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We propose a Fundamental Theorem of Asset Pricing and a Super-Replication Theorem in a modelindependent framework. We prove these theorems in the setting of finite, discrete time and a market consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a… (More)

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a… (More)

- Mathias Beiglböck, Pierre Henry-Labordère, Friedrich Penkner
- Finance and Stochastics
- 2013

In this paper we investigate model-independent bounds for exotic options written on a risky asset using infinite-dimensional linear programming methods. Based on arguments from the theory of MongeKantorovich mass-transport we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we… (More)

We present a unified approach to Doob’s Lp maximal inequalities for 1 ≤ p < ∞. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob’s… (More)

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our… (More)

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain… (More)

Let H be a countable subgroup of the metrizable compact abelian group G and f : H → T = R/Z a (not necessarily continuous) character of H . Then there exists a sequence (χn) ∞ n=1 of (continuous) characters of G such that limn→∞ χn(α) = f(α) for all α ∈ H and (χn(α)) ∞ n=1 does not converge whenever α ∈ G \ H . If one drops the countability and… (More)

It is well known and not difficult to prove that if C ⊆ Z has positive upper Banach density, the set of differences C −C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic. Jin’s result follows trivially from the first… (More)

We show that the left-monotone martingale coupling is optimal for any given performance function satisfying the martingale version of the Spence-Mirrlees condition, without assuming additional structural conditions on the marginals. We also give a new interpretation of the left monotone coupling in terms of Skorokhod embedding which allows us to give a… (More)

András Biró and Vera Sós prove that for any subgroup G of T generated freely by finitely many generators there is a sequence A ⊆ N such that for all β ∈ T we have (‖.‖ denotes the distance to the nearest integer) β ∈ G ⇒ ∑ n∈A ‖nβ‖ < ∞, β / ∈ G ⇒ lim sup n∈A,n→∞ ‖nβ‖ > 0. We extend this result to arbitrary countable subgroups of T. We also show that not… (More)