Effective Vs. Efficient Securities in Arbitrage-Free Markets with Bid-Ask Spreads : A Linear Programming Characterization

We consider a securities market with bid-ask spreads at any period, including liquidation. Although the minimum-cost super-replication problem is non-linear, we introduce an auxiliary problem that allows us to characterize no-arbitrage via linear programming techniques. Since no-arbitrage per se does not bound the bid-ask spread of a newly traded security, we introduce the notion of effective new security. We show that effectiveness restricts the no-arbitrage bid and ask prices of a new security to the interval defined by the minimum-cost problem. We discuss in details the cases in which the boundaries of this interval can be reached without violating no-arbitrage. We also compare effectiveness to efficiency as discussed in Jouini and Kallal (2001). We show that effectiveness is not sufficient for efficency, but is equivalent to the weaker notion of zero inefficiency cost