2023Activity reportProject-TeamMATHRISK

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 87, 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 79. 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 77. 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 99; (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 101). We also established some qualitative properties of the value function (monotonicity, strict convexity, smoothness) 100. (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 49 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   50, 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.  52 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  51 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 $M\approx 10$.

3.5.2 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 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 91, 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 92, 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 93, 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 70, 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 21, 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 27, 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.

3.5.4 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 24, 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 102, 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 84, 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 86, 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 85, 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 25 has been delivered to the Consortium on September 29 2023. It contains the following new implemented algorithms.

  I. Machine Learning algorithms and Risk Management:

  • Optimal Stopping via Randomized Neural Networks. C.Herrera, F.Krach, P.Ruyssen, J.Teichmann • Deep Learning-Based Least Square Forward-Backward Stochastic Differential Equation Solver for High-Dimensional Derivative Pricing. J.Liang Z.Xu P.Li Quantitative Finance, 21-8, 2021. • The Deep Parametric PDE Method: Application to Option Pricing. K.Glau L.Wunderlich Applied Mathematics and Computation, 432, 2022. • Computing XVA for American basket derivatives by Machine Learning techniques. L.Goudenege A.Molent A.Zanette • Backward Hedging for American Options with Transaction Costs. L.Goudenge A.Molent A.Zanette • Pricing high-dimensional American options by kernel ridge regression. W.Hu T.Zastawniak Quantitative Finance, 20-5, 2020. • KrigHedge: Gaussian Process Surrogates for Delta Hedging. M.Ludkovski Y.Saporito Applied Mathematical Finance, 28-4, 2021.

  II. Advanced numerical methods for Equity Derivatives:

  • Option Pricing using Quantum Computers. N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, S. Woerner, Quantum, 4-291, 2020. • The interpolated drift implicit Euler scheme Multilevel Monte Carlo method for pricing Barrier options and applications to the CIR and CEV models. M. Ben Derouich, A. Kebaier • Hybrid multifactor scheme for stochastic Volterra equations. S. E. Rømer • On the discrete-time simulation of the rough Heston model. A.Richard, X.Tan, F.Yang Siam Journal of Financial Mathematics, 14-1, 2023. • The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘ex- pensive’ distributions. L. A. Grzelak, J. A. S. Witteveen, M. Suarez-Taboada, C. W. Oosterlee Quantitative Finance, 19-2, 2019

• 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:
  Ecole des Ponts ParisTech, Université d'Udine

Default cascades in sparse heterogeneous financial networks.

A. Sulem, H. Amini, and their PhD student Z. Cao have studied 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 20. 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 57, 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.

Central Limit Theorems for Price-Mediated Contagion in Stochastic Financial Networks

In 55, we provide central limit theorems to analyze the combined effects of fire sales and default cascades on systemic risk within stochastic financial networks. The impact of prices is modeled through a specifically defined inverse demand function. Our study presents various limit theorems that delve into the dynamics of total shares sold and the equilibrium pricing of illiquid assets in a streamlined fire sales context. We show that the equilibrium prices of these assets demonstrate asymptotically Gaussian fluctuations. In our numerical experiments, we demonstrate how our central limit theorems can be applied to construct confidence intervals for the magnitude of contagion and the extent of losses due to fire sales.

Ruin Probabilities for Risk Processes in Stochastic Networks

In 56, We study multidimensional CramẂe study multidimensional Cramér-Lundberg risk processes where agents, located on a large sparse network, receive losses from their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.

8.2.1 Mean-field (Graphon) Backward Stochastic Differential Equations and systemic risk measures

Participants: A. Sulem, R. Chen, A. Minca, R. Dumitrescu, Z. Cao, H. Amini.

Agnès Sulem, Rui Chen, Andreea Minca, Roxana Dumitrescu have studied mean-field BSDEs with a generalized mean-field operator which can capture system influence with higher order interactions such as those occurring in an inhomogeneous random graph.

We interpret the BSDE solution as a dynamic global risk measure for a representative bank whose risk attitude is influenced by the system. This influence can come in a wide class of choices, including the average system state or average intensity of system interactions 22.

This opens the path towards using dynamic risk measures induced by mean-field BSDE as a complementary approach to systemic risk measurement.

Extensions to Graphon BSDEs with jumps are studied by H. Amini, A. Sulem, and their PhD student Z. Cao in 58. The use of graphons has emerged recently in order to analyze heterogeneous interaction in mean-field systems and game theory. Existence, uniqueness and stability of solutions under some regularity assumptions are established. We also prove convergence results for interacting mean-field particle systems with inhomogeneous interactions to graphon mean-field BSDE systems.

8.2.2 Stochastic Graphon Mean-field Games and approximate Nash Equilibria

Participants: A. Sulem, Z. Cao, H. Amini.

In 59, we study continuous stochastic games with inhomogeneous mean field interactions on large networks and explore their graphon limits. We consider a model with a continuum of players, where each player's dynamics involve not only mean field interactions but also individual jumps induced by a Poisson random measure. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We introduce the graphon game model based on a graphon controlled stochastic differential equation system with jumps, which can be regarded as the limiting case of a finite game's dynamic system as the number of players goes to infinity. Under some general assumptions, we establish the existence and uniqueness of Markovian graphon equilibria. We then 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.

8.2.3 Reinforcement Learning for Graphon Mean-Field Games

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

This is an ongoing work in collaboration with Mathieu Laurière (NYU Shangai). 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.

8.10.1 Approximations of Stochastic Differential Equations (SDEs)

8.10.2 Central limit theorem for the stratified resampling mechanism

In 41, R. Flenghi and B. Jourdain prove the joint convergence in distribution of q variables modulo one obtained as partial sums of a sequence of i.i.d. square integrable random variables multiplied by a common factor given by some function of an empirical mean of the same sequence. The limit is uniformy distributed over ${[0,1]}^q$. To deal with the coupling introduced by the common factor, we assume that the joint distribution of the random variables has a non zero component absolutely continuous with respect to the Lebesgue measure, so that the convergence in the central limit theorem for this sequence holds in total variation distance. While this result provides a generalization of Benford's law to a data adapted mantissa, the main motivation is the derivation of a central limit theorem for the stratified resampling mechanism.

The stratified resampling mechanism is one of the resampling schemes commonly used in the resampling steps of particle filters. In 40, R. Flenghi and B. Jourdain prove a central limit theorem for this mechanism under the assumption that the initial positions are independent and identically distributed and the weights proportional to a positive function of the positions such that the image of their common distribution by this function has a non zero component absolutely continuous with respect to the Lebesgue measure. This result relies on the convergence in distribution of the fractional part of partial sums of the normalized weights to some random variable uniformly distributed on $[0,1]$, which is established in hal-04338337. Under the conjecture that a similar convergence in distribution remains valid at the next steps of a particle filter which alternates selections according to the stratified resampling mechanism and mutations according to Markov kernels, they provide an inductive formula for the asymptotic variance of the resampled population after $n$ steps. They perform numerical experiments which support the validity of this formula.

8.10.3 Abstract Malliavin calculus and convergence in total variation

In 68, 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.

They approximate the invarient measure of a Markov process, solution of a stochastic equation with jumps by using a Euler scheme with decreasing step introduced by D Lamberton and G Pages in the early 2000 in the case of diffusion processes driven by Brownian motion. The novelty here is that They deal with jump processes. Under appropiate non degeneracy hypothesis, they have estimated the error in total variation distance and also proved convergence of the density functions.

A. Alfonsi, V. Bally and A Kohatzu Higa (Ritzumikan University) are working on a continuation of the above mentioned work on the approximation of the invarient measure for some non linear stochastic differential equations of Mc Kean Vlasov and Bozmann type.

8.10.4 Numerical approximation of American/Bermudean options

A. Alfonsi, J. Lelong and A. Kebaier are working on a numerical method to price American options based on the dual representation introduced by Rogers (2002).

8.10.5 Sewing Lemma

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 16. They are now working with L. Caramellino (Roma Univ) to extend their results by using the stochastic sewing lemma recently proposed by Khoa Lê (2020).

International visits to the team

• Hamed Amini, Associate Professor, University of Florida, Research stay.
• Xiao Wei, Associate Professor, China Institute for Actuarial Science(CIAS), Beijing Research stay in connection with Premia.
• Gudmund Pammer, Postdoctoral fellow, ETH Zurich, lecture.
• Antonino Zanette, rofessor, University of Udine, Reserach stays in connectiion with Premia.

10.1.1 Visits to international teams

11.1.1 Scientific events: organisation

• The members of MathRisk with Prof. Antonino Zanette (Univ of Udine) organized an international conference on numerical methods in finance, 14-16 June 2023, in Udine (Italy) to celebrate the 25th anniversary of the software Premia and the team MathRisk.
• J. Guyon organized the minisymposium "Volatility modeling in finance" during the 2023 ICIAM Conference, Tokyo, Aug 2023.
• J. Guyon organized the minisymposium "Recent Advances in Volatility modeling" during the 2023 SIAM Conference on Financial Mathematics & Engineering, Philadelphia, June 2023.
• A. Sulem and A. Zanette organized the Premia meeting for the delivery of the 25th release of the software to the Consortium. Talks by A. Zanette (Univ Udine), A. Molente (Univ Udine), A. Kbaier (Univ Evry), L. Goudenege (CNRS), 30 September 2023, INRIA Paris.
• A. Sulem organized the joint seminar MathRisk/LPSM 19 October 2023, INRIA Paris. Talks by Gudmund PAMMER (ETH Zurich), Mehdi TALBI (LPSM), Robert DENKERT (HU Berlin), A. ALFONSI (CERMICS/ENPC).

11.1.2 Journal editorship

11.1.3 Seminars and Conferences

• A. Alfonsi
  • 19 10 2023: "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation.", Séminaire commun MathRisk-LPSM
  • 09 11 2023: "How many inner simulations to compute conditional expectations with least-square Monte Carlo?", Séminaire LPSM.
  • 20 12 2023: "How many inner simulations to compute conditional expectations with least-square Monte Carlo?", Séminaire de la chaire Futures of quantitative finance.
• V. Bally
  • International Conference on Malliavin Calculus and Related Topics June 12, 2023 - June 16, 2023 Esch-sur-Alzette, Luxembourg Exposé: Construction of Boltzmann and Mc-Kean Vlasov Type Flows
  • A Random Walk in the Land of Stochastic Analysis and Numerical Probability, Conference in honor of Denis Talay 3-7 January 2022 - CIRM, Marseille. Exposé: Construction of Boltzmnn and Mc-Kean Vlasov Type Flows
• Z. Cao
  • MathRisk seminar, ENPC/CERMICS, January 30th 2023
  • SIAM Conference on Financial Mathematics and Engineering, Philadelphia, USA, June 6-9, 2023
  • Conference on stochastic control and financial engineering, Princeton, June 2023
  • SPA Conference, Lisbon , July 2023
• B. Jourdain
  • Workshop Mean Field interaction with singular kernels and their approximations, IHP Paris, 18-22 December 2023: Weak and strong error analysis for systems of particles with mean-field rank-based interaction in the drift
  • Talks in Financial and Insurance Mathematics, ETH Zürich, 12 October 2023: Convexity propagation and convex ordering of one-dimensional stochastic differential equations
  • Workshop SDEs with Low-regularity coefficients: Theory and Numerics, Torino, 21-22 September 2023: Convergence rate of the Euler-Maruyama scheme applied to diffusion processes with Lq-Lp drift coefficient and additive noise
  • Workshop A Random Walk in the Land of Stochastic Analysis and Numerical Probability, Marseille, 4-8 September 2023: Convergence rate of the Euler-Maruyama scheme applied to diffusion processes with Lq-Lp drift coefficient and additive noise
  • Workshop Stochastic processes, metastability and applications, Nancy, 31 May-2 June 2023: Central limit theorem for nonlinear functionals of empirical measures and fluctuations of mean-field interacting particle systems
  • Hong Kong - Singapore joint Seminar Series in Financial Mathematics/Engineering, 25 May 2023: Approximation of martingale couplings on the real line and stability in robust finance
• J. Guyon
  • QuantMinds International, London, Nov 2023, Minicourse
  • Research in Options 2023, Rio de Janeiro, Dec 2023, minicourse
  • Workshop "Frontiers in Stochastic Modelling for Finance", Palermo, October 2023.
  • Workshop "Stochastics around Finance", Kanazawa, August 2023.
  • 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Tokyo, August 2023.
  • 7th International Conference on Mathematics in Finance, Kruger National Park, South Africa, July 2023.
  • BNP Paribas, Paris, Seminar of the BNP Paribas chair "Futures of Quantitative Finance", June 2023.
  • MathSport International 2023, Budapest, June 2023.
  • MathRisk Conference on Numerical Methods in Finance, Udine, June 2023.
  • SIAM Conference on Financial Mathematics and Engineering, Philadelphia, June 2023.
  • Rough volatility workshop 2023, Isle of Skye, May 2023.
  • Quantitative Finance, conference in honor of Michael Demspter's 85th birthday, Cambridge, UK, April 2023.
  • Quantitative Finance Workshop 2023, Gaeta, April 2023.
  • Bank of America, New York, Invited seminar, April 2023.
  • Barclays Capital, New York, Invited seminar, March 2023.
  • Columbia University, New York, Columbia Mathematical Finance Seminar Series, March 2023.
  • Bloomberg, New York, Keynote speaker at BBQ (Bloomberg Quant Seminar), March 2023.
  • Imperial College London, Finance and Stochastics Seminar, March 2023.
  • BNP Paribas, Paris, Kickoff event of the BNP Paribas chair "Futures of Quantitative Finance", March 2023.
  • Capital Fund Management, Paris, EconoPhysiX seminar, February 2023.
  • Capital Fund Management, Paris, Seminar, January 2023.
• D. Lamberton
  • Régularité de la frontière d'exercice : une approche probabiliste. Groupe de travail modélisation stochastique et finance, Ecole des Ponts/Université Gustave Eiffel, January 16th, 2023.
  • Regularity results in optimal stopping: a probabilistic approach. A Random Walk in the Land of Stochastic Analysis and Numerical Probability, in honor of Denis Talay. Luminy, September 2023.
• K. Shao
  • Ceremade Young researchers’ days 2023, Paris, France. (June 1-2, 2023)
  • MathRisk Conference on Numerical Methods in Finance, Udine, Italy. (June 14-16, 2023)
  • Conference: New Monge Problems and Applications, Champs-sur-Marne, France. (September 14-15, 2023)

11.1.4 Scientific expertise

• A. Alfonsi

  Member of the council of the Bachelier Finance Society

• A. Sulem

  Member of the Nominating Committee of the Bachelier Finance Society

11.1.5 Research administration

• A. Alfonsi
  • Deputy director of the CERMICS.
  • In charge of the Master “Finance and Data” at Ecole des Ponts.
• V. Bally
  • Responsible of the Master 2, option finance, Université Gustave Eiffel
  • Member of the LAMA committee, UGE.

• B. Jourdain

  Deputy head of the Labex Bézout.

• A. Sulem
  • Member of the Scientific Committee of AMIES( Agence pour les Mathématiques en Interaction avec l'Entreprise et la Société)
  • Member of the Committee for INRIA international Chairs

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.
• V. Bally
  • Course "Taux d'Intêret" M2 Finance.
  • Course "Calcul de Malliavin et applications en finance" M2 Finance
  • Course "Analyse du risque" M2 Actuariat,
  • Course "Calcul Stochastiques" M2 Recherche
  • Course "Probabilités approfondies" M1
• J. Guyon
  • course "Probability Theory", 1st year ENPC
  • course "Volatility Modeling", Master MFD, 3rd year ENPC - UGE
  • course "Advanced Computational Methods in Finance", Master of Financial Engineering, Baruch College, City University of New York
  • J. Guyon, B. Liang : course "Nonlinear Option Pricing", Master MAFN, Columbia University
  • J. Guyon, F. Meunier : Project of the ENPC course TDLOG : Live probability calculator for the draws of the European football cups
• B. Jourdain
  • course "Mathematical finance", 2nd year ENPC
  • course "Monte-Carlo methods", 3rd year ENPC and Research Master MathÈmatiques et Application, university Gustave Eiffel
  • course "Machine Learning 1", MSC Data Science for Business, X-HEC
  • course "Monte-Carlo Markov chain methods and particle algorithms", Research Master ProbabilitÈs et ModËles AlÈatoires, Sorbonne UniversitÈ
• D. Lamberton
  • "Arbitrage, volatilité et gestion de portefeuille", master 2 course, Université Gustave Eiffel.
  • "Intégration et probabilités", L3 course, Université Gustave Eiffel.
  • "Modélisation et probabilités", L2 course and exercises, Université Gustave Eiffel.
• 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
  • Guido Gazzani (from May 2023); ENPC, Advisor: J. Guyon
  • Guillaume Szulda, ENPC, Advisor: A. Alfonsi (From January 2023 to December 2024)
• PhD defended
  • Zhonguyan Cao, "Dynamics and Stability of Complex Financial networks", Université Paris-Dauphine, Supervisor: Agnès Sulem, Dim Mathinnov doctoral allocation, defended September 20th 2023, Université Paris-Dauphine
  • Roberta Flenghi "Central limit theorems for nonlinear functionals of the empirical measure and for stratified resampling", Supervisor: B. Jourdain, defended on December 20 2023
  • Yfen Qin, "Regularity properties for non linear problems", Dim-Mathinnov doctoral allocation, supervisor: V. Bally, defended June 23 2023, Université Gustave Eiffel
• PhD in progress
  • Elise Devey (started October 2023), "Graphon Mean-Field Games and Renewable Energy Systems", Supervisor: Agnès Sulem, INRIA doctoral grant
  • Hervé Andrës (started in June 2021) "Dependence modelling in economic scenario generation for insurance", supervised by B. Jourdain
  • 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, supervised by A. Alfonsi
  • Kexin Shao (started in october 2021) "Martingale optimal transport and financial applications", supervised by B. Jourdain and A. Sulem
  • Edoardo Lombardo, “High order numerical approximation for some singular stochastic processes and related PDEs”, started in November 2020, International PhD, advisors: Aurélien 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
• Internship
  • Bryan Khan, UGE, "Applications of neural networks for optimal stopping problems", Master 2 report, Summer 2023, Advisor: D. Lamberton
  • Ben Mohamed Jihed, Ecole Polytechnique de Tunisie, 01/03/2023 to 31/08/2023, "Etude de la régularisation de modèles de McKean-Vlasov de volatilité stochastique locale singulière par des techniques de noyau régularisant et par Machine et Deep learning."" Advisor: A. Kbaier
  • Eya Kaabar, Ecole Polytechnique de Tunisie, 01/03/2023 to 31/08/2023 "Développement d'une approche Machine Learning pour la maximisation robuste d’utilité en finance quantitative et implémentation dans le logiciel Premia". Advisor: A. Kbaier
  • Adriano Todisco (Master MFD internship, April-August 2023), "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles", Advisor: J. Guyon
  • Caleb Ovo ENSTA Paris, Development and implementation in the software Premia of numerical methods for pricing and hedging financial derivatives presented in [1908.01602 - Solving high-dimensional optimal stopping problems using deep learning], [2202.02717 - Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing], [2210.04645 - Optimal Stopping with Trees] et [Quantitative Finance - Empirical deep hedging]. Advisor: L. Goudenege
  • Alessio Espa, Sorbonne Université, Development of quantum computing quantique in finance; Construction of a pricing framework using Qiskit; Comparison of efficiency with other techniques. Advisor: L. Goudenege

11.2.3 Juries

• A. Alfonsi
  • Member of the jury of the PhD thesis of Zhongyuan Cao “Systemic Risk, Complex Financial Networks and Graphon Mean Field Interacting Systems”.
  • Jury president of the PhD thesis of Wanqing Wang “Approximation et simulation des équations différentielles stochastiques rétrogrades réfléchies, applications en finance”.
  • Jury president of the PhD thesis of Mouna Ben Derouich “Improved Multi Level Monte Carlo Methods for Pricing Barrier Options in Finance”.
• J. Guyon PhD thesis examination of Long Zhao (Columbia University, April 2023): Martingale Schrodinger bridges and optimal semistatic portfolios.
• B. Jourdain
  • Reviewer for the Habilitation of Zhenjie Ren, defended on January 4, University Paris-Dauphine,
  • Reviewer for the PhD thesis of Thomas Cavallazzi, defended on June 23, University Rennes 1,
  • PhD of Yoan Tardy, defended on june 29, Sorbonne university,
  • PhD of Arnaud Descours, defended on October 20, University Clermont Auvergne, President of the jury
• A. Sulem
  • Reviewer for the PhD thesis of Jeremy Chichportich, defended on March 10th 2023, LPSM, Sorbonne Université.

11.3.1 Articles and contents

• J. Guyon: Un PSG-Bayern Munich en huitièmes de finale de la Ligue des champions ? Probable, mais…, Le Monde, December 18, 2023

11.3.2 Interventions

• J. Guyon: Live intervention (2 h 30 min) during the draws of the football European Cups on La chaine L'Equipe on December 18. football cup
• J. Guyon gave a one-hour interview to Risk magazine in August 2023, "Podcast: Julien Guyon on volatility modelling and World Cup draws". In this edition of Quantcast, Risk’s quantitative podcast, Julien discusses volatility modeling and option pricing as well as World Cup draws and formats. Available at volatility modeling

International journals

• 16 articleA.Aurélien Alfonsi and V.Vlad Bally. Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach).The Annals of Applied Probability335October 2023HALDOIback to text
• 17 articleA.Aurélien Alfonsi and A.Ahmed Kebaier. Approximation of Stochastic Volterra Equations with kernels of completely monotone type.Mathematics of Computation93346November 2023, 643-677HALDOI
• 18 articleA.Aurélien Alfonsi, B.Bernard Lapeyre and J.Jérôme Lelong. How many inner simulations to compute conditional expectations with least-square Monte Carlo?Methodology and Computing in Applied Probability253June 2023, 71HALDOIback to text
• 19 articleA.Aurélien Alfonsi and E.Edoardo Lombardo. High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids.IMA Journal of Numerical AnalysisAugust 2023HALDOIback to text
• 20 articleH.Hamed Amini, Z.Zhongyuan Cao and A.Agnès Sulem. Limit Theorems for Default Contagion and Systemic Risk.Mathematics of Operations ResearchDecember 2023HALDOIback to text
• 21 articleM.Mathias Beiglböck, B.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Stability of the Weak Martingale Optimal Transport Problem.The Annals of Applied Probability336BDecember 2023HALDOIback to text
• 22 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 Risk812023, 33-52HALDOIback to text
• 23 articleJ.Julien Guyon. Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle.Finance and Stochastics281November 2023, 27-79HALDOI
• 24 article J.Julien Guyon and M.Mehdi El Amrani. Does the Term-Structure of the At-the-Money Skew Really Follow a Power Law? Risk August 2023 HAL back to text back to text back to text back to text
• 25 articleJ.Julien Guyon and J.Jordan Lekeufack. Volatility is (mostly) path-dependent.Quantitative Finance239July 2023, 1221-1258HALDOIback to textback to textback to textback to textback to text
• 26 articleJ.Julien Guyon and S.Scander Mustapha. Neural Joint S&P 500/VIX Smile Calibration.Risk MagazineDecember 2023HALDOIback to textback to textback to text
• 27 articleB.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Lipschitz continuity of the Wasserstein projections in the convex order on the line.Electronic Communications in Probability28noneJanuary 2023HALDOIback to text
• 28 articleB.Benjamin Jourdain and S.Stéphane Menozzi. Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise.The Annals of Applied Probability2023HAL
• 29 articleB.Benjamin Jourdain and G.Gilles Pagès. Convex ordering for stochastic Volterra equations and their Euler schemes.Finance and Stochastics2023HALback to text

Edition (books, proceedings, special issue of a journal)

• 30 proceedingsH.Hamed AminiZ.Zhongyuan CaoA.Agnes SulemThe Default Cascade Process in Stochastic Financial Networks.ICAIF '23: 4th ACM International Conference on AI in FinanceACM2023, 227-234HALDOI

Doctoral dissertations and habilitation theses

• 31 thesisZ.Zhongyuan Cao. Systemic risk, complex financial networks and graphon mean field interacting systems.Université Paris sciences et lettresSeptember 2023HALback to text
• 32 thesisR.Roberta Flenghi. Central limit theorem over non-linear functionals of empirical measures and for the stratified resampling mechanism.Ecole des PontsDecember 2023HAL
• 33 thesisY.Yifeng Qin. Approximation of the stochastic differential equation with jumps and convergence in total variation distance.Université Gustave EiffelJune 2023HAL

Reports & preprints

• 34 miscA.Aurélien Alfonsi. Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation..February 2023HALback to text
• 35 miscA.Aurélien Alfonsi and N.Nerea Vadillo. Risk valuation of quanto derivatives on temperature and electricity.2023HALDOIback to text
• 36 miscH.Hamed Amini, Z.Zhongyuan Cao, A.Andreea Minca and A.Agnès Sulem. Ruin Probabilities for Risk Processes in Stochastic Networks.February 2023HALDOI
• 37 miscH.Hervé Andrès, A.Alexandre Boumezoued and B.Benjamin Jourdain. Implied volatility (also) is path-dependent.December 2023HALback to text
• 38 miscH.Hervé Andrès, A.Alexandre Boumezoued and B.Benjamin Jourdain. Signature-based validation of real-world economic scenarios.February 2023HAL
• 39 miscV.Vlad Bally and Y.Yifeng Qin. Approximation for the invariant measure with applications for jump processes (convergence in total variation distance).May 2023HAL
• 40 miscR.Roberta Flenghi and B.Benjamin Jourdain. Central limit theorem for the stratified resampling mechanism.August 2023HALback to text
• 41 miscR.Roberta Flenghi and B.Benjamin Jourdain. Convergence to the uniform distribution of vectors of partial sumsmodulo one with a common factor.August 2023HALback to text
• 42 miscB.Benjamin Jourdain and G.Gilles Pagès. Convex ordering of solutions to one-dimensional SDEs.December 2023HALback to text
• 43 miscB.Benjamin Jourdain and G.Gudmund Pammer. An extension of martingale transport and stability in robust finance.April 2023HALback to text
• 44 miscB.Benjamin Jourdain and K.Kexin Shao. Maximal Martingale Wasserstein Inequality.October 2023HALback to text
• 45 miscB.Benjamin Jourdain and K.Kexin Shao. Non-decreasing martingale couplings.April 2023HALback to text
• 46 miscY.Yifeng Qin. Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analysis in total variation distance).January 2023HAL