Consistent price systems and face-lifting pricing under transaction costs

Abstract

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion.Using the constructed price systems, we show, under very general assumptions, the following "face-lifting" result: the asymptotic superreplication price of a European contingent claim g(ST ) equalŝ g(S0), whereĝ is the concave envelope of g and St is the price of the asset at time t. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.

Fragility of arbitrage and bubbles in local martingale diffusion models

“…In the jargon of the transaction costs literature, (S, Q) is an ε-consistent price system, see e.g. Guasoni, Rásonyi and Schachermayer (2008).…”

“…Theorem 1.2 of Guasoni, Rásonyi and Schachermayer (2008) states that, for a process satisfying (CFS) there exists Q ∼ P and a Q-martingaleS (w.r.t. F X t ) which is ε-close to S. Thus, M(t) :=S(t)E[ dQ dP |F X t ] is a P-martingale (w.r.t.…”

“…FX t ), in particular, R ∈ M (S). Note that the construction in Guasoni, Rásonyi and Schachermayer (2008) assures thatŠ(T ) ∈ L 2 (R). A fortiori,S(t) is an L 2 (R)-martingale, hence sup t∈[0,T ] |S t | ∈ L 2 (R).…”

Fractional Brownian motion satisfies two-way crossing

“…The stickiness property itself is a particular case of the conditional full support property, which says that future trajectories of the process after any stopping time have full support among the space of continuous trajectories (for the topology of locally uniform convergence). Conditional full support was shown to hold for fBm by [Guasoni et al, 2008].…”

Convex Duality with Transaction Costs

“…First step: In this step, we use the conditional full support property of P in a similar way to Guasoni, Rasonyi and Schachermayer [15].…”