Option pricing in incomplete and nonlinear financial market models with default.

A. Sulem with M.C. Quenez and M. Grigorova have studied option pricing and hedging in nonlinear incomplete financial markets model with default. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow nonlinear dynamics with a nonlinear driver $f$, which encodes the imperfections or constraints of the market. A large class of imperfect market models can fit in this framework, including imperfections coming from different borrowing and lending interest rates, taxes on profits from risky investments, or from the trading impact of a large investor seller on the market prices and the default probability. Our market is incomplete, in the sense that not every contingent claim can be replicated by a portfolio. In this framework, we address in 13 the problem of pricing and (super)hedging of European options. By using a dynamic programming approach, we provide a dual formulation of the seller’s superhedging price as the supremum over a suitable set of equivalent probability measures $Q\in 𝒬$ of the non-linear ${ℰ}_Q^f$-expectation under $Q$ of the payoff. We also provide a characterization of this price as the minimal supersolution of a constrained BSDE with default. In 78, we study the superhedging problem for American options with irregular payoffs. We establish a dual formulation of the seller’s price in terms of the value of a non-linear mixed optimal control/stopping problem. We also characterize the seller's price process as the minimal supersolution of a reflected BSDE with constraints. We then prove a duality result for the buyer's price in terms of the value of a non-linear optimal control/stopping game problem. A crucial step in the proofs is to establish a non-linear optional and a non-linear predictable decomposition for processes which are ${ℰ}_Q^f$-strong supermartingales under $Q$, for all $Q\in 𝒬$. American option pricing in a non-linear complete market model with default is previously studied in 71. A complete analysis of BSDEs driven by a Brownian motion and a compensated default jump process with intensity process $\left({\lambda }_t\right)$ is achieved in 69. Note that these equations do not correspond to a particular case of BSDEs with Poisson random measure, and are particularly useful in default risk modeling in finance.

Optimal stopping.

The theory of optimal stopping in connection with American option pricing has been extensively studied in recent years. Our contributions in this area concern:

(i) The analysis of the binomial approximation of the American put price in the Black-Scholes model. We proved that the rate of convergence is, up to a logarithmic factor, of the order $1/n$, where $n$ is the number of discretization time points 87; (ii) The American put in the Heston stochastic volatility model. We have results about existence and uniqueness for the associated variational inequality, in suitable weighted Sobolev spaces, following up on the work of P. Feehan et al. (2011, 2015, 2016) (cf 89). We also established some qualitative properties of the value function (monotonicity, strict convexity, smoothness) 88. (iii) A probabilistic approach to the smoothness of the free boundary in the optimal stopping of a one-dimensional diffusion (work in progress with T. De Angelis)(University of Torino),

Stochastic control with jumps.

The 3rd edition of the book Applied Stochastic Control of Jump diffusions (Springer, 2019) by B. Øksendal and A. Sulem 15 contains recent developments within stochastic control and its applications. In particular, there is a new chapter devoted to a comprehensive presentation of financial markets modelled by jump diffusions, one on backward stochastic differential equations and risk measures, and an advanced stochastic control chapter including optimal control of mean-field systems, stochastic differential games and stochastic Hamilton-Jacobi-Bellman equations.

Asset Liability Management.

Life insurance contracts are popular and involve very large portfolios, for a total amount of trillions of euros in Europe. To manage them in a long run, insurance companies perform Asset and Liability Management (ALM) : it consists in investing the deposit of policyholders in different asset classes such as equity, sovereign bonds, corporate bonds, real estate, while respecting a performance warranty with a profit sharing mechanism for the policyholders. A typical question is how to determine an allocation strategy which maximizes the rewards and satisfies the regulatory constraints. The management of these portfolios is quite involved: the different cash reserves imposed by the regulator, the profit sharing mechanisms, and the way the insurance company determines the crediting rate to its policyholders make the whole dynamics path-dependent and rather intricate. A. Alfonsi et al. have developed in 46 a synthetic model that takes into account the main features of the life insurance business. This model is then used to determine the allocation that minimizes the Solvency Capital Requirement (SCR). In   47, numerical methods based on Multilevel Monte-Carlo algorithms are proposed to calculate the SCR at future dates, which is of practical importance for insurance companies. The standard formula prescribed by the regulator is basically obtained from conditional expected losses given standard shocks that occur in the future.

3.4.1 Numerical methods for Optimal transport

Optimal transport problems arise in a wide range of topics, from economics to physics. There exists different methods to solve numerically optimal transport problems. A popular one is the Sinkhorn algorithm which uses an entropy regularization of the cost function and then iterative Bregman projections. Alfonsi et al.  49 have proposed an alternative relaxation that consists in replacing the constraint of matching exactly the marginal laws by constraints of matching some moments. Using Tchakaloff's theorem, it is shown that the optimum is reached by a discrete measure, and the optimal transport is found by using a (stochastic) gradient descent that determines the weights and the points of the discrete measure. The number of points only depends of the number of moments considered, and therefore does not depend on the dimension of the problem. The method has then been developed in  48 in the case of symmetric multimarginal optimal transport problems. These problems arise in quantum chemistry with the Coulomb interaction cost. The problem is in dimension ${\left({ℝ}^3\right)}^M$ where $M$ is the number of electrons, and the method is particularly relevant since the optimal discrete measure weights only $N+2$ points, where $N$ is the number of moments constraint on the distribution of each electron. Numerical examples up to $M=100$ can be thus investigated while existing methods could not go beyond $M\approx 10$.

3.4.2 Mean-field systems

3.4.3 Martingale Optimal Transport

In mathematical finance, optimal transport problems with an additional martingale constraint are considered to handle the model risk, i.e. the risk of using an inadequate model. The Martingale Optimal Transport (MOT) problem introduced in 65 provides model-free hedges and bounds on the prices of exotic options. The market prices of liquid call and put options give the marginal distributions of the underlying asset at each traded maturity. Under the simplifying assumption that the risk-free rate is zero, these probability measures are in increasing convex order, since by Strassen's theorem this property is equivalent to the existence of a martingale measure with the right marginal distributions. For an exotic payoff function of the values of the underlying on the time-grid given by these maturities, the model-free upper-bound (resp. lower-bound) for the price consistent with these marginal distributions is given by the following martingale optimal transport problem : maximize (resp. minimize) the integral of the payoff with respect to the martingale measure over all martingale measures with the right marginal distributions. Super-hedging (resp. sub-hedging) strategies are obtained by solving the dual problem. With J. Corbetta, A. Alfonsi and B. Jourdain 5 have studied sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on ${𝐑}^d$, with $\nu$ dominating $\mu$. Their method is the first generic approach to tackle the martingale optimal transport problem numerically and it can also be applied to several marginals.

Martingale Optimal Transport provides thus bounds for the prices of exotic options that take into account the risk neutral marginal distributions of the underlying assets deduced from the market prices of vanilla options. For these bounds to be robust, the stability of the optimal value with respect to these marginal distributions is needed. Because of the global martingale constraint, stability is far less obvious than in optimal transport (it even fails in multiple dimensions). B. Jourdain has advised the PhD of W. Margheriti devoted to this issue and related problems. He also initiated a collaboration on this topic with M. Beiglböck, one of the founders of MOT theory. In 82, B. Jourdain and W. Margheriti exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $\nu$ in the convex order. The integral of $|x-y|$ with respect to each of these couplings is smaller than twice the ${𝒲}^1$ distance between $\mu$ and $\nu$. Moreover, for $\rho >1$, replacing $|x-y|$ and ${𝒲}_1$ respectively with ${|x-y|}^{\rho }$ and ${𝒲}_{\rho }^{\rho }$ does not lead to a finite multiplicative constant. In 27, they show that a finite constant is recovered when replacing ${𝒲}_{\rho }^{\rho }$ with the product of ${𝒲}_{\rho }$ times the centred $\rho$-th moment of the second marginal to the power $\rho -1$ and they study the generalisation of this stability inequality to higher dimension. In 28, they give a direct construction of the projection in adapted Wasserstein distance onto the set of martingale couplings of a coupling between two probability measures on the real line in the convex order which satisfies the barycentre dispersion assumption. Under this assumption, Wiesel had given a clear algorithmic construction of the projection for finitely supported marginals before getting rid of the finite support condition by a rather messy limiting procedure. In 20, with M. Beiglböck and G. Pammer they establish stability of martingale couplings in dimension one : when approximating in Wasserstein distance the two marginals of a martingale coupling by probability measures in the convex order, it is possible to construct a sequence of martingale couplings between these probability measures converging in adapted Wasserstein distance to the original coupling. In 64, they deduce the stability of the Weak Martingale Optimal Transport Problem with respect to the marginal distributions in dimension one which is important since financial data can give only imprecise information on these marginals. As application, this yields the stability of the superreplication bound for VIX futures and of the stretched Brownian motion. In 40, B. Jourdain et al. prove that, in dimension one, contrary to the minimum and maximum in the convex order, the Wasserstein projections of $\mu$ (resp. $\nu$) on the set of probability measures dominated by $\nu$ (resp. dominating $\mu$) in the convex order are Lipschitz continuous in $(\mu ,\nu )$ for the Wasserstein distance. The thesis of K. Shao (advisers: B. Jourdain, A. Sulem) focuses so far on optimal couplings for costs ${|y-x|}^{\rho }$ in dimension one.

3.5.1 Approximation of stochastic differential equations

3.5.2 Monte-Carlo and Multi-level Monte-Carlo methods

6.1.1 PREMIA

• Keywords:
  Financial products, Computational finance, Option pricing, Machine learning
• Scientific Description:
  Premia is a numerical platform for computational finance. It is designed for option pricing, hedging and financial model calibration. Premia is developed by the MathRisk project team in collaboration with a consortium of financial institutions. The Premia project keeps track of the most recent advances in the field of computational finance in a well-documented way. It focuses on the implementation of numerical analysis techniques for both probabilistic and deterministic numerical methods. An important feature of the platform Premia is the detailed documentation which provides extended references in option pricing. Premia contains various numerical algorithms: deterministic methods (Finite difference and finite element algorithms for partial differential equations, wavelets, Galerkin, sparse grids ...), stochastic algorithms (Monte-Carlo simulations, quantization methods, Malliavin calculus based methods), tree methods, approximation methods (Laplace transforms, Fast Fourier transforms...) These algorithms are implemented for the evaluation of vanilla and exotic options on equities, interest rate, credit, energy and insurance products. Moreover Premia provides a calibration toolbox for Libor Market model and a toolbox for pricing Credit derivatives. The latest developments of the software address evaluation of financial derivative products, risk management and computations of risk measures required by new financial regulation. They include the implementation of advanced numerical algorithms taking into account model dependence, counterparty credit risk, hybrid features, rough volatility and various nonlinear effects. A big effort has been put these last years on the development and implementation of deep learning techniques using neural network approximations, and Machine Learning algorithms in finance, in particular for high-dimensional American option pricing, high-dimensional PDEs, deep hedging,
• Functional Description:
  Premia is a software designed for quantitative finance, developed by the MathRisk project team in collaboration with a consortium of financial institutions presently composed of Crédit Agricole CIB and NATIXIS. The Premia project keeps track of the most recent advances in computational finance and focuses on the implementation of numerical techniques to solve financial problems. An important feature of the platform Premia is its detailed documentation which provides extended references in computational finance. Premia is a powerful tool to assist Research and Development professional teams in their day-to-day duty. It is also a useful support for academics who wish to perform tests on new algorithms or pricing methods. Besides being a single entry point for accessible overviews and basic implementations of various numerical methods, the aim of the Premia project is: - to elaborate a powerful testing platform for comparing different numerical methods between each other, - to build a link between professional financial teams and academic researchers, - to provide a useful teaching support for Master and PhD students in mathematical finance. The project Premia has started in 1999 and is now considered as a standard reference platform for quantitative finance among the academic mathematical finance community.
• Release Contributions:
  A big effort has been put these last years on the development and implementation of deep learning techniques using neural network approximations, and Machine Learning algorithms in finance, in particular for high-dimensional American option pricing, high-dimensional PDEs, deep hedging.The latest developments of the software address also the evaluation of financial derivative products, risk management and computations of risk measures by advanced numerical algorithms taking into account model dependence, counterparty credit risk (computations of XVA), hybrid features, rough stochastic volatility models and various new regulations. Nested Monte Carlo strategies with GPU optimizations, and Chebyshev Interpolation method for Parametric Option Pricing have been implemented. We have also developed our activity on insurance contracts, in particular on the computation of risk measures (Value at Risk, Condition Tail Expectation) of variable annuities contracts like GMWB (guaranteed minimum withdrawal benefit) including taxation and customers mortality modeling.
• News of the Year:

  Release 24 of the Premia software has been delivered to the Consortium in March 2022. It contains the following new implemented algorithms dedicated to Machine Learning in finance: • Deep Hedging. H.Buhler, L. Gonon, J. Teichmann, B. Wood Quantitative Finance, Volume 19-8, 2019. • Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions. Y.Chen J.W.L. Wan Quantitative Finance, 21-1, 2021 • Deep learning for ranking response surfaces with applications to optimal stopping problems. R.Hu Quantitative Finance, 21-1, 2021 • DGM: A deep learning algorithm for solving partial differential equations. J.Sirignano K.Spiliopoulos Journal of Computational Physics 375, 2018 • Financial option valuation by unsupervised learning with artificial neural networks. B. Salvador, C. W. Oosterlee, R. van der Meer • Equal Risk Pricing of Derivatives with Deep Hedging. A.Carbonneau F.Godin Quantitative Finance, 2020 • Quant GANs: Deep Generation of Financial Time Series. M.Wiese, R. Knobloch, R.Korn, P.Kretschmer Quantitative Finance, 20-9, 2020 • Artificial neural network for option pricing with and without asymptotic correction. H.Funahashi Quantitative Finance, 2020 • Pricing Bermudan options using regression trees/random forests. Z. El Filali Ech-Chafiq, P. Henry-Labordere, J.Lelong • Moving average options: Machine Learning and Gauss-Hermite quadrature for a double non-Markovian problem. L.Goudenge A.Molent A.Zanette European Journal of Operational Research 2022

  Pricing of Equity Derivatives: • American options in the Volterra Heston model. S.Pulido, E.Chevalier, E.Zuniga. • Sinh-Acceleration: Efficient Evaluation of Probability Distributions, Option Pricing, and Monte-Carlo Simulations. S. Boyarchenko, S.Levendorskii International Journal of Theoretical and Applied Finance 22-03,2019 • A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options. O. Kudryavtsev In: Karapetyants A.N., Pavlov I.V., Shiryaev A.N. (eds) Operator Theory and Har- monic Analysis. OTHA 2020. Springer Proceedings in Mathematics Statistics, vol 358.

• URL:
  http://­www.­premia.­fr
• Publications:
  hal-03436046, hal-01940715, hal-01873346, hal-03810106, hal-03526905, hal-03013606, hal-02183587
• Contact:
  Agnes Sulem
• Participants:
  Agnes Sulem, Antonino Zanette, Aurélien Alfonsi, Benjamin Jourdain, Jerome Lelong, Bernard Lapeyre, Ahmed Kebaier, Ludovic Goudenège
• Partners:
  Inria, Ecole des Ponts ParisTech

Default cascades in sparse heterogeneous financial networks.

A. Sulem and H. Amini are supervising a PhD student (Z. Cao) on the control of interbank contagion, dynamics and stability of complex financial networks, by using techniques from random graphs and stochastic control. We have obtained limit results for default cascades in sparse heterogeneous financial networks subject to an exogenous macroeconomic shock in 53. These limit theorems for different system-wide wealth aggregation functions allow us to provide systemic risk measures in relation with the structure and heterogeneity of the financial network. These results are applied to determine the optimal policy for a social planner to target interventions during a financial crisis, with a budget constraint and under partial information of the financial network. Banks can impact each other due to large-scale liquidations of similar assets or non-payment of liabilities. In 52, we present a general tractable framework for understanding the joint impact of fire sales and default cascades on systemic risk in complex financial networks. The effect of heterogeneity in network structure and price impact function on the final size of default cascade and fire sales loss is investigated.

7.4.1 Optimal transport

B. Jourdain and A. Tse have extended to non-linear functionals of the empirical measure of independent and identically distributed random vectors the central limit theorem which is well known for linear functionals. The main tool permitting this extension is the linear functional derivative, one of the notions of derivation on the Wasserstein space of probability measures that have recently been developed. In 39, R. Flenghi and B. Jourdain relax first the equal distribution assumption and then the independence property to be able to deal with the successive values of an ergodic Markov chain.

Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. In 40, B. Jourdain, W. Margheriti and G. Pammer prove that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, they provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.

7.4.2 Convex order

In 41, B. Jourdain and G. Pagès are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous ${ℝ}^d$-valued paths and for the monotonic convex order when $d=1$. Even if in general these solutions are neither semi-martingales nor Markov processes, they are able to exhibit conditions on their coefficients enabling the comparison. The approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. They consider two types of Euler schemes depending on the way the Volterra kernels are discretized. The conditions ensuring the comparison are slightly weaker for the first scheme than for the second one and this is the other way round for convergence. Moreover, they extend the integrability needed on the starting values in the existence and convergence results in the literature to be able to only assume finite first order moments, which is the natural framework for convex ordering.

7.4.3 Approximation of Boltzmann and Mc-Kean Vlasov equations

V. Bally and A. Alfonsi have studied existence, uniqueness and Euler approximation for jump type stochastic equations of Bolzmann and Mc Kean-Vlasov type 44. The specificity of their approach is to use a methodology based on stochastic flows and the sewing lemma which has been introduced in the rough path theory. Recently, V. Bally and his Phd student Y. Qin use Malliavin calculus for jump processes to prove convergence of the Euler scheme (for the above mentioned equations) in total variation distance 19(while in 44 only the Wasserstein distance is considered). Moreover, they study the approximation of the invariant measure by an adaptative Euler scheme, as introduced by Lamberton and Pagès and recently studied by Panloup and Pagès.

Joint SPX/VIX calibration.

J. Guyon and S. Mustapha calibrate neural stochastic differential equations jointly to S&P 500 smiles, VIX futures, and VIX smiles. Drifts and volatilities are modeled as neural networks. Minimizing a suitable loss allows them to fit market data for multiple S&P 500 and VIX maturities. A one-factor Markovian stochastic local volatility model is shown to fit both smiles and VIX futures within bid-ask spreads. The joint calibration actually makes it a pure path-dependent volatility model, confirming the findings in 79.

Validation of economic scenarios.

Real-world economic scenarios provide stochastic forecasts of economic variables like interest rates, equity stocks or indices, or inflation and are widely used in the insurance sector. Unlike risk-neutral scenarios, they aim to be realistic in view of the historical data and/or expert expectations of future incomes. In 37, H. Andrës, A. Boumezoued and B. Jourdain propose a new approach for the validation of real-world economic scenario motivated by insurance applications. This approach is based on the statistical test developed by Chevyrev and Oberhauser and relies on the notions of signature and maximum mean distance. This test allows to check whether two samples of stochastic processes paths come from the same distribution. Their contribution is to apply this test to two stochastic processes, namely the fractional Brownian motion and the Black-Scholes dynamics. They analyze its statistical power based on numerical experiments under two constraints: 1. they work in an asymetric setting in which they compare a large sample that represents simulated real-world scenarios and a small sample that mimics information from historical data, both with a monthly time step as often considered in practice and 2. they make the two samples identical from the perspective of validation methods used in practice, i.e. they impose that the marginal distributions of the two samples are the same at a given one-year horizon. By performing specific transformations of the signature, they obtain high statistical powers and demonstrate the potential of this validation approach for real-world economic scenarios. they also discuss several challenges related to the numerical implementation of this approach, and highlight its domain of validity in terms of distance between models and the volume of data at hand.

Interest rates for insurance: efficient calibration and derivatives valuation.

In view of the long lifetime of insurance contracts, major part of insurers/reinsurers assets portfolios is composed of bonds. Consequently, the main financial risk the insurers are exposed to is the interest rates risk. The thesis of S. Mehalla 92 (superviser B. Lapeyre) is dedicated to the study of efficient calibration procedures for interest rates models, such as as LIBOR Market Model, used by insurance/reinsurance undertakings. Financial models with stochastic volatility factor are studied. A new method to efficiently price swap rates derivatives under the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion is proposed (see 17). Fast calibration based on optimization algorithms are studied (see 16, 56).

Stochastic modeling of the Temperature for pricing weather derivatives

Insurance companies sell insurance contracts that give a protection against weather fluctuations, typically on rain or temperature. These contracts are managed then using weather derivatives. In 34, A. Alfonsi and his PhD student N. Vadillo Fernandez propose a new stochastic volatility model for the temperature in order to price derivatives on the climate (Heating Degree Day index). They develop a simple and efficient estimation of the model parameters on historical data and provide efficient numerical pricing methods.

7.7.1 Approximations of Stochastic Differential Equations (SDEs)

7.7.2 Numerical approximation of conditional expectations

A.Alfonsi, B. Lapeyre and J. Lelong have revisited the classical Monte-Carlo regression problem to approximate the conditional expectation $𝔼\left[Y\right|X]$ when it is possible to sample $Y$ given $X$ and then to generate $K\in {\mathbb{N}}^{*}$ values of $Y_{i,k}$ for a given value $X_i$ (See  32). They determine the optimal value of $K$ that minimizes the computation cost for a given precision.

7.7.3 Optimal transport and mean-field SDEs

In 23, O. Bencheikh and B. jourdain analyse the rate of convergence of a system of interacting particles with mean-field rank based interaction in the drift coefficient and constant diffusion coefficient.

7.7.4 Abstract Malliavin calculus and convergence in total variation

In 19, V. Bally and his PhD student Yifen Qin obtain total variation distance result between a jump-equation and its Gaussian approximation by Malliavin calculus techniques.

Neural networks and Machine Learning techniques for high dimensional American options.

The pricing of American option or its Bermudan approximation amounts to solving a backward dynamic programming equation, in which the main difficulty comes from the conditional expectation involved in the computation of the continuation value.

In 90, B. Lapeyre and J. Lelong study neural networks approximations of conditional expectations. They prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression on a finite-dimensional vector space is replaced by a neural network approximation, and illustrate the numerical efficiency of the method on several numerical examples. Its stability with respect to a change of parameters as interest rate and volatility is shown. The numerical study proves that training neural network with only a few chosen points in the grid of parameters permits to price efficiently for a whole range of parameters.

In 76, two efficient techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are proposed to compute the price of American basket options. Both techniques are based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. In 77, an efficient method is provided to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction techniques. Numerical tests show that the proposed algorithm is fast and reliable, and can handle American options on very large baskets of assets, overcoming the curse of dimensionality issue.

$•$Machine Learning in the Energy and Commodity Market. Evaluating moving average options is a computational challenge for the energy and commodity market, as the payoff of the option depends on the prices of underlying assets observed on a moving window. An efficient method for pricing Bermudan style moving average options is presented in 25, based on Gaussian Process Regression and Gauss-Hermite quadrature. This method is tested in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities, the Heston (non-Gaussian) model and the rough-Bergomi model, which involves a double non-Markovian feature, since the whole history of the volatility process impacts the future distribution of the process.

Investment strategies based on Machine learning algorithms. The goal of the thesis of H. Madmoun 91 (Supervisor: B. Lapeyre) is to show how portfolio allocation can benefit from the development of Machine Learning algorithms. Two investment strategies based on such algorithms are proposed. The first strategy (Low Turbulence Model) is an asset allocation method based on an interpretable low-dimensional representation framework of financial time series combining signal processing, deep neural networks, and bayesian statistics. The second one (Attention Based Ranking Model) is a stock picking strategy for large- cap US stocks.

9.1.1 Participation in International Programs

•   Edoardo Lombardo, international PhD grant between ENPC and Univ. Tor Vergata, Roma

9.2.1 Visits of international scientists

10.1.1 Scientific events: organisation

• A. Alfonsi:

  Co-organizer of the Mathrisk seminar “Méthodes stochastiques et finance”

  Co-organizer of the Bachelier (Mathematical Finance) seminar (IHP, Paris).

• V. Bally

  - Organizer of the seminar of the LAMA laboratory, Université Gustave Eiffel.

  - Organizer of the mini symposium " Stochastic Equations", XVth French-Romanian Colloquium on Applied Mathematics, Toulouse, 29 August - 2 September 2022

• A. Sulem

  - Co-organizer of the seminar INRIA-MathRisk /Université Paris Diderot LPSM “Numerical probability and mathematical finance”

  - Member of the scientific committee of the London-Paris Bachelier workshop (September 2022), IHP, Paris.

10.1.2 Journal

10.1.3 Invited talks

• A. Alfonsi
  • 06 01 2022: "A synthetic model for ALM in life insurance and numerical methods for SCR computation", Berlin Seminar on Mathematical Finance.
  • 22 03 2022: "Approximation of Optimal Transport problems with marginal moments constraints", Stochastic Mass Transports (22w5166), Banff.
  • 07 04 2022: "Approximation of Optimal Transport problems with marginal moments constraints", Séminaire commun MathRisk-LPSM
  • 21 04 2022: "A synthetic model for ALM in life insurance and numerical methods for SCR computation", Labex Bezout.
• V. Bally
  • Conference Kolmogorov Operators and their Applications; Cortona June 13-17, 2022. Talk : Construction of Boltzmann and Mc-Kean Vlasov Type Flows.
  • Conference Stochastic Analysis and Partial differential Equations, Barcelona 30 May - 03 June 2022. talk: Construction of Boltzmnn and Mc-Kean Vlasov Type Flows
• J. Guyon
  • Institut Henri Poincaré, Bachelier seminar, September 2022.
  • Ecole des Ponts ParisTech, Stochastic Methods in Finance seminar, October 2022.
  • ETH Zurich, Talks in Financial and Insurance Mathematics, October 2022.
  • WBS, 18th Quantitative Finance Conference, Dubrovnik, October 2022.
  • QuantMinds 2022 Conference, Barcelona, November 2022.
  • University of Oxford, Statistics and Machine Learning in Finance Seminar Series, December 2022.
• B. Jourdain
  • Worshop on Advances in Stochastic Control and Optimal Stopping, Marseille, 12 September 2022 : Approximation of martingale couplings on the real line in the weak adapted topology
  • 9th colloquium on BSDEs and Mean Field Systems, Annecy, 29 June 2022 : Central limit theorem over nonlinear functionals of empirical measures and fluctuations of interacting particle systems
  • Séminaire de probabilitÈs, Sorbonne université, 31 may 2022 : Stabilité du problème de transport optimal martingale faible
  • BIRS workshop Stochastic Mass Transports, 25 march 2022 : Approximation of martingale couplings on the real line in the weak adapted topology
  • Ritsumeikan University Seminar, 13 January 2022 : Convergence rate of the Euler-Maruyama scheme applied to diffusion processes with irregular drift coefficient and additive noise
• D. Lamberton
  • A probabilistic approach to the regularity of a free boundary. XVth French-Romanian Colloquium on Applied Mathematics, Toulouse, August 2022.
  • Regularity of the free boundary: a probabilistic approach. Advances in Stochastic Control and Optimal Stopping with Applications in Economics and Finance. Luminy, September 2022.
  • Régularité de la frontière libre d'un problème d'arrêt optimal : une approche probabiliste. Groupe de travail Mathématiques financières et actuarielles, probabilités numériques. Sorbonne Université, October 2022.

• A. Sulem

  Keynote speaker, 19th e-Summer School in Risk Finance and Stochastics 28/9 - 30/9/2022, Athens

10.1.4 Research administration

• A. Alfonsi
  • Deputy director of the CERMICS.
  • In charge of the Master “Finance and Data” at the Ecole des Ponts.
• V. Bally
  • Responsible of the Master 2, option finance, Université Gustave Eiffel
  • Member of the LAMA committee, UGE.
• A. Sulem
  • Member of the Scientific Committee of AMIES( Agence pour les Mathématiques en Interaction avec l'Entreprise et la Société)
  • Corresponding member of the Operational Committee for the assesment of Legal and Ethical risks (COERLE) at INRIA Paris research center
  • Member of the Committee for INRIA international Chairs
  • Member of the Nominating Committee of the Bachelier Finance Society

10.2.1 Supervision

• PhD defended :
  • Hachem Madmoun, defended at Ecole des Ponts Paris Tech, 30 November 2022, Adviser: B. Lapeyre.
• PhD in progress :
  • Roberta Flenghi (started in January 2021) "Central limit theorems for nonlinear functionals of the empirical measure of correlated random variables", supervised by B. Jourdain
  • HervÈ AndrËs (started in June 2021) "Dependence modelling in economic scenario generation for insurance", supervised by B. Jourdain
  • Kexin Shao (started in october 2021) "Martingale optimal transport and financial applications", supervised by B. Jourdain and A. Sulem
  • Zhonguyan Cao, "Dynamics and Stability of Complex Financial networks", Université Paris-Dauphine, Supervisor: Agnès Sulem, Dim Mathinnov doctoral allocation, started October 2020
  • Edoardo Lombardo, “High order numerical approximation for some singular stochastic processes and related PDEs”, started in November 2020, International PhD, advisors: Aurélie Alfonsi and Lucia Caramellino (Tor Vegata Roma University),
  • Nerea Vadillo Fernandez (CIFRE AXA Climate), “Risk valuation for weather derivatives in index-based insurance”, started in November 2020, supervised by A. Alfonsi
  • Yfen Qin, "Regularity properties for non linear problems", Dim-Mathinnov doctoral allocation, supervisor: V. Bally

10.2.2 Juries

• B. Jourdain
  • PhD of Antoine Bienvenu, defended on July 4, Sorbonne université,
  • PhD of Laetitia Della Maestra, defended on November 30, Université Paris-Dauphine
  • PhD of Mohan Yang, defended on December 8, Université Paris Cité, B. Jourdain reviewer

• J. Guyon

  PhD defense of William Lefebvre, LPSM, Université Paris-Cité, 09/12/2022,

• A. Sulem
  • President of the jury for the Prix Marc Yor 2022
  • Member of the Committee for the recruitment of an associate Professor in "Financial Mathematics, statistics, numerical probability", Université Paris-Cité,
  • President of the jury for the PhD defense of William Lefebvre, Méthodes de Contrôle stochastique appliquées à la construction de portefeuille, au contrôle avec retard et à la résolution d'EDPs, LPSM, Université Paris-Cité, 09/12/2022,
  • President of the jury for the PhD defense of Hachem Madmoun,Creating Investment Strategies based on Machine Learning Algorithms, ENPC, 30/11/2022
  • Reviewer for the PhD thesis of Zineb El Filali Ech-Chafiq, Applications de l'apprentissage automatique en finance, Univ. Grenoble Alpes, 27/09/2022
  • PhD of Maximilien Germain, Méthodes d’apprentissage automatique pour la résolution de problèmes de contrôle stochastique et d’équations aux dérivées partielles en grande dimension, LPSM, Université Paris-Cité, 20/06/2022

International journals

• 16 articleP.-E.Pierre-Edouard Arrouy, A.Alexandre Boumezoued, B.Bernard Lapeyre and S.Sophian Mehalla. Economic Scenario Generators: a risk management tool for insurance.MathematicS In Action1112022, 43--60
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• 17 articleP.-E.Pierre-Edouard Arrouy, B.Bernard Lapeyre, S.Sophian Mehalla and A.Alexandre Boumezoued. Jacobi Stochastic Volatility factor for the Libor Market Model.Finance and Stochastics26September 2022, 771–823
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• 18 articleV.Vlad Bally, L.Lucia Caramellino and A.Arturo Kohatsu-Higa. Using moment approximations to study the density of jump driven SDEs.Electronic Journal of Probability27January 2022
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• 19 articleV.Vlad Bally and Y.Yifeng Qin. Total variation distance between a jump-equation and its Gaussian approximation.Stochastic and Partial Differential Equations: Analyses and ComputationsAugust 2022
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• 20 articleM.Mathias Beiglböck, B.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Approximation of martingale couplings on the line in the weak adapted topology.Probability Theory and Related Fields1831-22022, 359–413
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• 21 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures.Journal of Approximation Theory2741056842022
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• 22 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise.SIAM Journal on Numerical Analysis6042022, 1701-1740
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• 23 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law.Annals of Applied Probability3262022, 4143–4185
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• 24 articleR.Rui Chen, R.Roxana Dumitrescu, A.Andreea Minca and A.Agnès Sulem. Mean-field BSDEs with jumps and dual representation for global risk measures.Probability, Uncertainty and Quantitative Risk2022
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• 25 articleL.Ludovic Goudenège, A.Andrea Molent and A.Antonino Zanette. Moving average options: Machine learning and Gauss-Hermite quadrature for a double non-Markovian problem.European Journal of Operational Research3032December 2022, 958-974
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• 26 articleB.Benjamin Jourdain and E.Ezéchiel Kahn. Strong solutions to a beta-Wishart particle system.Journal of Theoretical Probability3532022, 1574–1613
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• 27 articleB.Benjamin Jourdain and W.William Margheriti. Martingale Wasserstein inequality for probability measures in the convex order.Bernoulli2822022, 830-858
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• 28 articleB.Benjamin Jourdain and W.William Margheriti. One dimensional martingale rearrangement couplings.ESAIM: Probability and Statistics262022, 495-527
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• 29 articleB.Benjamin Jourdain and G.Gilles Pagès. Convex order, quantization and monotone approximations of ARCH models.Journal of Theoretical Probability3542022, 2480-2517
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• 30 articleB.Benjamin Jourdain and G.Gilles Pagès. Quantization and martingale couplings.ALEA : Latin American Journal of Probability and Mathematical Statistics192022
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Reports & preprints

• 31 miscA.Aurélien Alfonsi and A.Ahmed Kebaier. Approximation of Stochastic Volterra Equations with kernels of completely monotone type.January 2022
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• 32 misc A.Aurélien Alfonsi, B.Bernard Lapeyre and J.Jérôme Lelong. How many inner simulations to compute conditional expectations with least-square Monte Carlo? September 2022
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• 33 miscA.Aurélien Alfonsi and E.Edoardo Lombardo. High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids.September 2022
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• 34 miscA.Aurélien Alfonsi and N.Nerea Vadillo. A stochastic volatility model for the valuation of temperature derivatives.September 2022
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• 35 miscH.Hamed Amini, Z.Zhongyuan Cao and A.Agnès Sulem. Graphon Mean-Field Backward Stochastic Differential Equations With Jumps and Associated Dynamic Risk Measures.October 2022
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• 36 miscH.Hamed Amini, Z.Zhongyuan Cao and A.Agnes Sulem. The Default Cascade Process in Stochastic Financial Networks.October 2022
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• 37 miscH.Hervé Andrès, A.Alexandre Boumezoued and B.Benjamin Jourdain. Signature-based validation of real-world economic scenarios.July 2022
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• 38 miscA.Adrien Beguinet, V.Virginie Ehrlacher, R.Roberta Flenghi, M.Maria Fuente-Ruiz, O.Olga Mula and A.Agustin Somacal. Deep learning-based schemes for singularly perturbed convection-diffusion problems.May 2022
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• 39 miscR.Roberta Flenghi and B.Benjamin Jourdain. Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting.April 2022
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• 40 miscB.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Lipschitz continuity of the Wasserstein projections in the convex order on the line.September 2022
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• 41 miscB.Benjamin Jourdain and G.Gilles Pagès. Convex ordering for stochastic Volterra equations and their Euler schemes.November 2022
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