Asymptotic results with small transaction costs
H. Mete Soner (ETH Zurich, Switzerland)

Wednesday June 4, 14:30-15:00 | session 5.6 | Transaction Costs | room L

In the context of the multidimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. The asymptotic expansion is expressed in terms of a singular ergodic control problem, and our arguments are based on the theory of viscosity solutions, and the techniques of homogenization which leads to a system of corrector equations. We also provide some numerical results which illustrate the structure of the first order optimal controls.

Asymptotic arbitrage with small transaction costs
Irene Klein (University of Vienna, Austria)
Joint work with Emmanuel Lepinette and Lavinia Perez-Ostafe

Wednesday June 4, 15:00-15:30 | session 5.6 | Transaction Costs | room L

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 $j$.

Trading with small price impact
Johannes Muhle-Karbe (ETH Zürich, Switzerland)
Joint work with Ludovic Moreau and H. Mete Soner

Wednesday June 4, 15:30-16:00 | session 5.6 | Transaction Costs | room L

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fi xed transaction costs.