HENRY LABORDERE Pierre

In this paper, we introduce a primal-dual algorithm for solving (martingale) optimal transportation problems, with cost functions satisfying the twist condition, close to the one that has been used recently for training generative adversarial networks. As some additional applications, we consider anomaly detection and automatic generation of financial data.

Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas, options on realized variance or VIX options. These models differ strongly from the well-known local stochastic volatility models, in particular the instantaneous volatility-of-volatility of the associated naked SVMs is not modified, once calibrated to market instruments. They can be interpreted as a martingale version of the Schrodinger bridge. The numerical calibration is performed using a dynamic-like version of the Sinkhorn algorithm. We finally highlight a striking relation with Dyson non-colliding Brownian motions.

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We consider the classical problem of building an arbitrage-free implied volatility surface from bid-ask quotes. We design a fast numerical procedure, for which we prove the convergence, based on the Sinkhorn algorithm that has been recently used to solve efficiently (martingale) optimal transport problems.

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In this paper, we introduce a primal-dual algorithm for solving (martingale) optimal transportation problems, with cost functions satisfying the twist condition, close to the one that has been used recently for training generative adversarial networks. As some additional applications, we consider anomaly detection and automatic generation of financial data.

We consider the classical problem of building an arbitrage-free implied volatility surface from bid-ask quotes. We design a fast numerical procedure, for which we prove the convergence, based on the Sinkhorn algorithm that has been recently used to solve efficiently (martingale) optimal transport problems.

This thesis is devoted to the theoretical and numerical study of two nonlinear problems in the McKean sense in finance. In the first part, we address the problem of calibrating a model with local and stochastic volatility to take into account the prices of European vanilla options observed on the market. This problem results in the study of a nonlinear stochastic differential equation (SDE) in the McKean sense because of the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor with respect to the SDE solution. We obtain the existence of the process in the particular case where the stochastic volatility factor is a jump process with a finite number of states. We also obtain the weak convergence at order 1 of the time discretization of the nonlinear DHS in the McKean sense for general stochastic volatility factors. In the industry, the calibration is efficiently performed using a regularization of the conditional expectation by a Nadaraya-Watson type kernel estimator, as proposed by Guyon and Henry-Labordère in [JGPHL]. We also propose a half-time numerical scheme and study the associated particle system that we compare to the algorithm proposed by [JGPHL]. In the second part of the thesis, we focus on a problem of contract valuation with margin calls, a problem that appeared with the application of new regulations since the financial crisis of 2008. This problem can be modeled by an anticipatory stochastic differential equation (SDE) with dependence on the law of the solution in the generator. We show that this equation is well-posed and propose an approximation of its solution using standard linear SRDEs when the liquidation time of the option in case of default is small. Finally, we show that the computation of the solutions of these standard EDSRs can be improved using the multilevel Monte Carlo method introduced by Giles in [G].

We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplied scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrodinger equations.

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In this thesis we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally propose methods of numerical approximation.We first prove the existence of irreducible components intrinsic to martingale transports between two given measures, and the canonicity of these components. We then prove a duality result for the optimal martingale transport in any dimension, the point by point duality is no longer true but a form of quasi-safe duality is proved. This duality allows us to prove the possibility of decomposing the quasi-safe optimal transport into a series of subproblems of point by point optimal transports on each irreducible component. We finally use this duality to prove a martingale monotonicity principle, analogous to the famous monotonicity principle of classical optimal transport. We then study the local structure of optimal transports, deduced from differential considerations. We obtain a characterization of this structure using real algebraic geometry tools. We deduce the structure of martingale optimal transports in the case of Euclidean norm power costs, thus solving a conjecture dating back to 2015. Finally, we compare existing numerical methods and propose a new method that is shown to be more efficient and to deal with an intrinsic problem of the martingale constraint that is the convex order defect. We also give techniques to handle the numerical problems in practice.

We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [23], Watanabe [27] and McKean [18], by allowing for polynomial nonlinearity in the pair (u, Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to " small maturity " or " small nonlinearity " of the PDE. Our main ingredient is the automatic differentiation technique as in [15], based on the Malliavin integration by parts, which allows to account for the nonlin-earities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

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We propose an unbiased Monte-Carlo estimator for E[g(X t 1 , · · · , X tn)], where X is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Bismu-Elworthy-Li formula from Malliavin calculus, as exploited by Fournié et al. [14] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [3, Section 6.1] as an application of the parametrix method. MSC2010. Primary 65C05, 60J60. secondary 60J85, 35K10.

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This thesis is devoted to the study of the strong convergence properties of the Ninomiya and Victoir scheme. The authors of this scheme propose to approximate the solution of a stochastic differential equation (SDE), denoted $X$, by solving $d+1$ ordinary differential equations (ODE) on each time step, where $d$ is the dimension of the Brownian motion. The aim of this study is to analyze the use of this scheme in a multi-step Monte-Carlo method. Indeed, the optimal complexity of this method is directed by the order of convergence towards $0$ of the variance between the schemes used on the coarse and on the fine grid. This order of convergence is itself related to the strong order of convergence between the two schemes. We then show in chapter $2$, that the strong order of the Ninomiya-Victor scheme, denoted $X^{NV,eta}$ and of time step $T/N$, is $1/2$. Recently, Giles and Szpruch proposed a multi-step Monte Carlo estimator realizing $Oleft(epsilon^{-2}right)$ complexity using a modified Milstein scheme. In the same spirit, we propose a modified Ninomiya-Victoir scheme that can be coupled at high order $1$ with the Giles and Szpruch scheme at the last level of a multi-step Monte Carlo method. This idea is inspired by Debrabant and Rossler. These authors suggest using a high low order scheme at the finest discretization level. Since the optimal number of discretization levels of a multi-step Monte Carlo method is directed by the low error of the scheme used on the fine grid of the last discretization level, this technique allows to accelerate the convergence of the multi-step Monte Carlo method by obtaining a high low order approximation. The use of the $1$ coupling with the Giles-Szpruch scheme allows us to keep a multi-step Monte-Carlo estimator realizing an optimal complexity $Oleft( epsilon^{-2} right)$ while taking advantage of the $2$ low order error of the Ninomiya-Victoir scheme. In the third chapter, we are interested in the renormalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$. We show the stable law convergence to the solution of an affine SDE, whose source term is formed by the Lie brackets between the Brownian vector fields. Thus, when at least two Brownian vector fields do not commute, the limit is non-trivial. This ensures that the strong order $1/2$ is optimal. On the other hand, this result can be seen as a first step towards proving a central limit theorem for multi-step Monte-Carlo estimators. To do so, we need to analyze the stable law error of the scheme between two successive discretization levels. Ben Alaya and Kebaier proved such a result for the Euler scheme. When the Brownian vector fields commute, the limit process is zero. We show that in this particular case, the strong order is $1$. In chapter 4, we study the convergence to a stable law of the renormalized error $Nleft(X - X^{NV}right)$ where $X^{NV}$ is the Ninomiya-Victor scheme when the Brownian vector fields commute. We demonstrate the convergence of the renormalized error process to the solution of an affine SDE. When the dritf vector field does not commute with at least one of the Brownian vector fields, the strong convergence speed obtained previously is optimal.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

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We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

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This thesis presents three main research topics, the first two being independent and the last one indicating the relation of the first two problems in a concrete case.In the first part we focus on the martingale optimal transport problem in Skorokhod space, whose first goal is to study systematically the tension of martingale transport schemes. We first focus on the upper semicontinuity of the primal problem with respect to the marginal distributions. Using the S-topology introduced by Jakubowski, we derive the upper semicontinuity and show the first duality. We also give two dual problems concerning the robust overcoverage of an exotic option, and we establish the corresponding dualities, by adapting the principle of dynamic programming and the discretization argument initiated by Dolinsky and Soner.The second part of this thesis deals with the optimal Skorokhod folding problem. We first formulate this optimization problem in terms of probability measures on an extended space and its dual problems. Using the classical duality. convex approach and the optimal stopping theory, we obtain the duality results. We also relate these results to martingale optimal transport in the space of continuous functions, from which the corresponding dualities are derived for a particular class of payment functions. Next, we provide an alternative proof of the monotonicity principle established by Beiglbock, Cox and Huesmann, which allows us to characterize optimizers by their geometric support. We show at the end a stability result which contains two parts: the stability of the optimization problem with respect to the target marginals and the connection with another problem of the optimal folding.The last part concerns the application of stochastic control to the martingale optimal transport with the local time dependent payoff function, and to the Skorokhod folding. For the case of one marginal, we find the optimizers for the primal and dual problems via the Vallois solutions, and consequently show the optimality of the Vallois solutions, which includes the optimal martingale transport and the optimal Skorokhod folding. For the case of two marginals, we obtain a generalization of the Vallois solution. Finally, a special case of several marginals is studied, where the stopping times given by Vallois are well ordered.

We develop a weak exact simulation technique for a process X defined by a multi-dimensional stochastic differential equation (SDE). Namely, for a Lipschitz function g, we propose a simulation based approximation of the expectation E[g(X_{t_1}, \cdots, X_{t_n})], which by-passes the discretization error. The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are up-dated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Elworthy's formula from Malliavin calculus, as exploited by Fournie et al. for the simulation of the Greeks in financial applications.Unlike the exact simulation algorithm of Beskos and Roberts, our algorithm is suitable for the multi-dimensional case. Moreover, its implementation is a straightforward combination of the standard discretization techniques and the above mentioned automatic differentiation method.

This thesis deals with two areas of stochastic analysis and financial mathematics: the Malliavin calculus for Markov chains (Part I) and counterparty risk (Part II). Part I aims at studying the Malliavin calculus for Markov chains in continuous time. Two points are presented: proving the existence of the density for the solutions of a stochastic differential equation and computing the sensitivities of derivatives. Part II deals with current topics in the field of market risk, namely XVA (price adjustments) and multi-curve modeling.

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In this paper, we focus on model-free pricing and robust hedging of options depending on the local time, consistent with Vanilla options. This problem is classically approached by means of the Skorokhod embedding problem (SEP), which consists in representing a given probability on the real line as the distribution of a Brownian motion stopped at a chosen stopping time. By using the stochastic control approach initiated in Galichon, Henry-Labordere and Touzi, we recover the optimal hedging strategies and the corresponding prices given by Vallois' embeddings to the SEP. Furthermore, we extend the analysis to the two-marginal case. We provide a construction of two-marginal embedding and some examples for which the robust superhedging problem is solved. Finally, a special multi-marginal case is studied, where we construct a Markov martingale and compute its explicit generator. In particular, we provide a new example of fake Brownian motion.

In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula \cite{sab}. The geometry underlying this model is the hyperbolic manifold $\HH^{n+1}$ with $n$ the number of Libor forward rates.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

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This habilitation thesis focuses on three parts which are motivated by problems in mathematical finance: (1) martingale optimal transport, (2) asymptotic implied volatility for local and stochastic volatility models using short-time (geometrical) heat kernel expansion and (3) probabilistic numerical schemes for nonlinear parabolic second-order PDEs.

We give a study to the algorithm for semi-linear parabolic PDEs in Henry-Labordere (2012) and then generalize it to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren et al. (to appear) [5] and extended in Ekren et al. (2012) [6,7].

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

We consider the problem of superhedging 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 present a general duality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labordére and Touzi \cite{ght}, we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Azéma-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek \cite{hobson-klimmek} (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers \cite{brownhobsonrogers}. The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob\lój and Spoida \cite{OblSp}.

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry-LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

We generalize the algorithm for semi-linear parabolic PDEs in Henry-Labordére \cite{Henry-Labordere_branching} to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang \cite{EkrenKellerTouziZhang} and extended in Ekren, Touzi and Zhang \cite{EkrenTouziZhang1, EkrenTouziZhang2}.

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VIX options traded on the CBOE have become popular volatility derivatives. As S&P 500 vanilla options and VIX both depend on S&P 500 volatility dynamics, it is important to understand the link between these products. In this paper, we bound VIX options from vanilla options and VIX futures. This leads us to introduce a new martingale optimal transportation problem that we solve numerically. Analytical lower and upper bounds are also provided which already highlight some (potential) arbitrage opportunities. We fully characterize the class of marginal distributions for which these explicit bounds are optimal, and illustrate numerically that they seem to be optimal for the market-implied marginal distributions.

In this thesis, we treat two stochastic optimal control problems. Each problem corresponds to a part of this paper. The first problem is very specific, it is the valuation of American put contracts in the presence of discrete dividends (Part I). The second one is more general, since it is about proving the existence of a dynamic programming principle under probability constraints in a discrete time framework (Part II). Although the two problems are quite distinct, the dynamic programming principle is at the heart of both problems. The relation between the valuation of an American Put and a free boundary problem has been proved by McKean. The frontier of this problem has a clear economic meaning since it corresponds at any moment to the upper bound of the set of asset prices for which it is preferable to exercise one's right to sell immediately. The shape of this frontier in the presence of discrete dividends has not been solved to our knowledge. Under the assumption that the dividend is a deterministic function of the asset price at the time preceding its payment, we study how the frontier is modified. In the vicinity of the dividend dates, and in the model of Chapter 3, we know how to qualify the monotonicity of the frontier, and in some cases quantify its local behavior. In Chapter 3, we show that the smooth-fit property is satisfied at all dates except the dividend dates. In both Chapters 3 and 4, we give conditions to guarantee the continuity of this frontier outside the dividend dates. Part II is originally motivated by the optimal management of the production of a hydro-electric plant with a constraint in probability on the water level of the dam at certain dates. Using Balder's work on Young's relaxation of optimal control problems, we focus more specifically on solving them by dynamic programming. In Chapter 5, we extend the results of Evstigneev to the framework of Young's measurements. We then establish that it is possible to solve by dynamic programming some problems with constraints in conditional expectations. Thanks to the work of Bouchard, Elie, Soner and Touzi on stochastic target problems with controlled loss, we show in Chapter 6 that a problem with expectation constraints can be reduced to a problem with conditional expectation constraints. As a special case, we prove that the initial dam management problem can be solved by dynamic programming.

Vanilla calibration is a major problem in finance. We try to solve it for three classes of models: local and stochastic volatility models, the so-called "local correlation" model and a hybrid model of local volatility with stochastic rates. From a mathematical point of view, the calibration equation is a particularly complex nonlinear and integro-differential equation. In a first part, we prove the existence of solutions for this equation, as well as for its adjoint (simpler to solve). These results are based on fixed point methods in Hölder spaces and require classical theorems related to parabolic partial differential equations, as well as some a priori estimates in short time. The second part deals with the application of these existence results to the three financial models mentioned above. We also present the numerical results obtained by solving the edp. The calibration by this method is quite satisfactory. Finally, we focus on the algorithm used for the numerical solution: a predictor-corrector ADI scheme, which is modified to take into account the nonlinear character of the equation. We also describe an instability phenomenon of the edp solution that we try to explain from a theoretical point of view thanks to the so-called "Hadamard instability".

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