2025​​Activity reportTeamMATHRISK​​​‌

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​ 14 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 $Q\in \mathrm{𝒬}$ of the non-linear ${\mathrm{ℰ‌}}_Q^f$-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 88​​, 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 ${ℰ}_{\mathrm{Q‌‌}}^f$-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 80‌. A complete analysis‌​‌ of BSDEs driven by​​ a Brownian motion and​​​‌ a compensated default jump‌ process with intensity process‌​‌ $({\lambda }_t)$ is achieved in 78​​​‌. 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 $\mathrm{1‌‌}/n$, where​​ $n$ is the number​​​‌ of discretization time points‌ 104; (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​​​‌ 106). We also‌ established some qualitative properties‌​‌ of the value function​​ (monotonicity, strict convexity, smoothness)​​​‌ 105. (iii) A‌ probabilistic approach to the‌​‌ smoothness of the free​​ boundary in the optimal​​​‌ stopping of a one-dimensional‌ diffusion (work in collaboration‌​‌ with T. De Angelis)(University​​ of Torino) (see 59​​​‌),

Stochastic control with‌ jumps.

The 3rd edition‌​‌ of the book Applied​​ Stochastic Control of Jump​​​‌ diffusions (Springer, 2019) by‌ B. Øksendal and A.‌​‌ Sulem 16 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​‌ 50 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  51, 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.

3.5.1 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. 53 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 52 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​​ ${({ℝ}^3)‌}^M$ where $M$ is​ the number of electrons,​‌ and the method is​​ particularly relevant since the​​​‌ optimal discrete measure weights​ only $N+\mathrm{2‌}$ points, where $N$ is​​ the number of moments​​​‌ constraint on the distribution​ of each electron. Numerical​‌ examples up to $\mathrm{M}=100$ can be​​ thus investigated while existing​​​‌ methods could not go‌ beyond $M\approx \mathrm{10‌‌}$.

3.5.2 Mean-field systems​​

3.5.3‌​‌ 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 71 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 6​​ 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 95, 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 $|\mathrm{x‌}-y|$ with​​ respect to each of​​​‌ these couplings is smaller​ than twice the ${\mathrm{𝒲‌}}^1$ distance between $\mathrm{\mu }$ and $\nu$. Moreover,​​​‌ for $\rho >\mathrm{1}$, replacing $|\mathrm{x‌}-y|$ and​​ ${𝒲}_1$ respectively with​​​‌ ${|x-\mathrm{y}|}^{\rho }$ and ${\mathrm{𝒲‌}}_{\rho }^{\rho }$ does not​​ lead to a finite​​​‌ multiplicative constant. In 96​, they show that​‌ a finite constant is​​ recovered when replacing ${\mathrm{𝒲‌}}_{\rho }^{\rho }$ with the​ product of ${𝒲}_{\mathrm{\rho ‌}}$ times the centred $\mathrm{\rho }$-th moment of the​​​‌ second marginal to the​ power $\rho -\mathrm{1‌}$ and they study the​​ generalisation of this stability​​​‌ inequality to higher dimension.​ In 97, 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 69,​​​‌ 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 70‌​‌, 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 98​​​‌, 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 $\mathrm{\mu ‌‌}$) 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​​ ${|y-\mathrm{x‌}|}^{\rho }$ in dimension‌ one.

3.5.4​​​‌ Martingale Schrödinger problems

Calibration‌ problems in finance can‌​‌ be cast as Schrödinger​​​‌ problems. Due to the​ no-arbitrage condition, martingale Schrödinger​‌ problems must be considered.​​ To jointly calibrate S&P​​​‌ 500 (SPX) and VIX​ options, J. Guyon has​‌ introduced dispersion-constrained martingale Schrödinger​​ problems. In 89,​​​‌ he solved for the​ first time this 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​ SPX options, VIX futures,​‌ and VIX options. He​​ did so using a​​​‌ nonparametric, discrete-time, minimum-entropy approach.​ He established a strong​‌ duality theorem and characterized​​ the absence of joint​​​‌ SPX/VIX arbitrage. The minimum​ entropy jointly calibrating model​‌ is explicit in terms​​ of the dual Schrödinger​​​‌ portfolio, i.e., the maximizer​ of the dual problems,​‌ should it exist, and​​ is numerically computed using​​​‌ an extension of the​ Sinkhorn algorithm. Numerical experiments​‌ show that the algorithm​​ performs very well in​​​‌ both low and high​ volatility regimes.

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 107, 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 85, 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 87, 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 86, 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.

3.7.1 Approximation of stochastic‌ differential equations

3.7.2 Monte-Carlo​ and Multi-level Monte-Carlo methods​‌

7.1.1 PREMIA​​​‌

• Keywords:
  Computational finance, Quantum​ Finance, Monte-Carlo methods, Option​‌ pricing, Numerical probability, Machine​​ learning, Numerical algorithm
• 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.​​​‌ Moreover Quantum computing in​ Finance is explored, in​‌ particular option pricing using​​ quantum computers.
• 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:

  The new release‌​‌ Premia 27 has been​​ delivered to the Consortium​​​‌ on June 3 2025.‌ It contains the following‌​‌ new implemented algorithms in​​ Machine Learning, Neural networks,​​​‌ Risk Management:

  - Optimal‌ stopping with signatures. C.‌​‌ Bayer, P. Hager,S. Riedel,​​ J. Schoenmakers, The Annals​​​‌ of Applied Probability 33-1,‌ 2023 - Primal and‌​‌ dual optimal stopping with​​ signature. C. Bayer, L.​​​‌ Pelizzari, J. Schoenmakers arxiv.org/abs/2312.03444‌ - Pricing American options‌​‌ under rough volatility using​​ deep-signatures and signature-kernels. C.​​​‌ Bayer, L. Pelizzari, Zhou‌ arxiv.org/abs/2501.06758 - Deep calibration‌​‌ of rough stochastic volatility​​ models. C.Bayer B.Stemper -​​​‌ Optimal Damping with Hierarchical‌ Adaptive Quadrature for Efficient‌​‌ Fourier Pricing of Multi-Asset​​ Options in Levy Models,​​​‌ C. Bayer, C. Ben‌ Hammouda, A. Papapantoleon, M.‌​‌ Samet and R. Tempone​​ Journal of Computational Finance,​​​‌ Volume 27-3, 2023. -‌ Leveraging Machine Learning for‌​‌ High-Dimensional Option Pricing within​​ the Uncertain Volatility Model.​​​‌ L. Goudenege A. Molent,‌ A. Zanette arxiv.org/abs/2407.13213 -‌​‌ Robust Pricing of Equity-Indexed​​ Annuities under Uncertain Volatility​​​‌ and Stochastic Interest Rate,‌ L. Goudenege A. Molent,‌​‌ A. Zanette arxiv.org/abs/2407.13213 -​​ Neural Optimal Stopping Boundary.​​​‌ A. Max Reppen, H.‌ Mete Soner, Valentin Tissot-Daguette,‌​‌ Mathematical Finance, Volume 35-2,​​ 2025. - Estimating risks​​​‌ of option books using‌ neural-SDE market models. S.‌​‌ N. Cohen, C. Reisinger​​ and S. Wang, Journal​​​‌ of Computational Finance, Volume‌ 26-3, 2022. - Hedging‌​‌ option books using neural-SDE​​ market models. S. N.​​​‌ Cohen, C. Reisinger and‌ S. Wang, Applied Mathematical‌​‌ Finance 29-5, 2022. -​​ Arbitrage-free neural-SDE market models.​​​‌ Arbitrage-free neural-SDE market models.‌ S. N. Cohen, C.‌​‌ Reisinger and S. Wang,​​ Appl. Math. Finance 30-1,​​​‌ 2023. - Neural variance‌ reduction for stochastic differential‌​‌ equations, P. D. Hinds​​ and M. V. Tretyakov​​​‌ Journal of Computational Finance,‌ Volume 27-3, 2023.

• URL:‌​‌
  http://­www.­premia.­fr
• Publications:
  hal-03791594,​​ hal-03436046, hal-01940715,​​​‌ hal-01873346, hal-03810106,‌ hal-05421581, 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:​
  Ecole des Ponts ParisTech,​‌ Université d'Udine

8.2.1 Stochastic Graphon​​ Mean-field Games and approximate​​​‌ Nash Equilibria

The use​ of graphons has emerged​‌ recently in order to​​ analyze heterogeneous interaction in​​​‌ mean-field systems and game​ theory. Graphon BSDEs with​‌ jumps and associated dynamic​​ global risk measures are​​​‌ studied by H. Amini,​ A. Sulem, and Z.​‌ Cao in 21.​​ Existence, uniqueness and stability​​​‌ of solutions under some​ regularity assumptions are established.​‌ We also prove convergence​​ results for finite interacting​​​‌ mean-field particle systems with​ heterogeneous interactions to graphon​‌ mean-field BSDE systems. Finally,​​ we introduce the graphon​​​‌ dynamic risk measure induced​ by the solution of​‌ a graphon mean-field BSDE​​ system and study its​​​‌ properties. In particular, a​ dual representation theorem is​‌ provided in the convex​​ case.

In 23 we​​​‌ study continuous stochastic games​ with heterogeneous mean field​‌ interactions and jumps on​​ large networks and explore​​​‌ their limit counterparts. We​ introduce the graphon game​‌ model based on a​​ controlled graphon mean field​​​‌ stochastic differential equation system​ with jumps, which can​‌ be regarded as the​​ limiting case of a​​​‌ finite game dynamic system​ as the number of​‌ players goes to infinity.​​ We examine the case​​​‌ of controlled dynamics, with​ control terms present in​‌ the drift, diffusion, and​​ jump components. We focus​​​‌ on the study of​ Markovian controls and concentrate​‌ on the limit theory.​​ We provide convergence results​​​‌ on the state trajectories​ and their laws, transitioning​‌ from finite game systems​​ to graphon systems. We​​​‌ also study approximate equilibria​ for finite games on​‌ large networks, using the​​ graphon equilibrium as a​​​‌ benchmark. The rates of​ convergence are analyzed under​‌ various underlying graphon models​​ and regularity assumptions.

In​​ 22, we study​​​‌ continuous stochastic graphon games‌ with heterogeneous mean field‌​‌ interactions and jumps. We​​ consider a continuum of​​​‌ players, where each player's‌ dynamics involve graphon mean‌​‌ field interactions and individual​​ jumps induced by a​​​‌ Poisson random measure. We‌ investigate the case when‌​‌ the drift, diffusion, and​​ jump terms of the​​​‌ dynamics of the graphon‌ system are controlled by‌​‌ a stochastic process. The​​ graphon objective function presents​​​‌ non-linear mean field interactions‌ in the running and‌​‌ final rewards. We prove​​ the existence of a​​​‌ relaxed equilibrium of the‌ graphon mean field game‌​‌ via controlled martingale problems.​​ Under convexity assumptions and​​​‌ using measurable selection arguments,‌ strict Markovian equilibria are‌​‌ constructed from relaxed equilibria.​​ Under some additional monotonicity​​​‌ assumptions, we obtain the‌ uniqueness of the graphon‌​‌ equilibrium. We provide an​​ explicit solution for the​​​‌ graphon mean field equilibrium‌ for two simple examples‌​‌ in the linear-quadratic case.​​

8.2.2 Extended Graphon Mean​​​‌ Field Games in Discrete‌ Time

Participants: A. Sulem‌​‌, Z. Cao,​​ H. Amini, K.​​​‌ Shao.

This is‌ an ongoing work in‌​‌ collaboration with Mathieu Laurière​​ (NYU Shangai) and Gökçe​​​‌ Dayanıklı (University of Illinois).‌ In this paper, we‌​‌ study games involving a​​ continuum of heterogeneous players​​​‌ in the discrete-time setting‌ with finite state spaces‌​‌ and continuous action spaces.​​ We introduce a new​​​‌ model that incorporates joint‌ state-action interactions within the‌​‌ graphon-weighted aggregate, described by​​ a coupled forward-backward system.​​​‌ We rigorously establish the‌ existence of solutions to‌​‌ this system and demonstrate​​ a one-to-one correspondence between​​​‌ graphon Nash equilibria (GNE)‌ and the solutions of‌​‌ the forward-backward system. Additionally,​​ we prove the uniqueness​​​‌ of the GNE under‌ various structural assumptions. To‌​‌ illustrate the practical relevance​​ of our framework, we​​​‌ provide an example of‌ optimal investment with a‌​‌ relative performance objective and​​ solve it numerically using​​​‌ fixed-point iterations. See the‌ phD thesis of K.‌​‌ Shao 32.

This​​ is an ongoing work​​​‌ in collaboration with Mathieu‌ Laurière (NYU Shanghai). We‌​‌ develop theoretical and numerical​​ analysis of extended Graphon​​​‌ Mean Field Games (GMFG)‌ in a discrete-time setting.‌​‌ On the theoretical side,​​ we provide rigorous analysis​​​‌ on the existence of‌ approximated Nash equilibrium of‌​‌ the GMFG system by​​ considering joined state-action distribution,​​​‌ we also refined the‌ proof of existence by‌​‌ categorizing pure policies and​​ mixed policies. On the​​​‌ numerical side, we explore‌ some learning schemes (i.e.‌​‌ reinforcement learning) to study​​ graphon mean field equilibrium.​​​‌

8.5.1 Differentiability of optimal​‌ stopping boundaries

Participants: D.​​ Lamberton, T. De​​​‌ Angelis.

D. Lamberton​ and Tiziano De Angelis​‌ (University of Torino) are​​ working on the optimal​​​‌ stopping problem of a​ one dimensional diffusion in​‌ finite horizon. They develop​​ a probabilistic approach to​​​‌ the regularity of the​ associated free boundary problem,​‌ and derive a probabilistic​​ proof of the differentiability​​​‌ of the free boundary​ for the optimal stopping​‌ problem of a one-dimensional​​ diffusion in 59.​​​‌ They have new results​ concerning the second order​‌ mixed derivative of the​​ value function (work in​​​‌ progress).

8.5.2 Dual approach​ for hedging Bermudan options​‌

Participants: A. Alfonsi,​​ J. Lelong, A.​​​‌ Kebaier.

A. Alfonsi,​ J. Lelong and A.​‌ Kebaier develop in 18​​ a numerical method to​​​‌ price and hedge American​ and Bermudean options based​‌ on the dual representation​​ introduced by Rogers (2010).​​​‌ The key idea is​ to rewrite the dual​‌ formula as an excess​​ reward representation and to​​​‌ combine it with a​ strict convexification technique. The​‌ hedging strategy is then​​ obtained by using a​​​‌ Monte Carlo method, solving​ backward a sequence of​‌ least square problems. Convergence​​ results for the algorithm​​​‌ are obtained and tests​ on various Bermudan options​‌ are provided. Beyond giving​​ directly the hedging portfolio,​​​‌ the strength of the​ algorithm is to assess​‌ both the relevance of​​ including financial instruments in​​​‌ the hedging portfolio and​ the effect of the​‌ rebalancing frequency.

8.5.3 Computing​​ XVA for American basket​​ derivatives by machine learning​​​‌ techniques

Participants: A. Zanette‌, L. Goudenege.‌​‌

In 26, L.​​ Goudenège, A. Molent (Univ​​​‌ Udine) and A. Zanette‌ study American option pricing‌​‌ taking into account the​​ default risk. Total value​​​‌ adjustment (XVA) is the‌ change in value to‌​‌ be added to the​​ price of a derivative​​​‌ to account for the‌ bilateral default risk and‌​‌ the funding costs. In​​ this paper, we compute​​​‌ such a premium for‌ American basket derivatives whose‌​‌ payoff depends on multiple​​ underlyings. In particular, in​​​‌ our model, those underlyings‌ are supposed to follow‌​‌ the multidimensional Black-Scholes stochastic​​ model. In order to​​​‌ determine the XVA, we‌ follow the approach introduced‌​‌ by (Burgard and Kjaer​​ in SSRN Electronic J​​​‌ 7:1–19, 2010) and afterward‌ applied by (Arregui et‌​‌ al. in Appl Math​​ Comput 308:31–53, 2017), (Arregui​​​‌ et al. in Int‌ J Comput Math 96:2157–2176,‌​‌ 2019) for the one-dimensional​​ American derivatives. The evaluation​​​‌ of the XVA for‌ basket derivatives is particularly‌​‌ challenging as the presence​​ of several underlings leads​​​‌ to a high-dimensional control‌ problem. We tackle such‌​‌ an obstacle by resorting​​ to Gaussian Process Regression,​​​‌ a machine learning technique‌ that allows one to‌​‌ address the curse of​​ dimensionality effectively. Moreover, the​​​‌ use of numerical techniques,‌ such as control variates,‌​‌ turns out to be​​ a powerful tool to​​​‌ improve the accuracy of‌ the proposed methods. The‌​‌ paper includes the results​​ of several numerical experiments​​​‌ that confirm the goodness‌ of the proposed methodologies.‌​‌

8.6.1​​ Study of the Wasserstein​​​‌ projections in the convex‌ order

Participants: A. Alfonsi‌​‌, B. Jourdain.​​

In 36, A.​​​‌ Alfonsi and B. Jourdain‌ first show continuity of‌​‌ both Wasserstein projections in​​ the convex order when​​​‌ they are unique. They‌ also check that, in‌​‌ arbitrary dimension $d$,​​ the quadratic Wasserstein projection​​​‌ of a probability measure‌ $\mu$ on the set‌​‌ of probability measures dominated​​ by $\nu$ in the​​​‌ convex order is non-expansive‌ in $\mu$ and Hölder‌​‌ continuous with exponent 1/2​​ in $\nu$. When​​​‌ $\mu$ and $\nu$ are‌ Gaussian, they show that‌​‌ this projection is Gaussian​​ and also consider the​​​‌ quadratic Wasserstein projection of‌ $\nu$ on the set‌​‌ of probability measures dominating​​ $\mu$ in the convex​​​‌ order. In the case‌ when $d\ge \mathrm{2‌‌}$ and $\nu$ is not​​ absolutely continuous with respect​​​‌ to the Lebesgue measure‌ where uniqueness of the‌​‌ latter projection was not​​ known, they check that​​​‌ there is always a‌ unique Gaussian projection and‌​‌ characterize when non Gaussian​​ projections with the same​​​‌ covariance matrix also exist.‌ Still for Gaussian distributions,‌​‌ they characterize the covariance​​ matrices of the two​​​‌ projections. It turns out‌ that there exists an‌​‌ orthogonal transformation of space​​ under which the computations​​​‌ are similar to the‌ easy case when the‌​‌ covariance matrices of $\mathrm{\mu }$ and $\nu$ are diagonal.​​​‌

8.6.2 Convex comparison of‌ Gaussian mixtures

Participants: B.‌​‌ Jourdain, G. Pagès​​.

Motivated by the​​​‌ study of the propagation‌ of convexity by semi-groups‌​‌ of stochastic differential equations​​​‌ and convex comparison between​ the distributions of solutions​‌ of two such equations,​​ G. Pagès (LPSM) and​​​‌ B. Jourdain study the​ comparison for the convex​‌ order between a Gaussian​​ distribution and a Gaussian​​​‌ mixture. We give and​ discuss intrinsic necessary and​‌ sufficient conditions for convex​​ ordering. On the examples​​​‌ that we have worked​ out, the two conditions​‌ appear to be closely​​ related (see 27).​​​‌

8.6.3 Quadratic Wasserstein distance​ between Gaussian laws revisited​‌ with correlation

Participants: A.​​ Alfonsi, B. Jourdain​​​‌.

In 35,​ A. Alfonsi and B.​‌ Jourdain give a simple​​ derivation of the formula​​​‌ obtained in the eighties​ for the quadratic Wasserstein​‌ distance between two Gaussian​​ distributions on ${𝐑}^{\mathrm{d‌}}$ with respective covariance matrices​ ${\Sigma }_{\mu }$ and ${\mathrm{\Sigma ‌}}_{\nu }$. This derivation​​ relies on the existence​​​‌ of an orthogonal matrix​ $O$ such that ${\mathrm{O‌}}^{*}{\Sigma }_{\mu }\mathrm{O}$ and $O^{*}{\mathrm{\Sigma ‌}}_{\nu }O$ share the​ same correlation matrix and​‌ on the simplicity of​​ optimal couplings in the​​​‌ case with the same​ correlation matrix and therefore​‌ the same copula.

8.6.4​​ Martingale Optimal Transport

Participants:​​​‌ B. Jourdain, K.​ Shao.

In 29​‌, B. Jourdain and​​ K. Shao investigate non-decreasing​​​‌ martingale couplings. See also​ the phD thesis of​‌ K. Shao 32.​​

8.8.1 Nonlinear weak error​​​‌ expansion of McKean-Vlasov Stochastic‌ differential equations

Participants: B.‌​‌ Jourdain, A. Lê​​.

According to Talay​​​‌ and Tubaro, the weak‌ error between the solution‌​‌ to a stochastic differential​​ equation with smooth coefficients​​​‌ and its Euler-Maruyama scheme‌ can be expanded in‌​‌ powers of the time-step.​​ In 46, B.​​​‌ Jourdain and A.-D. Lê‌ generalize this result to‌​‌ the case when the​​ error is measured by​​​‌ a smooth functional on‌ the Wasserstein space of‌​‌ probability measures in place​​ of the linear functional​​​‌ given by the expectation‌ of a smooth function.‌​‌ Since this does not​​ complicate their analysis based​​​‌ on the master partial‌ differential equation, they even‌​‌ deal with the McKean-Vlasov​​ case when the coefficients​​​‌ of the stochastic differential‌ equation may depend on‌​‌ its current marginal distribution.​​

8.8.2 Stochastic Volterra Equations​​​‌

Participants: A. Alfonsi,‌ E. Abi Jaber,‌​‌ G. Szulda, B.​​ Jourdain, G. Pagès​​​‌.

8.8.3 An abstract framework​‌ for the approximation of​​ the invariant measure

Participants:​​​‌ A. Alfonsi, V.​ Bally, A. Kohatsu​‌ Higa.

In collaboration​​ with A. Kohatsu Higa​​​‌ (Ritsumeikan University, Japan), we​ establish a general framework​‌ to study the rate​​ of convergence of a​​​‌ Euler type approximation scheme​ with decreasing time steps​‌ to the invari- ant​​ measure, for a general​​​‌ class of stochastic systems​ (see 34). The​‌ error is measured in​​ general Wasserstein distances, which​​​‌ enables to encompass cases​ with non global contractivity​‌ conditions. Our main assumption​​ is a coupling property​​​‌ which is expressed in​ terms of the one-step​‌ approximation. We show that​​ the proposed set-up can​​​‌ be applied to a​ wide range of equations​‌ that may be law​​ dependent, such as Langevin​​ equations, reflected equations, Boltzmann​​​‌ type equations and for‌ a recent McKean Vlasov‌​‌ type model for neuronal​​ activity.

8.8.4 Sewing Lemma​​​‌

Participants: A. Alfonsi,‌ V. Bally, L.‌​‌ Caramellino.

A. Alfonsi​​ and V. Bally have​​​‌ proposed a new approach‌ based on the sewing‌​‌ lemma on the Wasserstein​​ Space to study existence​​​‌ and uniqueness of solutions‌ of the Boltzmann equation‌​‌ 48. They are​​ now working in 17​​​‌ with L. Caramellino (Roma‌ Univ) to extend their‌​‌ results by using the​​ stochastic sewing lemma recently​​​‌ proposed by Khoa Lê‌ (2020).

10.2.1​​​‌ Visits of international scientists‌

Participants: A. Zanette,‌​‌ H. Amini, Z.​​ Cao.

• A. Zanette,​​​‌ Univ of Udine 4‌ August to September 4‌​‌ to supervize Premia software​​ development
• H. Amini, University​​​‌ of Florida, Collaboratio with‌ A. Sulem and E.‌​‌ DEvey
• Z. Cao, Shanghai​​ University, Collaboration with A.​​​‌ Sulem

11.1.1 Scientific events: organisation​​

11.1.2​​ Journal

11.1.3​ Talks in Conferences and​‌ Workshops

• A. Alfonsi
  • 15​​ 07 2025: “Stochastic Volterra​​​‌ Equations on Convex Domains”,​ SIAM Conference on Financial​‌ Mathematics and Engineering, Miami.​​
  • 20 11 2025: “Wasserstein​​​‌ projections in the convex​ order: regularity and characterization​‌ in the quadratic Gaussian​​ case”, Workshop Geometry, duality​​​‌ and convexity in new​ OT problems, Orsay.
  • 03​‌ 12 2025: “Stochastic Volterra​​ Equations on Convex Domains”,​​​‌ SIAM Conference on Financial​ Mathematics and Engineering, Berlin​‌ Probability Colloquium.
• E. Devey​​
  • Ceremade YRD (Young Researcher​​​‌ Days), 03/06-05/06, Caen (France)​
  • PGMO Days, 18/11-19/11, EDF​‌ Saclay (France)
• J. Guyon​​

  • Conferences and workshops:

    -​​​‌ Research In Options 2025,​ Fundação Getulio Vargas, Rio​‌ de Janeiro, December 2025.​​

    - Workshop on rough​​​‌ volatility, Weierstrass Institute, Berlin,​ November 2025.

    - QuantMinds​‌ 2025, London, November 2025.​​

    - CBOE RMC Quant​​​‌ Conference (keynote speaker), Chicago,​ October 2025.

    - WBS​‌ 21st Quantitative Finance Conference,​​ Palermo, September 2025.

    -​​​‌ Advances in Mathematics of​ Randomness for Handling Risks​‌ in Finance and Insurance,​​ Marseille, September 2025.

    -​​​‌ SIAM Conference on Financial​ Mathematics and Engineering (plenary​‌ speaker), Miami, July 2025.​​

    - VCMF 2025, Vienna,​​​‌ July 2025.

    - AMaMeF​ 2025, Verona, June 2025.​‌

    - MathSport International 2025,​​ Luxembourg, June 2025.

    -​​​‌ AFMathConf2025, Brussels, February 2025.​

  • Seminars

    - Bloomberg, New​‌ York, Keynote speaker at​​ BBQ (Bloomberg Quant Seminar),​​​‌ October 2025.

    - Cornell​ Financial Engineering Manhattan, New​‌ York, CFEM & UBS​​ AI & Data research​​ seminar, October 2025.

    -​​​‌ University of California, Berkeley,‌ IEOR Mathematical Finance Seminar,‌​‌ May 2025.

    - Columbia​​ University, New York, Columbia​​​‌ Mathematical Finance Seminar Series,‌ March 2025.

• B. Jourdain‌​‌
  • LPSM working group on​​ Financial and actuarial mathematics,​​​‌ numerical probability, 27 november‌ 2025 : Wasserstein projections‌​‌ in the convex order​​
  • Workshop Optimal transport :​​​‌ stochastics, projections, and applications,‌ The Fields Institute Toronto,‌​‌ 5 november 2025 :​​ Wasserstein projections in the​​​‌ convex order
  • SINEQ Final‌ Conference, GSSI L'Aquila, 23‌​‌ october 2025 : Central​​ Limit Theorem for the​​​‌ Stratified Resampling Scheme
  • Futures‌ of Quantitative Finance Seminar,‌​‌ 24 september 2025 :​​ Implied volatility (also) is​​​‌ path-dependent
  • Diffusions in Warsaw,‌ University of Warsaw, 11‌​‌ september 2025 : Convexity​​ propagation and convex ordering​​​‌ of one-dimensional stochastic differential‌ equations
  • Applied Probability and‌​‌ Statistics seminar, TU Graz,​​ 10 april 2025 :​​​‌ Convex comparison of Gaussian‌ mixtures
  • Bachelier seminar, 4‌​‌ april 2025 : Convex​​ comparison of Gaussian mixtures​​​‌
• A. Sulem
  • Plenary speaker,‌ Conference in honor of‌​‌ Alain Bensoussan for his​​ 85th birthday, 06/2025, Shandong​​​‌ University: "Stochastic Graphon Mean-Field‌ Games with Jumps and‌​‌ associated equilibria".
  • Plenary speaker,​​ Conference in honor of​​​‌ B. Øksendal for his‌ 80 birthday, 09/2025, Norwegian‌​‌ Academy of Sciences, Oslo,​​ "Stochastic Graphon Mean-Field Games​​​‌ with Jumps and approximate‌ Nash equilibria of large‌​‌ network games."

11.1.4 Scientific​​ expertise

• A. Alfonsi

  Member​​​‌ of the council of‌ the Bachelier Finance Society‌​‌

11.1.5 Research administration

• J.​​ Guyon

  head of the​​​‌ Applied Probability team at‌ CERMICS, Ecole des Ponts‌​‌

• B. Jourdain

  Deputy head​​ of the federation Bézout​​​‌

• A. Sulem

  Member of‌ the Scientific Committee of‌​‌ AMIES( Agence pour​​ les Mathématiques en Interaction​​​‌ avec l'Entreprise et la‌ Société)

11.1.6 Academic responsabilities‌​‌

• A. Alfonsi

  - In​​ charge of the Master​​​‌ “Finance and Data” at‌ the Ecole des Ponts‌​‌ (until Sept. 2025).

  -Representant​​ of the Master “Probabilité​​​‌ et Finance” at Ecole‌ des Ponts.

11.2.1 Teaching

• A.‌ Alfonsi
  • “Probabilités”, first year‌​‌ course at the Ecole​​ des Ponts.
  • “Données Haute​​​‌ Fréquence en finance”, lecture‌ for the Master at‌​‌ UPEMLV.
  • “Mesures de risque”,​​ Master course of UPEMLV​​​‌ and Sorbonne Université.
  • Professeur‌ chargé de cours at‌​‌ Ecole Polytechnique.

• B. Jourdain​​

  - course "Probability theory",​​​‌ 1st year ENPC

  -‌ course "Mathematical finance", 2nd‌​‌ year ENPC

  - course​​ "Monte-Carlo methods", 3rd year​​​‌ ENPC and Research Master‌ Mathématiques et Application, University‌​‌ Gustave Eiffel

  - course​​ "Monte-Carlo Markov chain methods​​​‌ and particle algorithms", Research‌ Master Probabilités et Modèles‌​‌ Aléatoires, Sorbonne Université

  -​​ course "Machine Learning 1",​​​‌ MSC Data Science for‌ Business, X-HEC

  - course‌​‌ "Randomness", 1st year Ecole​​ Polytechnique

• J. Guyon

  -​​​‌ course "Probability Theory", 1st‌ year ENPC. Lead teacher.‌​‌

  - course "Volatility Modeling",​​ Master Mathematics for Finance​​​‌ and Data (MFD), 3rd‌ year ENPC - UGE‌​‌

  - course "Advanced calibration​​ methods and VIX derivatives",​​​‌ joint lecture of the‌ BNP Paribas chair Futures‌​‌ of Quantitative Finance, Master​​ Probabilités et Finance, Master​​​‌ M2MO, Master MFD (Sorbonne‌ Université, Université Paris Cité,‌​‌ and ENPC-UGE)

  - J.​​​‌ Guyon, B. Liang :​ course "Nonlinear Option Pricing",​‌ Master MAFN, Columbia University​​

  - course "Volatility Modeling",​​​‌ Master of Science in​ Financial Engineering, NYU

• A.​‌ Sulem

  Master of Mathematics,​​ Université du Luxembourg, Responsible​​​‌ of the course on​ "Numerical Methods in Finance",​‌ and lectures (22 hours)​​

11.2.2 Supervision

• Postdoral fellows​​​‌
  • Léo Parent (supervision :​ J. Guyon)
• PhD in​‌ progress
  • Elise Devey (started​​ October 2023), "Graphon Mean-Field​​​‌ Games and Renewable Energy​ Systems", Supervisor: Agnès Sulem,​‌ INRIA doctoral grant
  • Thibault​​ Jeannin (started in november​​​‌ 2024) "Calibration of pure​ path-dependent volatility models", supervised​‌ by J. Guyon and​​ B. Jourdain
  • Arthur Bourdon​​​‌ (started in november 2024)​ "Approximation and explication of​‌ insurance valuation computations by​​ Artificial Intelligence", supervised by​​​‌ B. Jourdain
  • Grégoire Ounnoughene​ (Oct 2025 - ),​‌ "Uncertain path-dependent volatility models"​​ (co-supervison: J. Guyon with​​​‌ A. Alfonsi and G.​ Loeper), CIFRE PhD thesis​‌ with BNP Paribas
  • Rémi​​ Surat (Oct 2025 -​​​‌ ), "Generative modeling of​ financial time series" (co-supervised​‌ J. Guyon with L.​​ Pillaud-Vivien and G. Loeper),​​​‌ CIFRE PhD thesis with​ BNP Paribas
  • Faten Ben​‌ Said (CIFRE EDF, co-advisor:​​ Julien Reygner), “Caractérisation et​​​‌ prise en compte des​ dépendances statistiques dans le​‌ cadre d'applications de dynamique​​ sédimentaire”, started in March​​​‌ 2023.
  • François Escolan (co-advisors:​ A. Alfonsi, Virginie Ehrlacher​‌ and Julien Reygner), “Mean-field​​ limit of stochastic particle​​​‌ systems on manifolds”, started​ in November 2024.
• PhD​‌ defended
  • Kexin Shao, "Martingale​​ optimal transport and Graphon​​​‌ mean-field games", supervised by​ A. Sulem and B.​‌ Jourdain, defended on March​​ 27 2025
  • Edoardo Lombardo​​​‌ (International PhD, co-advisors: A.​ Alfonsi and Lucia Caramellino,​‌ Tor Vergata, Roma), “High​​ order numerical approximation for​​​‌ some singular stochastic processes​ and related PDEs”, defended​‌ on January 22, 2025,​​ ENPC.
• Internship
  • Hassene Kallala,​​​‌ 2nd year, IMI Dept,​ ENPC (June-August 2025), quintic​‌ OU stochastic volatility model,​​ supervision: J. Guyon
  • Maxence​​​‌ Caucheteux (June to August​ 2025): “Espace d’état pour​‌ le processus markovien multifactoriel​​ associé à un processus​​​‌ de Volterra”. supervision: A.​ Alfonsi
  • Kevin Aoun, ENSTA,​‌ 26/05/2025 - 28/07/2025: implementation​​ of deep pricing algorithms​​​‌ in the Premia software;​ supervision: L. Goudenege.
  • Hassen​‌ Ben Jemaa, Ecole Polytechnique​​ de Tunisie, " Pricing​​​‌ of Bernudean Options in​ high dimensions using deep​‌ learning and high order​​ weak approximation", March 1-​​​‌ July 31 2025,
  • Mohamed​ Ben Saada, "Deep learning​‌ algorithms for Linear Quadratic​​ Mean-field games with common​​​‌ noise", Ecole Polytechnique de​ Tunisie, March 1- July​‌ 31 2025, Supervision: A.​​ Kebaier.
  • Wissai Haouami, "implementation​​​‌ of recent algorithms of​ option pricing using neural​‌ networks and implementation in​​ the Premia software", ENSTA,​​​‌ June - September 2025,​ supervision: L. Goudenege.

• Project​‌ supervision

  - Project of​​ the ENPC course TDLOG​​​‌ (2024-25): Draw simulator, league​ phase of the New​‌ UEFA Champions League Format,​​ supervision: J. Guyon

  -​​​‌ Project of the ENPC​ course TDLOG (2025-26): Simulation​‌ of the New UEFA​​ Champions League Format, supervision:​​​‌ J. Guyon

  - Project​ of the ENPC course​‌ TDLOG (2025-26): Luck index,​​ supervision: J. Guyon

  -​​​‌ IMI Department Project (2024-25):​ Simulation of the New​‌ UEFA Champions League Format,​​ importance of a game,​​ supervision: J. Guyon

11.2.3​​​‌ Juries

• A. Alfonsi
  • President‌ of the jury for‌​‌ the PhD thesis of​​ Natascha Hey “Trading with​​​‌ concave (cross-) impact”.
  • Examiner‌ of the PhD thesis‌​‌ of Alexis Houssard “Some​​ aspects of model risk​​​‌ management using singular stochastic‌ control and model-free approaches”.‌​‌

• J. Guyon

  PhD thesis​​ examination of Jules Delemotte​​​‌ (Ecole Polytechnique, December 10,‌ 2025): Smile dynamics, rough‌​‌ volatility, volatility with memory​​

• B. Jourdain
  • Reviewer for​​​‌ the PhD of Paul‌ Maurer defended on November‌​‌ 14 2025, University Côte​​ d'Azur
• A. Sulem
  • President​​​‌ of the committee for‌ the PhD thesis examination‌​‌ of Thomas Le Corre,​​ Distributed control of flexible​​​‌ loads in power grids‌, PSL ENS, 29/10/2025‌​‌
  • PhD thesis examination of​​ Anna De Crescenzo, "Heterogeneous​​​‌ mean-field systems", Université‌ Paris-Cité, 13/10/2025
  • PhD thesis‌​‌ examination of Yadh Hafsi,​​ " Inference and Control​​​‌ of Liquidity Risk "‌, Université Paris Saclay,‌​‌ President of the committee,​​ 09/09/2025
  • Member of the​​​‌ recruitment committee for n‌ assistant professor position in‌​‌ Applied Mathematics, " Mathematics​​ for Economy, Finance and​​​‌ game theory ", Université‌ Paris-Dauphine, Spring 2025.
  • Member‌​‌ of the committee for​​ the tenure of Eduardo​​​‌ Abi Jaber, Ecole Polytechnique,‌ 28 May 2025

11.3.1 Productions (articles,​​ videos, podcasts, serious games,​​​‌ ...)

Participants: J. Guyon‌.

J. Guyon has‌​‌ a strong expertise in​​ quantitative analysis in football​​​‌ (soccer). His main reserach‌ interest is fairness in‌​‌ sports, in particular the​​ fairness and efficiency of​​​‌ competition formats, ranking systems,‌ seeding systems, draw procedures,‌​‌ and match schedules 43​​.

• J. Guyon, cited​​​‌ in The Times for‌ his work on the‌​‌ 2026 FIFA World Cup​​ draw probabilities, December 5,​​​‌ 2025
• J. Guyon, Interview‌ in the French sports‌​‌ daily , L'Equipe,​​ December 5, 2025
• J.​​​‌ Guyon: Interview in the‌ French sports daily L'Equipe‌​‌, December 11, 2025​​
• Live intervention on La​​​‌ chaîne L’Équipe in the‌ TV program L’Équipe du‌​‌ soir, January 20, 2025.​​ L'Equipe du soir
• Interview​​​‌ in the French newspaper‌ Le Télégramme, January 21,‌​‌ 2025. Le Télégramme
• Interview​​ in the French newspaper​​​‌ L’Équipe, January 22, 2025.‌ L'Equipe
• Live intervention on‌​‌ La chaîne L’Équipe in​​ the TV program L’Équipe​​​‌ du soir, January 22,‌ 2025. L'Equipe du soir‌​‌
• Interview in the French​​ newspaper L’Équipe, January 23,​​​‌ 2025. L'Equipe
• Interview in‌ the French newspaper Le‌​‌ Télégramme, January 28, 2025.​​ Le Télégramme
• Interview in​​​‌ the French newspaper L’Équipe,‌ January 30, 2025. L'Equipe‌​‌

International journals​‌

• 17 articleA.Aurélien​​ Alfonsi, V.Vlad​​ Bally and L.Lucia​​​‌ Caramellino. Stochastic sewing‌ lemma on Wasserstein space‌​‌.Electronic Journal of​​ Probability30noneJanuary​​​‌ 2025HALDOIback‌ to text
• 18 article‌​‌A.Aurélien Alfonsi,​​ A.Ahmed Kebaier and​​​‌ J.Jérôme Lelong.‌ A pure dual approach‌​‌ for hedging Bermudan options​​.Mathematical Finance35​​​‌4October 2025,‌ 745-759HALDOIback‌​‌ to text
• 19 article​​A.Aurélien Alfonsi and​​​‌ G.Guillaume Szulda.‌ On non-negative solutions of‌​‌ stochastic Volterra equations with​​ jumps and non-Lipschitz coefficients​​​‌.Bernoulli314‌November 2025HALDOI‌​‌back to text
• 20​​ articleH.Hamed Amini​​​‌, Z.Zhongyuan Cao‌ and A.Agnès Sulem‌​‌. Fire Sales, Default​​ Cascades and Complex Financial​​​‌ Networks.Mathematics and‌ Financial Economics192‌​‌April 2025, 225-260​​HALDOIback to​​​‌ text
• 21 articleH.‌Hamed Amini, Z.‌​‌Zhongyuan Cao and A.​​Agnès Sulem. Graphon​​​‌ Mean-Field Backward Stochastic Differential‌ Equations With Jumps and‌​‌ Associated Dynamic Risk Measures​​.Finance and Stochastics​​​‌2025. In press.‌ HALDOIback to‌​‌ text
• 22 articleH.​​Hamed Amini, Z.​​​‌Zhongyuan Cao and A.‌Agnès Sulem. Markovian‌​‌ Equilibria of Stochastic Graphon​​ Games with Jumps.​​​‌Pure and Applied Functional‌ AnalysisSpecial Issue on‌​‌ Systems Theory, Control and​​ PDE dedicated to Professor​​​‌ Alain Bensoussan2025.‌ In press. HALDOI‌​‌back to text
• 23​​ articleH.Hamed Amini​​​‌, Z.Zhongyuan Cao‌ and A.Agnes Sulem‌​‌. Stochastic Graphon Mean​​ Field Games with Jumps​​​‌ and Approximate Nash Equilibria‌.SIAM Journal on‌​‌ Control and Optimization2026​​. In press. HAL​​​‌DOIback to text‌
• 24 articleH.Hervé‌​‌ Andrès and B.Benjamin​​ Jourdain. Existence, uniqueness​​​‌ and positivity of solutions‌ to the Guyon-Lekeufack path-dependent‌​‌ volatility model with general​​ kernels.International Journal​​​‌ of Theoretical and Applied‌ FinanceDecember 2025HAL‌​‌DOI
• 25 articleG.​​Guido Gazzani and J.​​​‌Julien Guyon. Pricing‌ and Calibration in the‌​‌ 4-Factor Path-Dependent Volatility Model​​.Quantitative Finance25​​​‌32025, 471-489‌HALDOIback to‌​‌ text
• 26 articleL.​​Ludovic Goudenège, A.​​​‌Andrea Molent and A.‌Antonino Zanette. Computing‌​‌ XVA for American basket​​ derivatives by machine learning​​​‌ techniques.Computational Management‌ Science22August 2025‌​‌, 541 - 569​​HALDOIback to​​​‌ text
• 27 articleB.‌Benjamin Jourdain and G.‌​‌Gilles Pagès. Convex​​ comparison of Gaussian mixtures​​​‌.Journal of Multivariate‌ Analysis2092025,‌​‌ 105448HALback to​​ text
• 28 articleB.​​​‌Benjamin Jourdain and G.‌Gilles Pagès. Convex‌​‌ ordering for stochastic Volterra​​ equations and their Euler​​​‌ schemes.Finance and‌ Stochastics292025,‌​‌ 1-62HALback to​​ text
• 29 articleB.​​​‌Benjamin Jourdain and K.‌Kexin Shao. Non-decreasing‌​‌ martingale couplings.ESAIM:​​ Probability and Statistics29​​​‌January 2025, 1-44‌HALDOIback to‌​‌ text

International peer-reviewed conferences​​

• 30 inproceedingsM.Margherita​​​‌ Castellano, L.Ludovic‌ Goudenège and F.Flore‌​‌ Nabet. Energy estimate​​​‌ of a Discrete Duality​ Finite Volume scheme for​‌ a phase-field model with​​ surfactants.DD29 -​​​‌ 29th International Conference on​ Domain Decomposition MethodsMilan,​‌ ItalyJune 2025HAL​​

Doctoral dissertations and habilitation​​​‌ theses

• 31 thesisE.​Edoardo Lombardo. Approximation​‌ and regularity results for​​ the Heston model and​​​‌ related processes.École​ des Ponts ParisTech; Università​‌ degli studi di Roma​​ "Tor Vergata" (1972-....)January​​​‌ 2025HAL
• 32 thesis​K.Kexin Shao.​‌ Martingale optimal transport and​​ graphon mean fieldgames.​​​‌Université Paris sciences et​ lettresMarch 2025HAL​‌back to textback​​ to text

Reports &​​​‌ preprints

• 33 miscE.​Eduardo Abi Jaber,​‌ A.Aurélien Alfonsi and​​ G.Guillaume Szulda.​​​‌ Weak solutions of stochastic​ volterra equations in convex​‌ domains with general kernels​​.June 2025HAL​​​‌DOIback to text​
• 34 miscA.Aurélien​‌ Alfonsi, V.Vlad​​ Bally and A.Arturo​​​‌ Kohatsu-Higa. Euler-type approximation​ for the invariant measure:​‌ An abstract framework.​​September 2025HALback​​​‌ to text
• 35 misc​A.Aurélien Alfonsi and​‌ B.Benjamin Jourdain.​​ Quadratic Wasserstein distance between​​​‌ Gaussian laws revisited with​ correlation.December 2025​‌HALback to text​​
• 36 miscA.Aurélien​​​‌ Alfonsi and B.Benjamin​ Jourdain. Wasserstein projections​‌ in the convex order:​​ regularity and characterization in​​​‌ the quadratic Gaussian case​.December 2025HAL​‌back to text
• 37​​ misc A.Aurélien Alfonsi​​​‌, A.Ahmed Kebaier​ and J.Jérôme Lelong​‌. How can the​​ dual martingale help solving​​​‌ the primal optimal stopping​ problem? February 2026 HAL​‌
• 38 miscH.Hervé​​ Andrès, A.Alexandre​​​‌ Boumezoued and B.Benjamin​ Jourdain. The implied​‌ volatility surface (also) is​​ path-dependent.October 2025​​​‌HALback to text​
• 39 miscF.Faten​‌ Ben Said, A.​​Aurélien Alfonsi, A.​​​‌Anne Dutfoy, C.​Cédric Goeury, M.​‌Magali Jodeau, J.​​Julien Reygner and F.​​​‌Fabrice Zaoui. A​ tree-based Polynomial Chaos expansion​‌ for surrogate modeling and​​ sensitivity analysis of complex​​​‌ numerical models.September​ 2025HAL
• 40 misc​‌A.Arthur Bourdon,​​ B.Benjamin Jourdain and​​​‌ H.Hervé Andrès.​ Linear independence properties of​‌ the signature components of​​ time-augmented stochastic processes.​​​‌January 2026HAL
• 41​ miscE.Elise Devey​‌. Approximately optimal distributed​​ controls for high-dimensional stochastic​​​‌ systems with pairwise interaction​ through controls.October​‌ 2025HALback to​​ text
• 42 miscP.​​​‌ K.Peter K. Friz​, B.Benjamin Jourdain​‌, T.Thomas Wagenhofer​​ and A.Alexandre Zhou​​​‌. On the Weak​ Error for Local Stochastic​‌ Volatility Models.August​​ 2025HALback to​​​‌ text
• 43 miscJ.​Julien Guyon, A.​‌Adle Ben Salem,​​ T.Thomas Buchholtzer and​​​‌ M.Mathieu Tanré.​ Drawing and Scheduling the​‌ UEFA Champions League League​​ Phase.2025HAL​​​‌DOIback to text​
• 44 miscJ.Julien​‌ Guyon, T.Thibault​​ Jeannin and B.Benjamin​​​‌ Jourdain. On the​ surjectivity of the conditional​‌ expectation given a real​​ random variable.August​​ 2025HALback to​​​‌ text
• 45 miscJ.‌Julien Guyon and L.‌​‌Léo Parent. The​​ Discrete-Time 4-Factor Path-Dependent Volatility​​​‌ Model: Calibration under P‌ and Q.2025‌​‌HALDOIback to​​ text
• 46 miscB.​​​‌Benjamin Jourdain and A.-D.‌Anh-Dung Le. Nonlinear‌​‌ weak error expansion of​​ McKean-Vlasov stochastic differential equations​​​‌.November 2025HAL‌back to text