Sticky Continuous Processes have Consistent Price Systems

Abstract

Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni [5].

Sticky processes, local and true martingales

“…In Guasoni et al [13], it was shown that processes with CFS can be approximated arbitrarily closely under the supremum norm by semi-martingales that admit equivalent martingale measures. In the subsequent paper Bender et al [2], the same result was obtained for continuous path processes that are merely sticky. In these papers, such approximation was possible because the stochastic processes were assumed to be continuous.…”

“…This definition is clearly equivalent to Definition 2.2 of Guasoni [10] where κ is assumed to be any deterministic number. In Lemma 3.1 of Bender et al [2] it was shown that, for processes with continuous paths, stickiness is equivalent to P( sup…”

“…Here In the recent paper Bender et al [2], it was shown that if a continuous path process is sticky then for any ε > 0 there exists a semi-martingaleS that admits an equivalent martingale measure such that…”

“…Property (NA2) was established in Guasoni and Rásonyi [12] for the class of continuous processes S satisfying the CFS-O property, see Remark 2.3. Using arguments of Bender et al [2], one could establish (NA2) for continuous sticky processes S. The essential novelty of Proposition 6.3 thus lies in allowing jumps for S.…”

No arbitrage and lead–lag relationships

“…Finally, we remark that Bender et al [3] have shown that the stickiness is sufficient for the existence of CPS as long as S is continuous: Proposition 2.6 ([3], Theorem 2.1). Suppose that S is continuous.…”