Abstract
We study, from the perspective of large financial markets, the asymptotic arbitrage 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 asymptotic arbitrage. For this reason, we follow a constructive approach to show first that a strong type of asymptotic arbitrage exists in the large market without transaction costs. 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 strategies, which leads to the desired result. Next, we introduce, in each small market, proportional transaction costs, and we construct, following a similar argument, a sequence of self-financing strategies providing a strong asymptotic arbitrage when the transaction costs converge fast enough to 0.
Highlights
Empirical studies of financial time series indicate that the statistical dependence of the logreturn increments decays slowly with the passage of time
A good example of a market model exhibiting this behaviour is the fractional Black–Scholes model, where the randomness of the risky asset is described by a fractional Brownian motion with Hurst parameter H > 1/2
The latter result contrasts with the fact that the fractional Black–Scholes model is free of arbitrage under arbitrarily small transaction costs
Summary
Empirical studies of financial time series indicate that the statistical dependence of the logreturn increments decays slowly with the passage of time (see [5,22]). In the present work, using more general self-financing trading strategies, we aim to construct, for an appropriate sequence of transaction costs, a strong AA (SAA), i.e., the possibility of getting arbitrarily rich with probability arbitrarily close to one while taking a vanishing risk. This problem can be viewed as a continuation of the study of AA initiated in [6], in the sense that our trading strategies are chosen beyond the 1-step setting of [6]. We end the paper with Appendices 1–3 providing some technical results and definitions used along the paper
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