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How to compute (super) hedging costs in rather general financial market models with transaction costs in discrete-time ? Despite the huge literature on this topic, most of results are characterizations of the super-hedging prices while it remains difficult to deduce numerical procedure to estimate them. We establish here a dynamic programming principle and we prove that it is possible to implement it under some conditions on the conditional supports of the price and volume processes for a large class of market models including convex costs such as order books but also non convex costs, e.g. fixed cost models.

In this paper, we introduce two new types of conditional random set taking values in a Banach space: the conditional interior and the conditional closure. The conditional interior is a version of the conditional core, as introduced by A. Truffert and recently developed by Lepinette and Molchanov, and may be seen as a measurable version of the topological interior. The conditional closure is a generalization of the notion of conditional support of a random variable. These concepts are useful for applications in mathematical finance and conditional optimization.

We consider the consumption-investment optimization problem for the financial market model with constant proportional transaction rates and Levy price process dynamics. Contrarily to the recent work of De Valliere (Financ Stoch 20:705–740, 2016), portfolio process trajectories are only left and right limited. This allows us to identify an optimal ladlag strategy, e.g. in the two dimensional case, as it is possible to suitably rebalance the portfolio processes when they jump out of the no-trade region in the solvency cone.

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.

The classical discrete time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing for convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate positions are converted into a single fixed asset, our framework turns into the no good deals setting. However, in general, the possibilities of assessing the risk with respect to any asset or a basket of the assets lead to a decrease of superhedging prices and the no arbitrage conditions become stronger. The mathematical technique relies on results for unbounded and possibly non-closed random sets in Euclidean space., 31 page. Accepted for publication in Finance and Stochastics

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; The function solution to the functional scheme derived from the Binomial tree financial model with local volatility converges to thesolution of a diffusion equation of type ht(t, x)+ x2σ2(t,x) hxx(t, x) = 0 as the number of discrete dates n → ∞. Contrarily to classical numerical methods, in particular finite difference methods, the principle is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.

When dealing with non linear trading costs, e.g. fixed costs, the usual tools from convex analysis are inadequate to characterize an absence of arbitrage opportunity as the mathematical model is no more convex. An unified approach is to describe a financial market model by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a large class of financial markets with transaction costs including non linear trading costs. The natural question is to which extent the results of the classical arbitrage theory are still valid when the model is not convex, in particular what does the existence of an equivalent separating probability measure mean ? Our contribution is a first attempt to characterise the absence of arbitrage opportunity in non convex financial market models.

The seminal Modigliani--Miller theorem (1958) is a cornerstone of corporate finance theory. It provides conditions under which changes in a firm's capital structure do not affect its fundamental value. A recent controversial debate around the relevancy of the Modigliani--Miller theorem regarding the banking sector has been raised since the 2008 financial crisis. In this paper, we provide an overview of the theorem with recent developments when considering several extensions of the initial model. We reformulate the Modigliani--Miller theorem under a Markowitz perspective. Under this approach, we consider the case of implicit government guarantees offered to banks. Our main result shows that a bank does not satisfy the Modigliani--Miller theorem; precisely, banks will favor leverage instead of equity.

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

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., 33 pages 6 figures

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.

Financial market models defined by a liquidation value process generalize the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. The solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex, e.g. in presence of proportional transaction costs and fixed costs. Therefore, the classical duality principle based on the Hahn–Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. Using an alternative method based on the concepts of essential supremum and maximum, we provide a characterization of European and American contingent claim prices under the absence of arbitrage opportunity of the second kind.

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, Rasonyi and Lepinette (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 cadlag case for further studies.

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for a sequence of financial markets with small proportional transaction costs λ n on market n, in terms of contiguity properties of sequences of equivalent probability measures induced by λ n -consistent price systems. These results are analogous to the frictionless case; compare (Kabanov and Kramkov in Finance Stoch. 2:143–172, 1998; Klein and Schachermayer in Theory Probab. Appl. 41:927–934, 1996). Our setting is simple, each market n contains two assets. The proofs use quantitative versions of the Halmos–Savage theorem (see Klein and Schachermayer in Ann. Probab. 24:867–881, 1996) and a monotone convergence result for nonnegative local martingales. Moreover, we study examples of models which admit a strong asymptotic arbitrage without transaction costs, but with transaction costs λ n >0 on market n; there does not exist any form of asymptotic arbitrage. In one case, (λ n ) can even converge to 0, but not too fast.

Local volatility models are popular as they can be calibrated to the market of European options by the simple Dupire formula. For such a model, we propose a modified Leland method which allows to approximately replicate a European contingent claim when the market is under proportional transaction costs. The convergence of the scheme is shown by means of a new strategy of proof based on partial differential equations (PDEs) techniques allowing us to obtain appropriate Greek estimations.

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X endowed by a random preference relation to the space L0(X) of X-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.

Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 ( R d ) of d -dimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. Applications of the introduced notion to a hedging problem under transaction costs and set-valued dynamic risk measures are given.

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.

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay-off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper, we suggest a modification of Leland's strategy ensuring that the approximation error vanishes in the limit.

We study the criteria of robust absence of arbitrage opportunity (RNA2) of the second kind as initially introduced by Rasony M. in the case of a continuous-time and infinite dimensional financial market model with proportional transaction costs allowing for bond market modeling. Robust no arbitrage criteria seems to be unavoidable to assure closedness of the set of attainable claims. This allows us to relate the (RNA2) condition to the richness of the solvency cones (Kt)t∈[0,T] of all solvent portfolio positions, or equivalently, to the existence of (many) consistent price systems starting from an arbitrary selector of the interior of the positive dual of K under efficient friction.

The paper deals with definition of supremal sets in a rather general framework where deterministic and random preference relations (preorders) and partial orders are defined by continuous multi-utility representations. It gives a short survey of the approach developed in (J. Math. Econ. 14(4–5):554–563, 2011 [4]), (J. Math. Econ. 49(6):478–487, 2013 [5]) with some new results on maximal sets.

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.

When dealing with non linear trading costs, e.g. fixed costs, the usual tools from convex analysis are inadequate to characterize an absence of arbitrage opportunity as the mathematical model is no more convex. An unified approach is to describe a financial market model by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a large class of financial markets with transaction costs including non linear trading costs. The natural question is to which extent the results of the classical arbitrage theory are still valid when the model is not convex, in particular what does the existence of an equivalent separating probability measure mean? Our contribution is a first attempt to characterise the absence of arbitrage opportunity in non convex financial market models.

We present a general financial market model defined by a liquidation value process. This approach generalizes the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. This allows to consider financial market models where proportional transaction costs and fixed costs coexist. In this case, the solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex. Therefore, the usual duality principle based on the Hahn-Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. We propose an alternative method to price European or American contingent claims under absence of arbitrage opportunity of the second kind.

We study the modified Leland’s strategy defined in Lepinette (Math. Finance 22(4):741–752, 2012) for hedging portfolios in the presence of a constant proportional transaction costs coefficient. We prove a limit theorem for the deviation between the real portfolio and the payoff. We identify the rate of convergence and the associated limit distribution. This rate can be improved using the modified strategy and non periodic revision dates.

Local volatility models are popular because they can be simply calibrated to the market of European options. For such models, we propose a modified Leland method which allows us to approximately replicate a European contingent claim when the market is under proportional transaction costs. The convergence of our scheme is shown by means of a new strategy of proof based on PDEs techniques allowing us to obtain appropriate Greeks estimations.

The capital structure of banks has become the focus of an extended debate among policy-makers, regulators and academics. The seminal Modigliani-Miller (1958) theorem is seen as supportive of regulators' drive to require higher equity capital to banks. This raises the question on to what extent does Modigliani-Miller theorem hold for banks. This article brings a new insight of the Modigliani-Miller theorem by considering the implicit government guarantee offered to banks. Our main theorem shows that a bank can no longer be considered as a classical firm and will favor leverage instead of equity.

The post-crisis financial reforms address the need for systemic regulation, focused not only on individual banks but also on the whole financial system. The regulator principal objective is to set banks' capital requirements equal to international minimum standards in order to mimimise systemic risk. Indeed, Basel agreement is designed to guide a judgement about minimum universal levels of capital and remains mainly microprudential in its focus rather than being macroprudential. An alternative model to Basel framework is derived where systemic risk is taken into account in each bank's dynamic. This might be a new departure for prudential policy. It allows for the regulator to compute capital and risk requirements for controlling systemic risk. Moreover, bank regulation is considered in a two-scale level, either at the bank level or at the system-wide level. We test the adequacy of the model on a data set containing 19 banks of 5 major countries from 2005 to 2012. We compute the capital ratio threshold per year for each bank and each country and we rank them according to their level of fragility. Our results suggest to consider an alternative measure of systemic risk that requires minimal capital ratios that are bank-specific and time-varying.

The paper deals with definition of supremal sets in a rather general framework where deterministic and random preference relations (preorders) and partial orders are defined by continuous multi-utility representations. It gives a short survey of the approach developed in [4], [5] with some new results on maximal sets.

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 rewrite as a non linear function G of the volume of traded assets, with G'(0) > 0. For a stock with Black-Scholes mid-price dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with h^n trading dates. Up to a well chosen regularization hn of the payoff function, we first introduce the frictionless replicating portfolio of h^n(S^n_1), where Sn is a fictive stock with enlarged local volatility dynamics. In the market with frictions, a proper modification of this portfolio strategy provides a terminal wealth, which converges in probability to the claim of interest 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 non-vanishing (non-linear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, that we pass through using an innovative approach based on the Malliavin calculus representation of the Greeks.

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.

This paper proves the Fundamental Theorem of Asset Pricing with transaction costs, when bid and ask prices follow locally bounded cadlag (right-continuous, left-limited) processes. The Robust No Free Lunch with Vanishing Risk (RNFLVR) condition for simple strategies is equivalent to the existence of a strictly consistent price system (SCPS). This result relies on a new notion of admissibility, which reflects future liquidation opportunities. The (RNFLVR) condition implies that admissible strategies are predictable processes of finite variation.The appendix develops an extension of the familiar Stieltjes integral for cadlag integrands and finite-variation integrators, which is central to modeling transaction costs with discontinuous prices.

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.

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (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, we study different notions of time consistency and we give examples of vector-valued risk measure processes.