On low dimensional case in the fundamental asset pricing theorem with transaction costs

Abstract

Summary The well-known Harrison–Plisse theorem claims that in the classical discrete time model of the financial market with finite Ω there is no arbitrage iff there exists an equivalent martingale measure. The famous Dalang–Morton–Willinger theorem extends this result for an arbitrary Ω. Kabanov and Stricker [KS01] generalized the Harrison–Pliska theorem for the case of the market with proportional transaction costs. Nevertheless the corresponding extension of the Kabanov and Stricker result to the case of non-finite Ω fails, the corresponding counter-example with 4 assets was constructed by Schachermayer [S04]. The main result of this paper is that in the special case of 2 assets the Kabanov and Stricker theorem can be extended for an arbitrary Ω. This is quite a surprising result since the corresponding cone of hedgeable claims ÂT is not necessarily closed.

Arbitrage in markets with bid-ask spreads

“…Actually it was the first serious paper on arbitrage in markets with transaction costs. Note that the result of Grigoriev (2005) also strengthens the one of Jouini and Kallal (1995) in the case of one risky asset.…”

“…The paper improves some of the results of Jouini and Kallal (1995) and Grigoriev (2005). We give necessary and sufficient conditions for the absence of arbitrage (e.g.…”

“…Moreover, robust no-arbitrage cannot be replaced by strict no-arbitrage due to a counter-example presented by Schachermayer (2004). Further in this direction, a very interesting and surprising theorem was proved by Grigoriev (2005) who generalised the result of Kabanov and Stricker (2001b) to arbitrary Ω in the special case of two assets.…”

“…The question whether one can extend this result to markets with a money account and arbitrary d risky assets remains open. However, Grigoriev (2005) suggested that the answer to this question seems to be negative. One of the purposes of our paper is to analyse the general model of market with bid-ask prices and a money account in order to research this issue.…”

Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty

“…With proportional transaction costs, the set of martingale measures is replaced by the set of consistent price systems (CPS's) or strictly consistent price systems (SCPS's). Equivalence between no-arbitrage and existence of a CPS is established by Kabanov and Stricker [12] for finite probability space Ω, and by Grigoriev [8] when the dimension is two. Such equivalence in general does no hold for infinite spaces and higher dimensions (see Section 3 of Schachermayer [18] and page 128-129 of Kabanov and Safarian [11] for counter examples).…”