Effective securities in arbitrage-free markets with bid-ask spreads at liquidation: 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. We introduce the notion of effective new security and 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 detail the cases in which the boundaries of this interval can be reached without violating no-arbitrage.