arXiv subjects

Nicolas Langrené

Publications and source records attributed to Nicolas Langrené.

At least 19 recordsLinked to original sources

arXiv2026

A deep learning approach for pricing convertible bonds with path-dependent reset and call provisions

This paper develops a deep learning framework for pricing convertible bonds with path-dependent downward reset and issuer call provisions governed by rolling-window triggers. We formulate the valuation problem as a path-dependent partial differential equation (PPDE) that captures both the historical stock-price path and the evolution of the conversion price. Model-specific PPDEs are derived under GBM, CEV, and Heston dynamics. Under suitable conditions, we establish the existence and uniqueness of a piecewise viscosity solution linked by contractual transmission conditions at monitoring dates. For computation, we construct a fixed-grid backward dynamic programming scheme and approximate its conditional expectations using neural networks, with $L^2$ convergence to the exact fixed-grid recursion as the approximation errors vanish. An application to the China CITIC Bank Convertible Bond produces stable prices across the three models and close agreement with the LSMC benchmark, but outperforms in dimensional scaling. The results show that contractual provisions have a greater valuation effect than the choice of underlying dynamics. The call provision reduces the bond value by truncating upside gains, whereas the downward reset provision increases it under the benchmark specification because improved conversion terms dominate the effect of earlier redemption. Delta and Gamma obtained by automatic differentiation of smooth local network approximations closely agree with central finite-difference estimates. The framework provides a flexible approach to pricing and sensitivity analysis for convertible bonds with complex path-dependent provisions.

q-fin.PR↗

arXiv2026

Signed random Fourier features for fast density estimation with indefinite kernels

Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of $N$ points incurs an $\mathcal{O}(N^{2})$ computational cost, which is prohibitive for large-scale datasets. Kernel approximation techniques can be applied to bring the computational cost down to $\mathcal{O}(N)$. The random Fourier features (RFF) technique, based on sampling from the spectral density of the kernel function, has become popular to speed up kernel estimators for machine learning applications. Unfortunately, it is restricted to positive definite kernels, while the majority of kernel functions popular in KDE, such as the parabolic kernel, do not satisfy this property. To overcome this limitation, this article introduces the signed random Fourier features (SRFF) technique. It is a generalization of RFF compatible with indefinite kernels whose inverse Fourier transform is absolutely integrable. The motivation for introducing this method is to speed up KDE in the case of multivariate compact kernels, which are generally not positive definite. We detail how to implement SRFF for both product kernels and isotropic kernels. For the class of Kuttner-Golubov kernels $K(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=(1-\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert ^α)^β\mathbf{1}_{\{\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert \leq1\}}$ where $\boldsymbol{x}_{i}\in\mathbb{R}^{d}$, $\boldsymbol{x}_{j}\in\mathbb{R}^{d}$, $α>0$, $β>0$, which includes the triangular, parabolic, biweight, triweight, and other kernel functions of interest for KDE as particular examples, we provide an explicit acceptance-rejection algorithm to sample from its signed spectral density. Our numerical tests on a dataset of one million points confirm the computational efficiency and accuracy of SRFF for large-scale KDE.

stat.CO↗

arXiv2026

Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework

We establish an explicit approximation formula for European put option prices within a general stochastic volatility model with time-dependent parameters. Our methodology is based on expansions of the mixing representation of the put option price as an expectation of the Black-Scholes formula, in which the resulting terms are calculated explicitly by Malliavin calculus. We obtain an explicit representation of the error generated by the expansion procedure, and bound it in terms of moments of functionals of the underlying volatility process. Under the assumption of piecewise-constant parameters, our approximation formulas become closed-form, and compatible with a proposed fast calibration scheme. Finally, we perform a numerical sensitivity analysis to investigate the quality of our approximation formula in the so-called Stochastic Verhulst model, and show that the errors are well within the acceptable range for application purposes.

q-fin.MF↗

arXiv2026

Deep Least Squares Monte Carlo methods for the valuation of variable annuities with guarantees

In general, the pricing of variable annuities with guarantees can be done by solving the corresponding optimal stochastic control problem if the contract withdrawal strategy is assumed to be optimal. This is typically solved as a dynamic programming problem using deterministic grid methods, which become computationally infeasible for more than a few state variables. In such situations, one needs to rely on simulation methods. The least-squares Monte Carlo (LSMC) method has become a popular simulation method for solving optimal stochastic control problems in quantitative finance over the last decades. In principle, the LSMC, originally developed for pricing Bermudan options, cannot be used directly for pricing variable annuities without simplifying assumptions because the underlying state variables are affected by the control decisions. This paper presents modifications of the LSMC algorithm that makes the pricing of general variable annuities feasible. For numerical illustrations, the pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategies is obtained with and without stochastic interest rates, using either polynomial regression or neural network regression in the LSMC algorithm. We found that the classical polynomial LSMC can give very accurate prices, at the cost of manual feature engineering, and with a standard deviation of the estimator that increases greatly when interest rates are made stochastic. By contrast, neural network LSMC gives slightly less accurate prices, requires more training time, but does not require manual feature engineering, and making interest rates stochastic makes no visible difference to its accuracy, suggesting a more stable and robust pricing performance of deep LSMC for higher-dimensional pricing problems.

q-fin.PR↗

arXiv2026

Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion

A fast simulation framework for stochastic Volterra processes based on Random Fourier Features (RFF) approximation of the kernel is developed. After recalling the main properties of Volterra processes and reviewing existing numerical simulation methods, an accelerated scheme is introduced that relies on a spectral representation of the kernel. A particular attention is devoted to sampling from the kernel spectral density using Hamiltonian Monte Carlo, whose efficiency and stability bring more convenience than alternative sampling procedures. Quantitative guarantees for the proposed method are established, including moment estimates and strong error bounds. The approach is further compared with the kernel approximation by sum of exponentials commonly used in the literature, emphasizing the broader generality of the present framework. As a primary application, Volterra processes associated with the Stationary fractional Brownian Motion (S-fBM) kernel are investigated. A spectral density representation is derived in closed form using hypergeometric functions, a condition for positive definiteness is established and explicit truncation as well as Monte Carlo error bounds are provided for the RFF approximation in this setting. Numerical experiments in dimensions one and two illustrate the accuracy of the kernel approximation, the reliable recovery of model parameters and the competitiveness of the accelerated simulation scheme in terms of computational efficiency and both weak and strong error performance.

q-fin.MF↗

arXiv2026

Scalable method for mean field control with kernel interactions via random Fourier features

We develop a scalable algorithm for mean field control problems with kernel interactions by combining particle system simulations with random Fourier feature approximations. The method replaces the quadratic-cost kernel evaluations by linear-time estimates, enabling efficient stochastic gradient descent for training feedback controls in large populations. We provide theoretical complexity bounds and demonstrate through crowd motion and flocking examples that the approach preserves control performance while substantially reducing computational cost. The results indicate that random feature approximations offer an effective and practical tool for high dimensional and large scale mean field control.

math.OC↗

arXiv2026

The dynamics of innovation diffusion: A survey of Bass-type models

This paper synthesises the existing research on the dynamics of innovation diffusion, with a focus on Bass-type models and their extensions. The theoretical foundation of innovation diffusion proposed by Rogers (1962) and the seminal work of Bass (1969) serve as a starting point for the analysis. We identify and examine various generalizations and stochastic extensions of the Bass model, including counting processes, diffusion processes, and uncertain processes, as well as parameter estimation techniques, from classical statistical techniques to more advanced techniques such as Bayesian filtering and metaheuristic optimisation. We finally explore alternative models of innovation diffusion, with a particular focus on agent-based models. This overview of the evolution of Bass-type models illustrates the progress made in innovation diffusion research over the past decades.

econ.GN↗

arXiv2026

A spectral mixture representation of isotropic kernels with application to random Fourier features

Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution for machine learning applications. 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 is usually applied to the Gaussian kernel because of its simplicity, since its spectral distribution is also Gaussian. Clearly, simple spectral sampling formulas would be desirable for broader classes of kernels. In this paper, we show that the spectral distribution of positive definite isotropic kernels in $\mathbb{R}^{d}$ for all $d\geq1$ can be decomposed as a scale mixture of $α$-stable random vectors, and we identify the mixing distribution as a function of the kernel. This constructive decomposition provides a simple and ready-to-use spectral sampling formula for many multivariate positive definite shift-invariant kernels, including exponential power kernels, and generalized Cauchy kernels, as well as newly introduced kernels such as the generalized Matérn, Tricomi, and Fox $H$ kernels. In particular, we retrieve the fact that the spectral distributions of these kernels, which can only be explicited in terms of the Fox $H$ special function, are scale mixtures of the multivariate Gaussian distribution, along with an explicit mixing distribution formula. This result has broad applications for support vector machines, kernel ridge regression, Gaussian processes, and other kernel-based machine learning techniques for which the random Fourier features technique is applicable.

cs.LG↗

arXiv2026

Squintability and Other Metrics for Assessing Projection Pursuit Indexes, and Guiding Optimization Choices

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 proximity. Here, measures for calculating the smoothness and squintability properties of the PP index are defined. These are used to investigate the performance of a recently introduced swarm-based algorithm, Jellyfish Search Optimizer (JSO), for optimizing PP indexes. The performance of JSO in detecting the target pattern (pipe shape) is compared with existing optimizers in PP. Additionally, JSO's performance on detecting the sine-wave shape is evaluated using different PP indexes (hence different smoothness and squintability) across various data dimensions (d = 4, 6, 8, 10, 12) and JSO hyper-parameters. We observe empirically that higher squintability improves the success rate of the PP index optimization, while smoothness has no significant effect. The JSO algorithm has been implemented in the R package, `tourr`, and functions to calculate smoothness and squintability measures are implemented in the `ferrn` package.

stat.CO↗

arXiv2025

Quantformer: from attention to profit with a quantitative transformer trading strategy

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, this paper introduces quantformer, an enhanced neural network architecture based on transformer, to build investment factors. By transfer learning from sentiment analysis, quantformer not only exploits its original inherent advantages in capturing long-range dependencies and modeling complex data relationships, but is also able to solve tasks with numerical inputs and accurately forecast future returns over a given period. This work collects more than 5,000,000 rolling data of 4,601 stocks in the Chinese capital market from 2010 to 2023. The results of this study demonstrate the model's superior performance in predicting stock trends compared with other 100-factor-based quantitative strategies. Notably, the model's innovative use of transformer-like model to establish factors, in conjunction with market sentiment information, has been shown to enhance the accuracy of trading signals significantly, thereby offering promising implications for the future of quantitative trading strategies.

q-fin.MF↗

arXiv2025

Fast Gaussian process inference by exact Matérn kernel decomposition

To speed up Gaussian process inference, a number of fast kernel matrix-vector multiplication (MVM) approximation algorithms have been proposed over the years. In this paper, we establish an exact fast kernel MVM algorithm based on exact kernel decomposition into weighted empirical cumulative distribution functions, compatible with a class of kernels which includes multivariate Matérn kernels with half-integer smoothness parameter. This algorithm uses a divide-and-conquer approach, during which sorting outputs are stored in a data structure. We also propose a new algorithm to take into account some linear fixed effects predictor function. Our numerical experiments confirm that our algorithm is very effective for low-dimensional Gaussian process inference problems with hundreds of thousands of data points. An implementation of our algorithm is available at https://gitlab.com/warin/fastgaussiankernelregression.git.

stat.ML↗

arXiv2025

Unleashing the potential of prompt engineering for large language models

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 as GPT-4o and Claude-3, and in Vision-Language Models (VLMs), with models such as CLIP and ALIGN. Prompt engineering is the process of structuring inputs, which has emerged as a crucial technique to maximize the utility and accuracy of these models. This paper explores both foundational and advanced methodologies of prompt engineering, including techniques such as self-consistency, chain-of-thought, and generated knowledge, which significantly enhance model performance. Additionally, it examines the prompt method of VLMs through innovative approaches such as Context Optimization (CoOp), Conditional Context Optimization (CoCoOp), and Multimodal Prompt Learning (MaPLe). Critical to this discussion is the aspect of AI security, particularly adversarial attacks that exploit vulnerabilities in prompt engineering. Strategies to mitigate these risks and enhance model robustness are thoroughly reviewed. The evaluation of prompt methods is also addressed through both subjective and objective metrics, ensuring a robust analysis of their efficacy. This review also reflects the essential role of prompt engineering in advancing AI capabilities, providing a structured framework for future research and application.

cs.CL↗

arXiv2024

Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks

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 accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.

q-fin.CP↗

arXiv2024

A Tidy Framework and Infrastructure to Systematically Assemble Spatio-temporal Indexes from Multivariate Data

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 their structure affects their values and variation in values. Here we discuss a modular data pipeline recommendation to assemble indexes. It is universally applicable to index computation and allows investigation of index behavior as part of the development procedure. One can compute indexes with different parameter choices, adjust steps in the index definition by adding, removing, and swapping them to experiment with various index designs, calculate uncertainty measures, and assess indexes robustness. The paper presents three examples to illustrate the pipeline framework usage: comparison of two different indexes designed to monitor the spatio-temporal distribution of drought in Queensland, Australia; the effect of dimension reduction choices on the Global Gender Gap Index (GGGI) on countries ranking; and how to calculate bootstrap confidence intervals for the Standardized Precipitation Index (SPI). The methods are supported by a new R package, called tidyindex.

stat.CO↗

arXiv2024

cubble: An R Package for Organizing and Wrangling Multivariate Spatio-temporal Data

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 studies, it is important to analyse different aspects of the spatio-temporal data simultaneouly, like for instance, temporal trends of multiple variables across locations. In order to facilitate the study of different portions or combinations of spatio-temporal data, we introduce a new data structure, cubble, with a suite of functions enabling easy slicing and dicing on the different components spatio-temporal components. The proposed cubble structure ensures that all the components of the data are easy to access and manipulate while providing flexibility for data analysis. In addition, cubble facilitates visual and numerical explorations of the data while easing data wrangling and modelling. The cubble structure and the functions provided in the cubble R package equip users with the capability to handle hierarchical spatial and temporal structures. The cubble structure and the tools implemented in the package are illustrated with different examples of Australian climate data.

stat.CO↗

arXiv2021

Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility

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. The difficulties then faced are simplifying a number of expectations induced by the Taylor expansion. Under the assumption of piecewise-constant parameters, we derive closed-form pricing formulas and devise a fast calibration scheme. Furthermore, we perform a numerical error and sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term generated by the Taylor expansion.

q-fin.MF↗

arXiv2021

Visual Diagnostics for Constrained Optimisation with Application to Guided Tours

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 developing new indexes in the literature, less has been done on understanding the optimisation. Index functions can be noisy, might have multiple local maxima as well as an optimal maximum, and are constrained to generate orthonormal projection frames, which complicates the optimization. In addition, projection pursuit is primarily used for exploratory data analysis, and finding the local maxima is also useful. The guided tour is especially useful for exploration, because it conducts geodesic interpolation connecting steps in the optimisation and shows how the projected data changes as a maxima is approached. This work provides new visual diagnostics for examining a choice of optimisation procedure, based on the provision of a new data object which collects information throughout the optimisation. It has helped to diagnose and fix several problems with projection pursuit guided tour. This work might be useful more broadly for diagnosing optimisers, and comparing their performance. The diagnostics are implemented in the R package, ferrn.

stat.CO↗

arXiv2021

Deep Semi-Martingale Optimal Transport

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 solved using adversarial networks. Both methods are mesh-free and therefore mitigate the curse of dimensionality. We test the performance and accuracy of our methods on several examples up to dimension 10. We also apply the first algorithm to a portfolio optimization problem where the goal is, given an initial wealth distribution, to find an investment strategy leading to a prescribed terminal wealth distribution.

math.OC↗