## Critical transaction costs and 1-step asymptotic arbitrage in fractional binary markets

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Author

Cordero, Fernando

Perez-Ostafe, Lavinia

Date

2014Type

- Working Paper

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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 Black-Scholes 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 (N) c , 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 (N) c converges to zero. However, the true behavior of λ (N) c is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that λ (N) c 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 exist Show more

Publication status

publishedJournal / series

arXivPages

Publisher

Cornell UniversityOrganisational unit

03845 - Teichmann, Josef
Notes

Submitted on 30 July 2014.More

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