Nicolas Langrené

• PhD Université Paris Cité
• Associate Professor at Beijing Normal-Hong Kong Baptist University

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N^2) operations,...

Multivariate spatio-temporal data refers to multiple measurements taken across space and time. For many analyses, spatial and time components can be separately studied: for example, to explore the temporal trend of one variable for a single spatial location, or to model the spatial distribution of one variable at a given time. However for some stud...

This comprehensive review delves into the pivotal role of prompt engineering in unleashing the capabilities of Large Language Models (LLMs). The development of Artificial Intelligence (AI), from its inception in the 1950s to the emergence of advanced neural networks and deep learning architectures, has made a breakthrough in LLMs, with models such...

The projection pursuit (PP) guided tour optimizes a criterion function, known as the PP index, to gradually reveal projections of interest from high-dimensional data through animation. Optimization of some PP indexes can be non-trivial, if they are non-smooth functions, or when the optimum has a small “squint angle”, detectable only from close prox...

Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution. This famous technique, known as Random Fourier Features (RFF), is in principle applicable to any such kernel whose spectral distribution can be identified and simulated. In practice, however, it...

Indexes are useful for summarizing multivariate information into single metrics for monitoring, communicating, and decision-making. While most work has focused on defining new indexes for specific purposes, more attention needs to be directed towards making it possible to understand index behavior in different data conditions, and to determine how...

The purpose of this Erratum is to remedy a minor mistake in Theorems 5.4 and Corollary 5.3 in the article ‘Closed-form approximations for option pricing under stochastic volatility’ [K. Das and N. Langrené, Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility, Stochastics 94(5) (2022), pp. 74...

In traditional quantitative trading practice, navigating the complicated and dynamic financial market presents a persistent challenge. Fully capturing various market variables, including long-term information, as well as essential signals that may lead to profit remains a difficult task for learning algorithms. In order to tackle this challenge, th...

We obtain an explicit approximation formula for European put option prices within a general stochastic volatility model with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula, reparameterising the volatility process and then performing a number of expansions. The bulk of th...

Indexes are useful for summarizing multivariate information into single metrics for monitoring, communicating, and decision-making. While most work has focused on defining new indexes for specific purposes, more attention needs to be directed towards making it possible to understand index behavior in different data conditions, and to determine how...

In this paper, we introduce two novel methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplish...

https://www.researchgate.net/publication/368829953_Simultaneous_upper_and_lower_bounds_of_American_option_prices_with_hedging_via_neural_networks

Transient stability assessment traditionally includes performing computationally expensive time-domain simulations to examine the dynamic stability of power systems for a limited number of the most critical contingencies. Recently, data-driven solutions have been introduced to augment the situational awareness in real-time over a wide range of oper...

The construction and operation of linear infrastructure has major impacts on biodiversity through loss of habitat, increased mortality and loss of connectivity. In particular, minimising the impact of roads which pass through ecologically sensitive areas on surrounding species at the construction and operational phases is critical for conservation....

This paper presents several numerical applications of deep learning-based algorithms for discrete-time stochastic control problems in finite time horizon that have been introduced in [Huré et al. 2021]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance ite...

We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. Th...

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck...

This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. In addition, our paper dedicates in proposing various numerical algorithms to solve for the value function, incl...

The problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets, is revisited. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N2) operations, which...

A guided tour helps to visualise high-dimensional data by showing low-dimensional projections along a projection pursuit optimisation path. Projection pursuit is a generalisation of principal component analysis, in the sense that different indexes are used to define the interestingness of the projected data. While much work has been done in develop...

We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second method is based on the dual formulation of the problem, which we express as a saddle point problem, and is so...

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To ov...

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regr...

To support N-1 pre-fault transient stability assessment, this paper introduces a new data collection method in a data-driven algorithm incorporating the knowledge of power system dynamics. The domain knowledge on how the disturbance effect will propagate from the fault location to the rest of the network is leveraged to recognise the dominant condi...

https://www.researchgate.net/publication/353652840_Robust_utility_maximization_under_model_uncertainty_via_a_penalization_approach

The retirement systems in many developed countries have been increasingly moving from defined benefit towards defined contribution system. In defined contribution systems, financial and longevity risks are shifted from pension providers to retirees. In this paper, we use a probabilistic approach to analyse the uncertainty associated with superannua...

We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. Th...

This paper studies a portfolio allocation problem, where the goal is to prescribe the wealth distribution at the final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which we solve by a gradient descent algorithm. This involves solving an associated HJB and Fokker--Planck equation by a finite dif...

In this paper, we develop a deep neural network approach to solve a lifetime expected mortality-weighted utility-based model for optimal consumption in the decumulation phase of a defined contribution pension system. We formulate this problem as a multi-period finite-horizon stochastic control problem and train a deep neural network policy represen...

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To over...

This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differentia...

https://www.researchgate.net/publication/342733634_Markovian_approximation_of_the_rough_Bergomi_model_for_Monte_Carlo_option_pricing

https://www.researchgate.net/publication/344410894_Portfolio_optimization_with_a_prescribed_terminal_wealth_distribution

The construction and operation of linear infrastructure has major impacts on biodiversity through loss of habitat, increased mortality and loss of connectivity. In particular, minimising the impact of roads which pass through ecologically sensitive areas on surrounding species at the construction and operational phases is critical for conservation....

The number of major tailings dam failures has doubled over the past 20 years, culminating in the tragic accident at Brumadinho in Brazil where about 300 people lost their lives. In this context, there is a growing demand from mining companies, institutional investors and policymakers alike for updated mining project assessment tools taking account...

Kernel density estimation and kernel regression are powerful but computationally expensive techniques: a direct evaluation of kernel density estimates at $M$ evaluation points given $N$ input sample points requires a quadratic $\mathcal{O}(MN)$ operations, which is prohibitive for large scale problems. For this reason, approximate methods such as b...

The volatility of concern in conventional volatility-managed strategies such as volatility-targeting strategy and mean-variance optimization is the expected conditional volatility. However for investors, it is the realized volatility that is important, because there is only one realization in the market. Simply managing the conditional volatility m...

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorabl...

The retirement systems in many developed countries have been increasingly moving from Defined Benefit (DB) towards the Defined Contribution (DC) system. In such systems, financial and longevity risks are shifted from the pension provider to the retiree. This paper uses a probabilistic approach to analyse the uncertainty associated with the retireme...

We present a simulation-and-regression method for solving dynamic portfolio optimization problems in the presence of general transaction costs, liquidity costs and market impact. This method extends the classical least squares Monte Carlo algorithm to incorporate switching costs, corresponding to transaction costs and transient liquidity costs, as...

This paper presents several numerical applications of deep learning-based algorithms that have been introduced in [Hur+21]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteratio...

https://www.researchgate.net/publication/346671204_Deep_neural_networks_algorithms_for_stochastic_control_problems_on_finite_horizon_numerical_applications

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regr...

Kernel density estimation and kernel regression are powerful but computationally expensive techniques: a direct evaluation of kernel density estimates at $M$ evaluation points given $N$ input sample points requires a quadratic $\mathcal{O}(MN)$ operations, which is prohibitive for large scale problems. For this reason, approximate methods such as b...

In the equity and foreign exchange (FX) markets, there has been a shift towards using non-affine pricing models as these have been shown to produce more realistic volatility distributions and more accurately capture market dynamics. One such non-affine model is the Inverse Gamma model, which we have incorporated into a Local-Stochastic Volatility (...

https://emap.fgv.br/seminarios/recent-advances-monte-carlo-methods-stochastic-control-problems

We consider the problem faced by mining companies to set their extraction rate over time in order to meet their production targets, from a real option perspective. As part of their strategic planning, mining companies usually set a production target for a given, short-term to medium-term, time horizon. The extraction rate is constrained by this tar...

Mining operations are affected by significant uncertainty in commodity prices, combined with geological uncertainties (both in quantity and quality of the available reserves). Technical difficulties and costs associated with ore extraction together with a highly uncertain environment present significant risks for profitability of mineral projects....

To achieve good crop yields, farmers are aware of the importance of good rainfall during the growing season. However, their choice of crop to plant may not necessarily be the optimal choice when climate uncertainty exists. Indeed, current cropping allocations may be driven more by historical trends and tradition than by the future forecasts of clim...

The flexibility to revise managerial and/or operational decisions over time in response to uncertain market conditions can significantly increase the value of a project. In order to maximise the project value, the operational decisions need to be made sequentially, in an optimal manner, in response to the evolution of uncertainties. Although dynami...

This paper introduces the Inverse Gamma (IGa) stochastic volatility model with time-dependent parameters, defined by the volatility dynamics $dV_{t}=\kappa_{t}\left(\theta_{t}-V_{t}\right)dt+\lambda_{t}V_{t}dB_{t}$. This non-affine model is much more realistic than classical affine models like the Heston stochastic volatility model, even though bot...

We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility, possibly degenerate. Our back...

This thesis deals with the numerical solution of general stochastic control problems, with notable applications for electricity markets. We first propose a structural model for the price of electricity, allowing for price spikes well above the marginal fuel price under strained market conditions. This model allows to price and partially hedge elect...

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations and local basis regressions to solve non-stationary optimal multiple switching problems in infinite horizon. We provide the rate of convergence of the method in terms of the time step used to discretize the problem, of the regression...

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approxi...

We develop a structural risk-neutral model for energy market modifying along several directions the approach introduced in Aid et al. (2009). In particular a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. T...

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations and local basis regressions to solve non-stationary optimal multiple switching problems in infinite horizon. We provide the rate of convergence of the method in terms of the time step used to discretize the problem, of the size of th...

In this paper, we will consider a midterm offer demand equilibrium model for electricity. This kind of model can be used for investment opportunity studies, wherein new assets are valued against obtained marginal costs on a restricted set of uncertainty scenarios. In order to correctly value peak-load assets, realistic marginal costs are required a...