Beatrice Acciaio

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.

Dynamic risk measures

This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.

Optimal risk sharing with non-monotone monetary functionals

We consider the problem of sharing pooled risks among n economic agents endowed with non-necessarily monotone monetary functionals. In this framework, results of characterization and existence of optimal solutions are easily obtained as extensions from the convex risk measures setting. Moreover, the introduction of the best monotone approximation of non-monotone functionals allows us to compare the original problem with the one which involves only ad hoc monotone criteria. The explicit calculation of optimal risk sharing rules is provided for particular cases, when agents are endowed with well-known preference relations.

Existence of Radial Solutions for Quasilinear Elliptic Equations with Singular Nonlinearities

Abstract We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f(u) = 0 in ℝn, n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and grows subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptions are made on f.

Are law-invariant risk functions concave on distributions?

Abstract While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay between convexity of law-invariant risk functions on random variables and convexity/concavity of their counterparts on distributions. We show that, given a law-invariant convex risk measure, on the level of distributions, if at all, concavity holds true. In particular, this is always the case under the additional assumption of comonotonicity.

The space of outcomes of semi-static trading strategies need not be closed

Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a fixed date T

CHARACTERIZATION OF MAX-CONTINUOUS LOCAL MARTINGALES VANISHING AT INFINITY

T

On the Minimization of Area Among Chord-Convex Sets

are available for static trading. This causes problems for optimal investment, and stands in sharp contrast to the purely dynamic case classically considered in mathematical finance.

On the Lower Arbitrage Bound of American Contingent Claims

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [NY06] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [Kar14], and is obtained by means of similar tools.

Modelling the default risk in large credit portfolios

In this paper we study the problem of minimizing the area for the chord-convex sets of given size, that is, the sets for which each bisecting chord is a segment of length at least 2. This problem has been already studied and solved in the framework of convex sets, though nothing has been said in the non-convex case. We introduce here the relevant concepts and show some first properties.