Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads

Abstract

In a general, finite-dimensional securities market model with bidask spreads, we characterize absence of arbitrage opportunities both by linear programming and in terms of martingales. We first show that absence of arbitrage is equivalent to the existence of solutions to the linear programming problems that compute the minimum costs of super-replicating the feasible future cashflows. Via duality, we show that absence of arbitrage is also equivalent to the existence of underlying frictionless (UF) state-prices. We then show how to transform the UF state-prices into state-price densities, and use them to characterize absence of arbitrage opportunities in terms of existence of a securities market with zero bid-ask spreads whose price process lies inside the bid-ask spread. Finally, we argue that our results extend those of Naik (1995) and Jouini and Kallal (1995) to the case of intermediate dividend payments and positive bid-ask spreads on all assets.

American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

“…, T . This result also follows from Kabanov and Stricker [11], Ortu [18], Kabanov, Rásonyi and Stricker [12,13], Tokarz [26], and Schachermayer [25].…”

Options under Proportional Transaction Costs: An Algorithmic Approach to Pricing and Hedging

“…The following result, obtained by Jouini and Kallal [12], who used a slightly different notion of arbitrage, referred to as 'free lunch' in their work, is also valid under the above definition of an arbitrage opportunity, as shown in Tokarz [34]. See also Ortu [25]. Theorem 2.1 (Jouini and Kallal [12]) There is no arbitrage opportunity if and only if P = ∅.…”

Effective securities in arbitrage-free markets with bid–ask spreads at liquidation: a linear programming characterization

“…5 Hereafter x y ðtÞX0 means Pðx y ðtÞX0Þ ¼ 1; x y ðtÞ40 means x y ðtÞX0 and Pðx y ðtÞ40Þ40; x y ðtÞb0 means Pðx y ðtÞ40Þ ¼ 1: 6 See in particular Naik (1995) and Ortu (2001). 7 Recall that L is the total number of nodes, and s T is the number of terminal nodes.…”

“…taking one strategy for each cell of this partition, we use the sum of their cashflow to super-replicate any given future payoff at the minimum cost. Our auxiliary program extends the linear programming characterization of no-arbitrage of Naik (1995) and Ortu (2001) to the case of bid-ask spreads at liquidation. Moreover, by linear duality, we are able to characterize the cases in which strict super-replication is costminimizing.…”

“…Still in a binomial model but without bid-ask spreads at liquidation, Edirisinghe et al (1993) show how to reformulate the super-replication problem as a linear programming one. Naik (1995) and Ortu (2001) analyze the general event-tree framework without bid-ask spreads at liquidation and use the linearized super-replication problem and its dual to provide alternative characterizations of no-arbitrage.…”