Optimal consumption portfolio and no-arbitrage with nonproportional transaction costs

In this paper we consider a finite-state financial market with non-proportional transaction cost and bid-ask spreads. The transaction cost consists of two parts: a fixed cost and a proportional cost to the size of transaction. We show that the existence of an optimal consumption policy implies that the market has no strong arbitrage; the opposite, however, is not true, i.e., no strong arbitrage does not imply the existence of an optimal consumption policy. This is in sharp contrast with the case of proportional transaction cost and other cases reported in the literature, where no strong arbitrage is equivalent to the existence of an optimal consumption policy. We also study the relationship between weak arbitrage and strong arbitrage. Different from the market with proportional transaction cost, we find that these two forms of arbitrage are equivalent unless the fixed cost is zero. A necessary and sufficient condition for the existence of an optimal consumption policy is also obtained.