arXiv subjects
Publications and source records attributed to Laurence Carassus.
We consider an investor who, while maximizing his/her expected utility, also compares the outcome to a reference entity. We recall the notion of personal equilibrium and show that, in a multistep, generically incomplete financial market model such an equilibrium indeed exists, under appropriate technical assumptions.
We study a general robust utility maximization problem in a discrete-time frictionless market. The investor is assumed to have a possibly infinite, random, nonconcave, and nondecreasing utility function defined on the whole real line. She also faces model ambiguity on her beliefs about the market, which is modelled through a set of priors. We assume that the utility and the prices are projective functions of the path, while the graphs of the local priors are projective sets. Our other assumptions are stated on a prior-by-prior basis and correspond to generally accepted assumptions in the literature on markets without ambiguity. Under the set-theoretic axiom of Projective Determinacy (PD), our main result is the existence of an optimal investment strategy when the utility function is also upper-semicontinuous. We further provide several counterexamples justifying our assumptions.
We study projective functions. We prove that projective functions generalise lower and upper-semianalytic ones while being stable by composition and difference. We show that the class of projective functions is closed under sums, differences, products, finite suprema and infima, sections and compositions. Assuming the set-theoretical axiom of Projective Determinacy, we also prove measurable selection results, stability under integration, and the existence of $ε$-optimal selectors. Finally, we illustrate how these results are important in the context of model uncertainty.
Drawing on set theory, this paper contributes to a deeper understanding of the structural condition of mathematical finance under Knightian uncertainty. We adopt a projective framework in which all components of the model -- prices, priors and trading strategies -- are treated uniformly in terms of measurability. This contrasts with the quasi-sure setting of Bouchard and Nutz, in which prices are Borel-measurable and graphs of local priors are analytic sets, while strategies and stochastic kernels inherit only universal measurability. In our projective framework, we establish several characterizations of the robust no-arbitrage condition, already known in the quasi-sure setting, but under significantly more elegant and consistent assumptions. These characterisations have important applications, in particular, the existence of solutions to the robust utility maximization problem. To do this, we work within the classical Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), augmented by the axiom of Projective Determinacy (PD). The (PD) axiom, a well-established axiom of descriptive set theory, guarantees strong regularity properties for projective sets and projective functions.
When uncertainty is modelled by a set of non-dominated and non-compact probability measures, a notion of essential supremum for a family of real-valued functions is developed in terms of upper semi-analytic functions. We show how the properties postulated on the initial functions carry over to their quasi-sure essential supremum. We propose various applications to financial problems with frictions. We analyse super-replication and prove a bi-dual characterization of the super-hedging cost. We also study a weak no-arbitrage condition called Absence of Instantaneous Profit (AIP) under which prices are finite. This requires new results on the aggregation of quasi-sure statements.
We study a robust utility maximization problem in a general discrete-time frictionless market under quasi-sure no-arbitrage. The investor is assumed to have a random and concave utility function defined on the whole real-line. She also faces model ambiguity on her beliefs about the market, which is modeled through a set of priors. We prove the existence of an optimal investment strategy using only primal methods. For that we assume classical assumptions on the market and on the random utility function as asymptotic elasticity constraints. Most of our other assumptions are stated on a prior-by-prior basis and correspond to generally accepted assumptions in the literature on markets without ambiguity. We also propose a general setting including utility functions with benchmark for which our assumptions are easily checked.
In a context of illiquidity, the reservation price is a well-accepted alternative to the usual martingale approach which does not apply. However, this price is not available in closed form and requires numerical methods such as Monte Carlo or polynomial approximations to evaluate it. We show that these methods can be inaccurate and propose a deterministic decomposition of the reservation price using the Lambert function. This decomposition allows us to perform an improved Monte Carlo method, which we name Lambert Monte Carlo (LMC) and to give deterministic approximations of the reservation price and of the optimal strategies based on the Lambert function. We also give an answer to the problem of selecting a hedging asset that minimizes the reservation price and also the cash invested. Our theoretical results are illustrated by numerical simulations.
We investigate an expected utility maximization problem under model uncertainty in a one-period financial market. We capture model uncertainty by replacing the baseline model $\mathbb{P}$ with an adverse choice from a Wasserstein ball of radius $k$ around $\mathbb{P}$ in the space of probability measures and consider the corresponding Wasserstein distributionally robust optimization problem. We show that optimal solutions converge to a strategy with minimal norm when uncertainty is increasingly large, i.e. when the radius $k$ tends to infinity.
We consider a global market constituted by several submarkets, each with its own assets and numéraire. We provide theoretical foundations for the existence of equivalent martingale measures and results on superreplication prices which allows to take into account difference of features between submarkets.
In the frictionless discrete time financial market of Bouchard and Nutz (2015), we propose a full characterization of the quasi-sure super-replication price: as the supremum of the mono-prior super-replication prices, through an extreme prior and through martingale measures.
In a discrete time setting, we study the central problem of giving a fair price to some financial product. For several decades, the no-arbitrage conditions and the martingale measures have played a major role for solving this problem. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: The prices are expressed using Fenchel conjugate and bi-conjugate without using any no-arbitrage condition.The super-hedging problem resolution leads endogenously to a weak no-arbitrage condition called Absence of Instantaneous Profit (AIP) under which prices are finite. We study this condition in details, propose several characterizations and compare it to the no-arbitrage condition.
We study the most famous example of a large financial market: the Arbitrage Pricing Model, where investors can trade in a one-period setting with countably many assets admitting a factor structure. We consider the problem of maximising expected utility in this setting. Besides establishing the existence of optimizers under weaker assumptions than previous papers, we go on studying the relationship between optimal investments in finite market segments and those in the whole market. We show that certain natural (but nontrivial) continuity rules hold: maximal satisfaction, reservation prices and (convex combinations of) optimizers computed in small markets converge to their respective counterparts in the big market.
This paper formulates an utility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty. The investors preferences are described by strictly increasing concave random functions defined on the positive axis. We prove that under suitable conditions the multiple-priors utility indifference prices of a contingent claim converge to its multiple-priors superreplication price. We also revisit the notion of certainty equivalent for random utility functions and establish its relation with the absolute risk aversion.
We consider infinite dimensional optimization problems motivated by the financial model called Arbitrage Pricing Theory. Using probabilistic and functional analytic tools, we provide a dual characterization of the super-replication cost. Then, we show the existence of optimal strategies for investors maximizing their expected utility and the convergence of their reservation prices to the super-replication cost as their risk-aversion tends to infinity.
In a discrete time and multiple-priors setting, we propose a new characterisation of the condition of quasi-sure no-arbitrage which has become a standard assumption. This characterisation shows that it is indeed a well-chosen condition being equivalent to several previously used alternative notions of no-arbitrage and allowing the proof of important results in mathematical finance. We also revisit the so-called geometric and quantitative no-arbitrage conditions and explicit two important examples where all these concepts are illustrated.
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenchel conjugate and bi-conjugate. The super-hedging problem leads endogenously to a weak condition of NA called Absence of Immediate Profit (AIP). We propose several characterizations of AIP and study the relation with the classical notions of no-arbitrage. We also give some promising numerical illustrations.
In the frictionless discrete time financial market of Bouchard et al.(2015) we consider a trader who, due to regulatory requirements or internal risk management reasons, is required to hedge a claim $ξ$ in a risk-conservative way relative to a family of probability measures $\mathcal{P}$. We first describe the evolution of $π_t(ξ)$ - the superhedging price at time $t$ of the liability $ξ$ at maturity $T$ - via a dynamic programming principle and show that $π_t(ξ)$ can be seen as a concave envelope of $π_{t+1}(ξ)$ evaluated at today's prices. Then we consider an optimal investment problem for a trader who is rolling over her robust superhedge and phrase this as a robust maximisation problem, where the expected utility of inter-temporal consumption is optimised subject to a robust superhedging constraint. This utility maximisation is carrried out under a new family of measures $\mathcal{P}^u$, which no longer have to capture regulatory or institutional risk views but rather represent trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique.
This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under non-dominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.