Strong asymptotic arbitrage in the large fractional binary market

Abstract

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.

Asymptotic arbitrage in fractional mixed markets

“…In this paper we aim to go a step further and study the sensitivity to arbitrage of the mixed fractional Black-Scholes model when the Brownian component asymptotically vanishes. In [3,4] it was argued that a good way of seeing the sensitivity to arbitrage of a market when one of its parameters converges to zero (or infinity), is to consider the family of markets indexed by the corresponding parameter and to use methods from large financial markets. To be precise, we study the asymptotic arbitrage opportunities in the sequence of mixed fractional Black-Scholes models when the scaling factor in front of the Brownian motion converges to zero.…”