Abstracts

We extend the theory of large financial markets to a sequence of financial markets with small proportional transaction costs $c(n)$ on market $n$. Our setting is simple: each market $n$ contains two assets, a risky one and a risk-free one. We give a characterization of the absence of asymptotic arbitrage of the first and of the second kind and of strong asymptotic arbitrage with transaction costs $c(n)$ on market $n$ in terms of contiguity properties of sequences of equivalent probability measures induced by $c(n)$-consistent price systems. Our results are in complete analogy with the respective results of the frictionless case, see Kabanov \& Kramkov (1998): 'Asymptotic arbitrage in large financial markets' and Klein \& Schachermayer (1996a): 'Asymptotic arbitrage in non-complete large financial markets'. The main tools of the proofs are the quantitative versions of the Halmos-Savage Theorem of Klein \& Schachermayer (1996b), and a result on monotone convergence of nonnegative local martingales. Moreover, we study examples showing that the introduction of transaction costs influences the existence of asymptotic arbitrage opportunities: we present models which admit a strong asymptotic arbitrage without transaction costs, but if we impose appropriate transaction costs $c(n)>0$ on each market n the arbitrage disappears, i.e., there does not exist any form of asymptotic arbitrage. In one example we can even choose transaction costs $c(n)$ converging to 0, such that still there does not exist any form of asymptotic arbitrage as long as the convergence to 0 is not too fast (we give a lower bound). The results could be generalized to a sequence of markets with $d(n)$ assets in market $n$ in the following way: it is only possible to vary the portfolio between asset $i$ and $j$ by going from asset $i$ to the risk-free asset first and then from there to asset $j$.