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Strong asymptotic arbitrage in the large fractional binary market

机译：大型分数二元市场中的强渐近套利

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摘要

We study, from the perspective of large financial markets, the asymptotic arbitrage (AA) opportunities in a sequence of binary markets approximating the fractional Black-Scholes model. This approximating sequence was introduced by Sottinen and named fractional binary market. The large financial market under consideration does not satisfy the standard assumptions of the theory of AA. For this reason, we follow a constructive approach to show first that a strong AA (SAA) exists in the frictionless case. Indeed, with the help of an appropriate version of the law of large numbers and a stopping time procedure, we construct a sequence of self-financing trading strategies leading to the desired result. Next, we introduce, in each small market, proportional transaction costs, and we show that a slight modification of the previous trading strategies leads to a SAA when the transaction costs converge fast enough to 0.
机译：从大型金融市场的角度出发，我们研究了近似二元分数Black-Scholes模型的二元市场序列中的渐进套利（AA）机会。该近似序列由Sottinen引入，并命名为分数二进制市场。所考虑的大型金融市场不符合AA理论的标准假设。由于这个原因，我们遵循一种建设性的方法，首先证明在无摩擦情况下存在一个强大的AA（SAA）。确实，借助适当的大数定律和停止时间程序，我们构建了一系列自筹资金的交易策略，从而获得了预期的结果。接下来，我们在每个小市场中介绍成比例的交易成本，并且表明当交易成本收敛到足够快至0时，对先前交易策略的轻微修改会导致SAA。

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