Asymptotic Arbitrage For Some Large Fractional Markets

A large financial market is represented by a sequence of "models" and, though each of them is arbitrage free, investors may obtain nonrisky profits in the limit. The study of asymptotic arbitrage in continuous time was initiated by Kabanov and Kramkov, followed by Klein and Schachermayer (for frictionless large markets) and by Lépinette, Klein and Perez-Ostafe (for markets with transaction costs). In this project, we advance the study of large financial markets, this time given by a sequence of markets approximating the fractional Black-Scholes model. As in Lépinette, Klein and Perez-Ostafe, each market model of the approximating sequence is subject to transaction costs. We consider two such approximations. The first one is given by a sequence of binary models as defined by Sottinen. Starting from the results of Cordero, Klein and Ostafe, we want to characterize, for each binary market model of the considered sequence, the smallest transaction cost starting from which one can construct a consistent price system (CPS). Knowing this, we aim to characterize the absence of asymptotic arbitrage by the rate of convergence to zero of the transaction costs. The second large financial market is given by a sequence of mixed fractional Black-Scholes models, which are again subject to transaction costs. We cover two cases depending on whether the Hurst parameter is bigger than or not. Our starting point in characterizing the absence of asymptotic arbitrage is given by the work of Cheridito.

The project deals with the study of arbitrage opportunities in the presence of transaction costs in a sequence of markets (called large financial market) approximating the socalled fractional Black Scholes (fBS) model. Two such approximations are examined: 1) a sequence of binary markets and 2) a sequence of mixed fBS models. When working with sequences of financial markets, the usual notion of arbitrage is replaced by the concept of asymptotic arbitrage (AA). We therefore aim to characterize the existence of AA under transaction costs in the two chosen large financial markets.1) We consider the sequence of binary markets, which was constructed by Sottinen and named fractional binary markets (fBMs). In a binary market without transaction costs, the absence of arbitrage can be written as a family of conditions on the nodes of a binary tree. We call arbitrage point a node in the binary tree which does not satisfy such a condition and arbitrage path a path that crosses at least one arbitrage point. We provide an indepth analysis of the asymptotic proportion of arbitrage points and of arbitrage paths in the fBMs. When transaction costs are taken into account, we use a constructive approach to determine the size of the transaction costs needed to eliminate the arbitrage from these models, first referring to each fixed fBM and then considering the whole sequence of fBMs as a large financial market. We construct, using only 1step trading strategies, an AA when the transaction costs converge fast enough to zero, whereas for constant transaction costs, we show that no such opportunity exists. Finally, using trading strategies beyond the 1step setting we construct a strong AA without transaction costs, but also when the transaction costs converge to zero fast enough.2) We consider a family of mixed processes given as the sum of a fractional Brownian motion and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyse the sequence of BlackScholes models driven by such mixed processes without transaction costs from the perspective of large financial markets. More precisely, we show the existence of a strong AA when the scaling factor converges to zero.