arXiv subjects

Jan Obłój

Publications and source records attributed to Jan Obłój.

16 recordsLinked to original sources

arXiv2026

Graph Causal Optimal Transport and Wasserstein Distances

We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.

math.PR↗

arXiv2025

Robust Pricing and Hedging of American Options in Continuous Time

We consider the robust pricing and hedging of American options in a continuous time setting. We assume asset prices are continuous semimartingales, but we allow for general model uncertainty specification via adapted closed convex constraints on the volatility. We prove the robust pricing-hedging duality. When European options with given prices are available for static trading, we show that duality holds against richer models where these options are traded dynamically. Our proofs rely on probabilistic treatment of randomised stopping times and suitable measure decoupling, and on optimal transport duality. In addition, similarly to the approach of Aksamit et al. (2019) in discrete time, we identify American options with European options on an enlarged space.

q-fin.MF↗

arXiv2021

Local times and Tanaka--Meyer formulae for càdlàg paths

Three concepts of local times for deterministic c{à}dl{à}g paths are developed and the corresponding pathwise Tanaka--Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not depend on the choice of the sequence of grids. This is a new result also for c{à}dl{à}g semimartingales, which may be related to previous results of Nicole El~Karoui and Marc Lemieux.

math.PR↗

arXiv2018

Two explicit Skorokhod embeddings for simple symmetric random walk

Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centered distribution $μ$ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma-Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of $μ$.

math.PR↗

arXiv2018

Robust framework for quantifying the value of information in pricing and hedging

We investigate asymmetry of information in the context of robust approach to pricing and hedging of financial derivatives. We consider two agents, one who only observes the stock prices and another with some additional information, and investigate when the pricing--hedging duality for the former extends to the latter. We introduce a general framework to express the superhedging and market model prices for an informed agent. Our key insight is that an informed agent can be seen as a regular agent who can restrict her attention to a certain subset of possible paths. We use results of Hou & Obłój on robust approach with beliefs to establish the pricing--hedging duality for an informed agent. Our results cover number of scenarios, including information arriving before trading starts, arriving after static position in European options is formed but before dynamic trading starts or arriving at some point before the maturity. For the latter we show that the superhedging value satisfies a suitable dynamic programming principle, which is of independent interest.

q-fin.MF↗

arXiv2018

Pointwise Arbitrage Pricing Theory in Discrete Time

We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain an abstract (pointwise) Fundamental Theorem of Asset Pricing and Pricing--Hedging Duality. Our results are general and in particular include so-called model independent results of Acciao et al. (2016), Burzoni et al. (2016) as well as seminal results of Dalang et al. (1990) in a classical probabilistic approach. Our analysis is scenario--based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

q-fin.MF↗

arXiv2017

Dual attainment for the martingale transport problem

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] showed existence for the natural stronger pointwise duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in C^{2-ε}$, $ε>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.

math.PR↗

arXiv2017

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 Beiglböck et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.

math.OC↗

arXiv2017

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time-space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.

math.PR↗

arXiv2017

Structure of martingale transports in finite dimensions

We study the structure of martingale transports in finite dimensions. We consider the family $\mathcal{M}(μ,ν) $ of martingale measures on $\mathbb{R}^N \times \mathbb{R}^N$ with given marginals $μ,ν$, and construct a family of relatively open convex sets $\{C_x:x\in \mathbb{R}^N \}$, which forms a partition of $\mathbb{R}^N$, and such that any martingale transport in $\mathcal{M}(μ,ν) $ sends mass from $x$ to within $\overline{C_x}$, $μ(dx)$--a.e. Our results extend the analogous one-dimensional results of M. Beiglböck and N. Juillet (2016) and M. Beiglböck, M. Nutz, and N. Touzi (2015). We conjecture that the decomposition is canonical and minimal in the sense that it allows to characterise the martingale polar sets, i.e. the sets which have zero mass under all measures in $\mathcal{M}(μ,ν)$, and offers the martingale analogue of the characterisation of transport polar sets proved in M. Beiglböck, M. Goldstern, G. Maresch, and W. Schachermayer (2009).

math.PR↗

arXiv2017

Pathwise Stochastic Calculus with Local Times

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the Itô-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

math.PR↗

arXiv2016

The maximum maximum of a martingale with given $n$ marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Obłój and Spoida [An iterated Azéma-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labordère and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

math.PR↗

arXiv2015

On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

We study the joint laws of a continuous, uniformly integrable martingale, its maximum, and its minimum. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Azéma & Yor, Perkins, Jacka and Cox & Obłój, this allows us to completely characterise the upper and lower bounds on all possible exit/no-exit probabilities, subject to a given terminal law of the martingale. In addition, we determine some further properties of these bounds, considered as functions of the maximum and minimum.

math.PR↗

arXiv2014

An Iterated Azéma-Yor Type Embedding for Finitely Many Marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $μ_1,...,μ_n$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalisation of the Azema and Yor (1979) solution. In particular, we recover the stopping boundaries obtained by Brown et al. (2001) and Madan and Yor (2002). Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the n stopping times. This is used in Henry-Labordere et al. (2013) to show that the construction maximises the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

math.PR↗

arXiv2012

On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation

We study the class of Azéma-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Azéma-Yor process. The proofs exploit group structure of the set of Azéma-Yor processes, indexed by functions, which we introduce. We investigate in detail Azéma-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between average value at risk, drawdown function, Hardy-Littlewood transform and its inverse. In particular, we construct Azéma-Yor martingales with a given terminal law and this allows us to rediscover the Azéma-Yor solution to the Skorokhod embedding problem. Finally, we characterize Azéma-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.

math.PR↗

arXiv2008

Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping

We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_τ\mathbb{E}[F(L_τ)-\int _0^τβ(B_s) ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

math.PR↗