英文标题：
《Critical transaction costs and 1-step asymptotic arbitrage in fractional
  binary markets》
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作者：
Fernando Cordero and Lavinia Perez-Ostafe
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最新提交年份：
2014
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英文摘要：
  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 $o(1/\\sqrt{N})$. Next, we characterize the asymptotic behavior of the smallest transaction costs $\\lambda_c^{(N)}$, 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 $\\lambda_c^{(N)}$ converges to zero. However, the true behavior of $\\lambda_c^{(N)}$ is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that $\\lambda_c^{(N)}$ 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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中文摘要：
我们研究了在一系列近似分数布莱克-斯科尔斯模型的二元市场中存在交易成本的套利机会。这个近似序列由Sottinen构造，命名为分数二元市场。由于在无摩擦的情况下，这些市场允许套利，我们的目标是确定从这些模型中消除套利所需的交易成本的大小。为了获得更深入的了解，我们首先只考虑一步交易策略，我们证明了当交易成本为$o（1/\\sqrt{N}）$时，套利机会会出现。接下来，我们刻画了最小交易成本$\\lambda_c^{（N）}$的渐近行为，称为“临界”交易成本，从该交易成本开始套利消失。由于分数Black-Scholes模型在任意小的交易成本下是无套利的，因此可以预期$\\lambda_c^{（N）}$收敛到零。然而，$\\lambda_c^{（N）}$的真实行为与这种直觉相反。更准确地说，在一系列新的交易策略的帮助下，我们证明了$\\lambda_ ^{（N）}$收敛到一。我们解释了这种明显的矛盾，并得出结论，将分数二元市场视为一个大型金融市场并研究其渐进套利机会是合适的。最后，当交易成本为$o（1/N^H）$时，我们在这个大市场上构造了一个$1$步渐近套利，而对于恒定的交易成本，我们证明了不存在这样的机会。
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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