [10], [13], [14]. The present note is focused on the condition no asymptotic free lunch (NAFL) which is the large financial market analogue of NFL, see [11].…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…These results can be understood as one-sided versions of the FTAP for large financial markets as contiguity corresponds to the property of absolute continuity of measures in the classical model. Criteria for the general situation (where M n is not a singleton) look more involved, see [13], [14] and in a different formulation [10]. E.g., in [10] it was shown that the above contiguity conditions can be replaced by (P n ) (Q n ) and (Q n ) (P n ), whereQ n (A) = sup Q∈M n Q(A), Q n (A) = inf Q∈M n Q(A).…”

Section: Introductionmentioning

confidence: 99%

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No asymptotic free lunch reviewed in the light of Orlicz spaces

Klein

2008

Lecture Notes in Mathematics

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Section: Definitions and Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

No asymptotic free lunch reviewed in the light of Orlicz spaces

Klein

2008

Lecture Notes in Mathematics

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“…In the latter case, the notion of arbitrage is replaced by the concept of "asymptotic arbitrage", which was introduced by Kabanov and Kramkov in [8] and [9] and further studied in [11] and [12] for frictionless markets and in [13] and [10] for the case of transaction costs.…”

Section: Introductionmentioning

confidence: 99%

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Cordero

Perez-Ostafe

2015

Int. J. Theor. Appl. Finan.

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Abstract. We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional BlackScholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these markets admit arbitrage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order o(1/ √ N ). Next, we characterize the asymptotic behavior of the smallest transaction costs λ, called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black-Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that λ converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/N H ), whereas for constant transaction costs, we prove that no such opportunity exists.

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“…Likewise, SAA condition is equivalent to the entire separation of (P n ) and ( Q n ). The results of [20], [21], [14], cited above and concerning the semimartingale market models, are reproved in Sect. 4.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic arbitrage and numéraire portfolios in large financial markets

Rokhlin

2007

Finance Stoch

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Abstract. This paper deals with the notion of a large financial market and the concepts of asymptotic arbitrage and strong asymptotic arbitrage (both of the first kind), introduced in [13], [14]. We show that the arbitrage properties of a large market are completely determined by the asymptotic behavior of the sequence of the numéraire portfolios, related to small markets. The obtained criteria can be expressed in terms of contiguity, entire separation and Hellinger integrals, provided these notions are extended to sub-probability measures. As examples we consider market models on finite probability spaces, semimartingale and diffusion models. Also a discrete-time infinite horizon market model with one log-normal stock is examined.

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A quantitative and a dual version of the Halmos-Savage theorem with applications to mathematical finance

Cited by 33 publications

References 13 publications

No asymptotic free lunch reviewed in the light of Orlicz spaces

No asymptotic free lunch reviewed in the light of Orlicz spaces

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Asymptotic arbitrage and numéraire portfolios in large financial markets

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