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In this paper, a general framework is developed for continuoustime financial market models defined from simple strategies through conditional topologies that avoid stochastic calculus and do not necessitate semimartingale models. We then compare the usual no-arbitrage conditions of the literature, e.g. the usual no-arbitrage conditions NFL, NFLVR and NUPBR and the recent AIP condition. With appropriate pseudo-distance topologies, we show that they hold in continuous time if and only if they hold in discrete time. Moreover, the super-hedging prices in continuous time coincide with the discrete-time super-hedging prices, even without any no-arbitrage condition.

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., arXiv admin note: text overlap with arXiv:1807.04612

The NA condition is one of the pillars supporting the classical theory of financial mathematics. We revisit this condition for financial market models where a dynamic risk-measure defined on [Formula: see text] is fixed to characterize the family of acceptable wealths that play the role of nonnegative financial positions. We provide in this setting a new version of the fundamental theorem of asset pricing and we deduce a dual characterization of the super-hedging prices (called risk-hedging prices) of a European option. Moreover, we show that the set of all risk-hedging prices is closed under NA. At last, we provide a dual representation of the risk-measure on [Formula: see text] under some conditions.

International audience; We provide an equivalent characterisation of absence of arbitrage opportunity NA for the Bid and Ask financial market model analog to the Dalang--Morton--Willinger theorem formulated for discrete-time financial market models without friction. This result completes the Grigoriev theorem for conic models in the two dimensional case by showing that the set of all terminal liquidation values is closed under NA..

International audience; Random sets and random preorders naturally appear in financial market modelling with transaction costs. In this paper , we introduce and study a concept of essential minimum of a family of vector-valued random variables, i.e. the set of all minimal elements with respect to some random preorder. We provide some conditions under which the essential minimum is not empty and we present two applications in optimisation to Mathematical Finance and Economics.

International audience; In this paper, we revisit the discrete-time partial hedging problem of contingent claims with respect to a dynamic risk-measure defined by its acceptance sets. A natural and sufficient weak no-arbitrage condition is studied to characterize the minimal risk-hedging prices. The method relies only on conditional optimization techniques. In particular, we do not need robust representation of the risk-measure and we do not suppose the existence of a risk-neutral probability measure. Numerical experiments illustrate the efficiency of the method.

We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore, we introduce the generalised conditional expectation of random closed sets and show that it is sandwiched between the conditional core and the conditional convex hull. The results rely on measurability properties of not necessarily closed random sets considered from the point of view of the families of their selections. Furthermore, we develop analytical tools suitable to handle random convex (not necessarily compact) sets in Banach spaces; these tools are based on considering support functions as functions of random arguments. The paper is motivated by applications to assessing multivariate risks in mathematical finance., 31 page, this work is a substantial extension of a part from arXiv:1605.07884

International audience; In this paper, we propose an approximation method based on Picard iterations deduced from the Doléans–Dade exponential formula. Our method allows to approximate trajectories of Markov processes in a large class, e.g. solutions to non-Lipchitz SDEs. An application to the pricing of Asian-style contingent claims in the CEV model is presented and compared to other methods of the literature.

We consider a fractional version of the Heston model where the two standard Brownian motions are replaced by two fractional Brownian motions with Hurst parameter H ∈ (1/2, 1). We show that the stochastic differential equation admits a unique positive solution by adapting and generalizing some results of Y. Hu, D. Nualart and X. Song on singular equations driven by rough paths. Moreover, we show that the fractional version of the variance, which is a version of the fractional Cox-Ingersoll-Ross model, is still a mean-reverting process.

The classical discrete time model of transaction costs relies on the assumption that the increments of the feasible portfolio process belong to the solvency set at each step. We extend this setting by assuming that any such increment belongs to the sum of an element of the solvency set and the family of acceptable positions, e.g. with respect to a dynamic risk measure.We formulate several no risk arbitrage conditions and explore connections between them. If the acceptance sets consist of non-negative random vectors, that is the underlying dynamic risk measure is the conditional essential infimum, we extend many classical no arbitrage conditions in markets with transaction costs and provide their natural geometric interpretation. The mathematical technique relies on results for unbounded and possibly non-closed random sets in the Euclidean space.

This paper is dedicated to the replication of a convex contingent claim h(S 1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h n of the payoff function, we first introduce the frictionless replicating portfolio of $h^{n}(S^{n}_{1})$ , where S n is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in $\mathbb{L}^{2}$ to the claim h(S 1) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, R\'asonyi and L\'epinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general c\`adl\`ag case for further studies., Comment: 41 pages

Introduced by Artzner, Delbaen, Eber and Heath (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini, Meddeb and Touzi (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results and examples of vector valued risk measure processes.

International audience; We consider two quasi-linear initial-value Cauchy problems on Rd: a parabolic system and an hyperbolic one. They both have a rst order non-linearity of the form (t; x; u) ru, a forcing term h(t; x; u) and an initial condition u0 2 L1(Rd) \ C1(Rd), where (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t; x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but recursive parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.