Mathias Beiglböck

Model-Independent Bounds for Option Prices: A Mass Transport Approach

In this paper we investigate model-independent bounds for exotic options written on a risky asset. Based on arguments from the theory of Monge-Kantorovich 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 prove that there is no duality gap.

A Model‐Free Version of the Fundamental Theorem of Asset Pricing and the Super‐Replication Theorem

We propose a Fundamental Theorem of Asset Pricing and a Super-Replication Theorem in a model-independent 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 super-linearly growing payoff-function, e.g., a power option. This condition is not needed when sufficiently many vanilla options maturing at the horizon T are traded in the market.

Multiplicative structures in additively large sets

Previous research extending over a few decades has established that multiplicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an infinite sequence must contain all of the initial products from another infinite sequence. And, as a corollary of a result of Renling Jin, we show that if A and B have positive upper Banach density, then A + B contains all of the initial products from an infinite sequence. We also show that if a set has a complement which is not additively piecewise syndetic, then any translate of that set is both additively and multiplicatively large in several senses.We investigate whether a subset of N with bounded gaps--a syndetic set--must contain arbitrarily long geometric progressions. We believe that we establish that this is a significant open question.

Complete Duality for Martingale Optimal Transport on the Line

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage

We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem which states that the class of stochastic processes

Some new results in multiplicative and additive Ramsey theory

S

A short proof of the Doob–Meyer theorem

allowing for a useful integration theory consists precisely of those processes which can be written in the form

Root to Kellerer

S=M+A

Geometry of distribution-constrained optimal stopping problems

, where

CLONES FROM IDEALS

M