Martingale Optimal Transport Problems Research Articles

A multi-marginal c-convex duality theorem for martingale optimal transport

A convex duality result for martingale optimal transport problems with two marginals was established in Beiglböck et al. (2013). In this paper we provide a generalization of this result to the multi-period setting.

Stability of the weak martingale optimal transport problem

Stability of the weak martingale optimal transport problem

Continuity of the martingale optimal transport problem on the real line

Continuity of the martingale optimal transport problem on the real line

A Deep Learning Approach to Data-Driven Model-Free Pricing and to Martingale Optimal Transport

We introduce a novel and highly tractable supervised learning approach based on neural networks that can be applied for the computation of model-free price bounds of, potentially high-dimensional, financial derivatives and for the determination of optimal hedging strategies attaining these bounds. In particular, our methodology allows to train a single neural network offline and then to use it online for the fast determination of model-free price bounds of a whole class of financial derivatives with current market data. We show the applicability of this approach and highlight its accuracy in several examples involving real market data. Further, we show how a neural network can be trained to solve martingale optimal transport problems involving fixed marginal distributions instead of financial market data.

Instability of martingale optimal transport in dimension d≥2

Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in general frameworks such as the one of Polish spaces. However, for the martingale transport problem several works based on different strategies established stability results for Ronly. We show that the restriction to dimension d=1 is not accidental by presenting a sequence of marginal distributions on R2 for which the martingale optimal transport problem is neither stable w.r.t. the value nor the set of minimizers. Our construction adapts to any dimension d≥2. For d≥2 it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in d=1.

A construction of the left-curtain coupling

In a martingale optimal transport (MOT) problem mass distributed according to the law μ is transported to the law ν in such a way that the martingale property is respected. Beiglböck and Juillet (On a problem of optimal transport under marginal martingale constraints, Annals of Probability, 44(1):42-106, 2016) introduced a solution to the MOT problem which they baptised the left-curtain coupling. The left-curtain coupling has been widely studied and shown to have many applications, including to martingale inequalities and the model-independent pricing of American options. Beiglböck and Juillet proved existence and uniqueness, proved optimality for a family of cost functions, and proved that when μ is a continuous distribution, mass at x is mapped to one of at most two points, giving lower and upper functions. Henry-Labordère and Touzi (An explicit martingale version of Brenier’s theorem, Finance and Stochastics, 20:635-668, 2016) showed that the left-curtain coupling is optimal for an extended family of cost functions and gave a construction of the upper and lower functions under an assumption that μ and ν are continuous, together with further simplifying assumptions of a technical nature. In this article we construct these upper and lower functions in the general case of arbitrary centred measures in convex order, and thereby give a complete construction of the left-curtain coupling. In the case where μ has atoms these upper and lower functions are to be interpreted in the sense of a lifted martingale.

One Dimensional Martingale Rearrangement Couplings

We are interested in martingale rearrangement couplings. As introduced by Wiesel in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Frechet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in Jourdain and Margheriti [Electr. J. Probab. (2020)] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fréchet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglböck and Juillet and which involve the uniform distribution on [0,1] in addition to the two marginals. We last discuss the stability in adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginal distributions.

Shadow martingales – a stochastic mass transport approach to the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(\nu^\alpha)_{\alpha \in [0,1]}$ with $\nu^\alpha(\mathbb{R})=\alpha$ that are submeasures of $\mu _0$, called a parametrization of $\mu_0$. Then, for any $\alpha$ we define an evolution $(\eta^\alpha_t)_{t\geq 0}$ of the measure $\nu^\alpha=\eta^\alpha_0$ across our peacock by setting $\eta^\alpha_t$ equal to the obstructed shadow of $\nu^\alpha$ in $(\mu _s)_{s \in [0,t]}$. We identify conditions on the parametrization $(\nu^\alpha)_{\alpha\in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(\nu_{\text{lc}}^\alpha)_{\alpha \in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

On the stability of the martingale optimal transport problem: A set-valued map approach

Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer (2019) and Wiesel (2019). We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck et al. (2021), we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w.r.t. the marginals.

Dispersion-Constrained Martingale Schrödinger Problems and the Exact Joint S&P 500/VIX Smile Calibration Puzzle

We solve for the first time (*) a longstanding puzzle of quantitative finance that has often been described as the Holy Grail of volatility modeling: build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures, and VIX options. So far the best attempts, which used parametric continuous-time (jump-)diffusion models on the SPX, only produced approximate fits. We use a very different, nonparametric and discrete-time approach. Given a VIX future maturity T1, we consider the set P of all joint probability measures on the SPX at T1 the VIX at T1, and the SPX at T2 = T1 + 30 days which are perfectly calibrated to the full SPX smiles at T1 and T2, and the full VIX smile at T1, and which also satisfy the martingality constraint on the SPX as well as the requirement that the VIX is the implied volatility of the 30-day log-contract on the SPX. We first consider robust hedging in this setting. By casting the superreplication problem as what we call a dispersion-constrained martingale optimal transport problem, we establish a strong duality theorem and, as a result, prove that the absence of joint SPX/VIX arbitrage is equivalent to the set P of jointly calibrating models being nonempty. Should they arise, joint arbitrages are identified using classical linear programming. In the absence of joint arbitrage, we then provide a solution to the joint calibration puzzle by solving what we call a dispersion-constrained martingale Schrodinger problem: we choose a reference measure and build the unique jointly calibrating model that minimizes the relative entropy. We establish several dual versions of the problem, one of which has a natural financial interpretation in terms of exponential utility indifference pricing, and prove absence of duality gaps. The minimum entropy jointly calibrating model is explicit in terms of what we call the dual Schrodinger portfolio, i.e., the maximizer of the dual problems, should it exist. We numerically compute this Schrodinger portfolio using an extension of the Sinkhorn algorithm, in the spirit of De March and Henry-Labordere (2019). Our numerical experiments show that the algorithm performs very well in both low and high volatility regimes. Along the way, we provide new variants, as well as a new proof, of strong duality theorems for the classical Schrodinger problem and for a mixed Schrodinger-Monge-Kantorovich problem (also known as entropic optimal transport problem) that has recently attracted a lot of attention in the optimal transport community, which are interesting in themselves. Our methodology applies not only to the VIX, but also to any index computed as a function of the price of an option on some underlying asset. (*) A first, much shorter version of this article was published in Risk in April 2020. This version includes new theorems, proofs, analysis, and numerical tests. In particular it develops a theory of dispersion-constrained martingale Schrodinger problems and proves strong duality results for them.

Martingale transport with homogeneous stock movements

We study a variant of the martingale optimal transport problem in a multi-period setting to derive robust price bounds on a financial derivative. On top of marginal and martingale constraints, we introduce a time-homogeneity assumption, which restricts the variability of the forward-looking transitions of the martingale across time. We provide a dual formulation in terms of superhedging and discuss relaxations of the time-homogeneity assumption by adding market frictions. In financial terms, the introduced time-homogeneity corresponds to a time-consistency condition for call prices, given the state of the stock. The time homogeneity assumption leads to improved price bounds since market data from many time points can be incorporated effectively. The approach is illustrated with two numerical examples.

Transport Plans with Domain Constraints

This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain its Kantorovich duality. A corollary of this result is a monotonicity principle which gives a geometric way of identifying the optimizer.

Computational methods for martingale optimal transport problems

We develop computational methods for solving the martingale optimal transport (MOT) problem—a version of the classical optimal transport with an additional martingale constraint on the transport’s dynamics. We prove that a general, multi-step multi-dimensional, MOT problem can be approximated through a sequence of linear programming (LP) problems which result from a discretization of the marginal distributions combined with an appropriate relaxation of the martingale condition. Further, we establish two generic approaches for discretising probability distributions, suitable respectively for the cases when we can compute integrals against these distributions or when we can sample from them. These render our main result applicable and lead to an implementable numerical scheme for solving MOT problems. Finally, specialising to the one-step model on real line, we provide an estimate of the convergence rate which, to the best of our knowledge, is the first of its kind in the literature.

A Benamou–Brenier formulation of martingale optimal transport

We introduce a Benamou–Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker–Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated.

Dual attainment for the martingale transport problem

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [Ann. Probab. 45 (2017) 3038–3074] established such existence for weak (quasi-sure) duality, [Finance Stoch. 17 (2013) 477–501] 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 $c(x,\cdot)\in C^{2-\varepsilon}$, $\varepsilon>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.

SAMPLING OF ONE-DIMENSIONAL PROBABILITY MEASURES IN THE CONVEX ORDER AND COMPUTATION OF ROBUST OPTION PRICE BOUNDS

For [Formula: see text] and [Formula: see text] two probability measures on the real line such that [Formula: see text] is smaller than [Formula: see text] in the convex order, this property is in general not preserved at the level of the empirical measures [Formula: see text] and [Formula: see text], where [Formula: see text] (resp., [Formula: see text]) are independent and identically distributed according to [Formula: see text] (resp., [Formula: see text]). We investigate modifications of [Formula: see text] (resp., [Formula: see text]) smaller than [Formula: see text] (resp., greater than [Formula: see text]) in the convex order and weakly converging to [Formula: see text] (resp., [Formula: see text]) as [Formula: see text]. According to  Kertz & Rösler(1992) , the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For [Formula: see text] and [Formula: see text] in this set, this enables us to define a probability measure [Formula: see text] (resp., [Formula: see text]) greater than [Formula: see text] (resp., smaller than [Formula: see text]) in the convex order. We give efficient algorithms permitting to compute [Formula: see text] and [Formula: see text] (and therefore [Formula: see text] and [Formula: see text]) when [Formula: see text] and [Formula: see text] have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

Existence of Transport Plans with Domain Constraints

Let $\Omega$ be one of $\X^{N 1},C[0,1],D[0,1]$: product of Polish spaces, space of continuous functions from $[0,1]$ to $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$, respectively. We first consider the existence of a probability measure $P$ on $\Omega$ such that $P$ has the given marginals $\alpha$ and $\beta$ and its disintegration $P_x$ must be in some fixed $\Gamma(x) \subset \kP(\Omega)$, where $\kP(\Omega)$ is the set of probability measures on $\Omega$. The main application we have in mind is the martingale optimal transport problem when the martingales are assumed to have bounded volatility/quadratic variation. We show that such probability measure exists if and only if the $\alpha$ average of the so-called $G$-expectation of bounded continuous functions with respect to the measures in $\Gamma$ is less than their $\beta$ average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer.

Martingale Optimal Transport with Stopping

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in [A. M. G. Cox and S. Kallblad, SIAM J. Control Optim., 55 (2017), pp. 3409--3436] allows us to obtain an equivalent infinite dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

Sampling of Probability Measures in the Convex Order and Approximation of Martingale Optimal Transport Problems

In this paper, motivated by the approximation of Martingale Optimal Transport problems, we are interested in sampling methods preserving the convex order for two probability measures µ and ν on ℝd, with ν dominating µ. When (Xi)1≤i≤I (resp. (Yj)1≤j≤J ) are independent and identically distributed according µ (resp. ν), in general µI = 1/1 ΣIi=1δXi and νJ = 1/J ΣJj=1δYj are not rankable for the convex order. We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. We first consider the one dimensional case d = 1, where, according to Kertz and Rosler, the set of probability measures with a finite first order moment is a lattice for the increasing and the decreasing convex orders. Given µ and ν in this set, we define µ ∨ ν (resp. µ ∧ v) as the supremum (resp. infimum) of µ and ν for the decreasing convex order when ∫ℝ xµ(dx) ≤ ∫ℝ xν(dx) and for the increasing convex order otherwise. This way, µ∨v (resp. µ∧ν) is greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ ) when µ and ν are convex combinations of Dirac masses. In general dimension, when µ and ν have finite moments of order ϱ≥ 1, we define the projection µ ⋏ϱν (resp. µ ⋎ϱν) of µ (resp. ν) on the set of probability measures dominated by ν (resp. larger than µ) in the convex order for the Wasserstein distance with index ϱ. When ϱ= 2, µI ⋏2 νJ can be computed efficiently by solving a quadratic optimization problem with linear constraints. It turns out that, in dimension d = 1, the projections do not depend on ϱ and their quantile functions are explicit in terms of those of µ and ν, which leads to efficient algorithms for convex combinations of Dirac masses. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate Martingale Optimal Transport problems.

Sampling of probability measures in the convex order by Wasserstein projection

Motivated by the approximation of Martingale Optimal Transport problems, we study sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When $(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$ (resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convex order. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than $\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly converging to $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertz and R\osler (1992), the set of probability measures with a finite first order moment is a lattice for the increasing and the decreasing convex orders. From this result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greater than $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficient algorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and $\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$ and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection $\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) of $\mu$ (resp. $\nu$) on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$) in the convex order for the Wasserstein distance with index $\rho$. When $\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving a quadratic optimization problem with linear constraints. It turns out that, in dimension 1, the projections do not depend on $\rho$ and their quantile functions are explicit, which leads to efficient algorithms for convex combinations of Dirac masses. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate Martingale Optimal Transport problems.