Thin times and random times' decomposition

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration $\mathbb F$. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all $\mathbb F$-stopping times. Then, for a given random time $\tau$, we introduce ${\mathbb F}^\tau$, the smallest right-continuous filtration containing $\mathbb F$ and making $\tau$ a stopping time, and we show that, for a thin time $\tau$, each $\mathbb F$-martingale is an ${\mathbb F}^\tau$-semimartingale, i.e., the hypothesis $({\mathcal H}^\prime)$ for $(\mathbb F, {\mathbb F}^\tau)$ holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well

Non-Arbitrage Under Additional Information for Thin Semimartingale Models

This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) when the stock price processes are quasi-left-continuous (do not jump on predictable stopping times). Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and the progressive enlargement with a random time. For this flow of information, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. This also generalizes \cite{choulli/deng} to the case when the jump times are not ordered in anyway. Furthermore, for the current context, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration.Comment: This paper develops the part of thin and single jump processes mentioned in our earlier version: "Non-arbitrage up to random horizon and after honest times for semimartingale models", Available at: arXiv:1310.1142v1. arXiv admin note: text overlap with arXiv:1404.041

Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglb\"ock et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.Comment: 29 page

Non-Arbitrage under a Class of Honest Times

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge- proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-tau" local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.Comment: 31 pages. arXiv admin note: text overlap with arXiv:1310.114

Hubbard Hamiltonian in the dimer representation. Large U limit

We formulate the Hubbard model for the simple cubic lattice in the representation of interacting dimers applying the exact solution of the dimer problem. By eliminating from the considerations unoccupied dimer energy levels in the large U limit (it is the only assumption) we analytically derive the Hubbard Hamiltonian for the dimer (analogous to the well-known t-J model), as well as, the Hubbard Hamiltonian for the crystal as a whole by means of the projection technique. Using this approach we can better visualize the complexity of the model, so deeply hidden in its original form. The resulting Hamiltonian is a mixture of many multiple ferromagnetic, antiferromagnetic and more exotic interactions competing one with another. The interplay between different competitive interactions has a decisive influence on the resulting thermodynamic properties of the model, depending on temperature, model parameters and assumed average number of electrons per lattice site. A simplified form of the derived Hamiltonian can be obtained using additionally Taylor expansion with respect to $x=\frac{t}{U}$ (t-hopping integral between nearest neighbours, U-Coulomb repulsion). As an example, we present the expansion including all terms proportional to t and to $\frac{t^2}U$ and we reproduce the exact form of the Hubbard Hamiltonian in the limit $U\to \infty$. The nonperturbative approach, presented in this paper, can, in principle, be applied to clusters of any size, as well as, to another types of model Hamiltonians.Comment: 26 pages, 1 figure, LaTeX; added reference

Arbitrages in a Progressive Enlargement Setting

This paper completes the analysis of Choulli et al. Non-Arbitrage up to Random Horizons and after Honest Times for Semimartingale Models and contains two principal contributions. The first contribution consists in providing and analysing many practical examples of market models that admit classical arbitrages while they preserve the No Unbounded Profit with Bounded Risk (NUPBR hereafter) under random horizon and when an honest time is incorporated for particular cases of models. For these markets, we calculate explicitly the arbitrage opportunities. The second contribution lies in providing simple proofs for the stability of the No Unbounded Profit with Bounded Risk under random horizon and after honest time satisfying additional important condition for particular cases of models

Rohrs, Bob

Bob Rohrs was drafted into the Army in November 1965. For 1 year, he served in the 5th Battalion of the 71st Artillery in Vietnam. Date of interview: April 12, 2007Length of interview: 83:4