Mathias Beiglböck

Bio: Mathias Beiglböck is an academic researcher from University of Vienna. The author has contributed to research in topics: Martingale (probability theory) & Mathematical finance. The author has an hindex of 26, co-authored 99 publications receiving 2559 citations. Previous affiliations of Mathias Beiglböck include Vaal University of Technology & Vienna University of Technology.

Model-independent bounds for option prices—a mass transport approach

Abstract: 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 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.

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

Abstract: 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

Abstract: 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 superlinearly 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.

On a problem of optimal transport under marginal martingale constraints

Abstract: The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb{E} [c(X_{1},X_{2})]$ by varying the joint distribution $(X_{1},X_{2})$ where the marginal distributions of the random variables $X_{1}$ and $X_{2}$ are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that $(X_{i})_{i=1,2}$ is a martingale, that is, $\mathbb{E} [X_{2}|X_{1}]=X_{1}$. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this “monotone martingale” is supported by the graphs of two functions $T_{1},T_{2}:\mathbb{R} \to\mathbb{R}$.

On a problem of optimal transport under marginal martingale constraints

Abstract: The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb {E}[c(X_1,X_2)]$ by varying the joint distribution $(X_1,X_2)$ where the marginal distributions of the random variables $X_1$ and $X_2$ are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that $(X_i)_{i=1,2}$ is a martingale, that is, $\mathbb {E}[X_2|X_1]=X_1$. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this "monotone martingale" is supported by the graphs of two functions $T_1,T_2:\mathbb {R}\to \mathbb {R}$.

Computational Optimal Transport

Abstract: Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

Abstract: Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Model-independent bounds for option prices—a mass transport approach

Abstract: 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 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.