اوراق بهادار موثر در بازارهای آزاد آربیتراژ با اسپرد قیمت خرید و فروش در انحلال شرکت: خصوصیات برنامه ریزی خطی

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.

The valuation of securities via super-replication in the presence of market frictions and its interplay with no-arbitrage is one of the most active research areas in finance theory. The topic has been analyzed both in discrete-time, starting from Bensaid et al. (1992) and Jouini and Kallal (1995), and in continuous-time, dating back to Cvitanic and Karatzas (1993).1 This paper follows the discrete-time, event-tree approach and offers two main contributions. First, we show how to employ linear programming techniques to characterize no-arbitrage in markets with bid–ask spreads. With respect to the existing literature the contribution is that our linear programming approach works also with bid–ask spreads at liquidation. Second, we supply a linear programming-based proof of the fact that no-arbitrage per se imposes only an upper bound on the bid and a lower bound on the ask price of a new security. We then introduce the notion of effective new security and show that this notion characterizes the new securities whose bid–ask spreads are bounded. In a seminal paper, Bensaid et al. (1992) incorporate bid–ask spreads in the standard binomial option pricing model and solve the super-replication problem via dynamic programming. 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. In this paper we address the general event-tree framework with bid–ask spreads also at liquidation. The presence of bid–ask spreads at liquidation arises in many practical applications. A European call option, for instance, is typically settled at maturity either with delivery of the underlying, or by cash, or at the discretion of the short position. In a world without bid–ask spreads at maturity, these different types of settlement are payoff-equivalent. In actual markets, however, bid–ask spreads are present also at maturity and different settlement provisions produce different payoff profiles. Bid–ask spreads at liquidation introduce a non-linearity in the otherwise linear super-replication problem. Indeed, investors typically aggregate their long and short positions with the same broker. This implies that rather than the cumulative long and short positions separately, what matters at the moment of final liquidation are the net positions held in each security. With bid–ask spreads at liquidation, this makes the terminal payoff, and hence the super-replication problem, non-linear in the intertemporal trading strategies. To deal with this non-linearity, we construct an auxiliary linear program with the same value function as the original problem, such that any solution to the super-replication problem is a linear transformation of a solution to the auxiliary linear program. To construct this auxiliary program we first partition the set of feasible trading strategies according to the sign of the net positions at liquidation. Then, 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 cost-minimizing. In particular, we show that this occurs if and only if the minimum cost of super-replication strictly exceeds the value assigned to a given claim by any strictly positive linear pricing rule compatible with no-arbitrage. Based on our linear programming approach we discuss the interplay between no-arbitrage and the notion of effectiveness of a new security introduced in the market. We first employ linear programming duality to formally derive a fact pointed out by Jouini and Kallal (1999), namely that no-arbitrage survives the introduction of a new security if and only if its bid price does not exceed the minimum cost to cover a short position and its ask price is greater than the maximum that can be borrowed against a liability equal to the payoff from a long position. 2 We then introduce the notion of effectiveness to characterize the new securities whose bid–ask spread will be bounded. In words, a newly traded security is long (short)-effective when it is optimal to take a long (short) position in the new security to super-replicate some future cashflow at the minimum cost. Intuitively, our notion of effectiveness conveys the idea that the new security improves the super-hedging capabilities of the investors. Our approach allows us to interpret the typical interval bounds for the bid and ask prices of a new security as generated by the interaction of no-arbitrage and effectiveness. In particular, effectiveness forces the no-arbitrage ask price of a new security to be smaller than the minimum cost incurred to super-replicate the payoff from a long position, and the no-arbitrage bid price to be larger than the maximum that can be borrowed against a liability not exceeding the one generated by a short position. Our results identify explicitly the cases in which reaching the boundaries of the interval gives rise to arbitrage opportunities. These cases occur in fact when the initial bid–ask spread vanishes, the price of the new security collapses to one of the extremes of the interval and strict super-replication is cost-optimal. The rest of the paper is structured as follows. In the next section we introduce the basic notation and definitions. In Section 3 we supply our linear programming characterization of no-arbitrage with bid–ask spreads at liquidation. In Section 4 we introduce the notion of effective new security and compare the pure no-arbitrage bounds with those that must hold for an effective new security. We also compare our notion of effective security to Jouini and Kallal's (2001) notion of efficient trading strategy. Section 5 concludes by addressing the possibility of extending our results to more general frameworks. All proofs are in Appendix A.

To conclude, we briefly discuss two questions regarding the possibility of extending our results to more general frameworks. The first questions concerns the extent to which our linear programming characterization of no-arbitrage with bid–ask spreads at liquidation carries on to an infinite set of states. In particular, the relevant case is the one with an infinite number of states at the last but one date T-1T-1. Although it is still possible to formally write our auxiliary linear program LP[m]LP[m] by constructing the matrices MkMk and GkGk as in the proof of Theorem 1, in this case our procedure would generate a continuum of matrices 19MkMk and GkGk, a fact with two consequences on our results. First, the only part of the proof of Theorem 1 that works as it is that π(m)π(m) exceeds the value of LP[m]LP[m]. The proof of the reverse inequality, which relies on the possibility of mapping linearly any feasible strategy of LP[m]LP[m] into a strategy with the same cost and feasible for P[m]P[m], may fail to hold with an infinite set of states since the linear map may be undefined. Second, in an infinite dimensional setting the equality between the value function of the primal and the dual problem may fail without additional assumptions, that is, our linear programming approach may exhibit a ‘duality gap’. 20 These and other related problems arising in an infinite dimensional setting constitute an interesting argument for future research. A second question concerns the possibility that the new security be traded at every date, and not just at the initial date, as we assume in Section 4. In this case, the open question consists in supplying the relations that must hold between the entire sequence {cA(t),cB(t)}{cA(t),cB(t)} of ask and bid prices of the new security and suitably defined super-replication problems in the original market for no-arbitrage to survive in the extended market. These conditions are easily determined in the special case of zero bid–ask spreads on all securities. This is due to a basic property of frictionless event-tree securities markets: no-arbitrage holds on the entire tree if and only if it holds on all its one-period sub-trees (see e.g., Dalang et al., 1990). Therefore, in the frictionless case, the extended market is arbitrage-free if and only if in all one-period sub-trees the beginning-of-period price of the new security is bounded above by the minimum cost to super-replicate its end-of-period price, and below by the maximum that can be borrowed against a liability not exceeding its end-of-period price.21 With bid–ask spreads, these conditions translate into requiring that in all one-period sub-trees (a) the beginning-of-period bid price of the new security be bounded above by the minimum cost to super-replicate the end-of-period ask price, and (b) the beginning-of-period ask price be bounded below by the maximum that can be borrowed against a liability not exceeding the end-of-period bid price. 22 Unfortunately, these conditions, although necessary, fail to be sufficient for no-arbitrage to hold in the extended market. The reason for this lies in a fundamental discontinuity introduced by the presence of bid–ask spreads, namely, that arbitrage opportunities may very well be banned from all one-period sub-trees, and still survive in the form of multi-period arbitrage strategies (see e.g., Pham and Touzi, 1999). In the presence of bid–ask spreads, therefore, if we want to obtain bounds on the entire sequence of ask and bid prices of the new security based on one-period super-replication problems of the original market, these problems need to be parametrized in a way that makes them intertemporally connected in a much more stringent way than in the frictionless case. We believe that this subject constitute another interesting topic for future research.