arXiv subjects

Stéphane Crépey

Publications and source records attributed to Stéphane Crépey.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

Crépey, Frikha, and Louzi (2025) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The best complexity of the scheme is in O($\varepsilon^{-\frac52}$), $\varepsilon>0$ being a prescribed accuracy, which is suboptimal compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity ofthe Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its best complexity is in O($\varepsilon^{-2}|\ln{\varepsilon}|^\frac52$). Our theoretical analysis is exemplified through numerical experiments.

q-fin.RM↗

arXiv2026

A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation

We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the value-at-risk (VaR) and expected shortfall (ES) of a financial loss, which can only be computed via simulations conditionally on the realisation of future risk factors. Thus the problem of estimating its VaR and ES is nested in nature and can be viewed as an instance of stochastic approximation problems with biased innovations. In this framework, for a prescribed accuracy $\varepsilon$, the optimal complexity of a nested stochastic approximation algorithm is shown to be of the order $\varepsilon^{-3}$. To estimate the VaR, our MLSA algorithm attains an optimal complexity of the order $\varepsilon^{-2-δ}$, where $δ\in(0,1)$ is some parameter depending on the integrability degree of the loss, while to estimate the ES, the algorithm achieves an optimal complexity of the order $\varepsilon^{-2}|\ln{\varepsilon}|^2$. Numerical studies of the joint evolution of the error rate and the execution time demonstrate how our MLSA algorithm regains a significant amount of the performance lost due to the nested nature of the problem.

q-fin.CP↗

arXiv2026

Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall

Crépey, Frikha, and Louzi (2025) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example.

q-fin.RM↗

arXiv2026

The Recalibration Conundrum: Hedging Valuation Adjustment for Callable Claims

The dynamic hedging theory only makes sense in the setup of one given model, whereas the practice of dynamic hedging is just the opposite, with models fleeing after the data through daily recalibration. This is quite of a quantitative finance paradox. In this paper we revisit Burnett (2021) \& Burnett and Williams (2021)'s notion of hedging valuation adjustment (HVA), originally intended to deal with dynamic hedging frictions, in the direction of recalibration and model risks. Specifically, we extend to callable assets the HVA model risk approach of B{é}n{é}zet and Cr{é}pey (2024). The classical way to deal with model risk is to reserve the differences between the valuations in reference models and in the local models used by traders. However, while traders' prices are thus corrected, their hedging strategies and their exercise decisions are still wrong, which necessitates a risk-adjusted reserve. We illustrate our approach on a stylized callable range accrual representative of huge amounts of structured products on the market. We show that a model risk reserve adjusted for the risk of wrong exercise decisions may largely exceed a basic reserve only accounting for valuation differences.

q-fin.RM↗

arXiv2025

Sensitivity Analysis of emissions Markets: A Discrete-Time Radner Equilibrium Approach

Emissions markets play a vital role in emissions reduction by incentivizing firms to minimize costs. However, their effectiveness heavily depends on the decisions of policymakers, future economic activity, and the availability of abatement technologies. This study investigates how variations in regulatory standards, firms' abatement costs, and emission levels influence allowance prices and firms' abatement efforts. This is done in a Radner equilibrium framework that incorporates intertemporal decision-making and uncertainty, enabling a comprehensive analysis of market dynamics and outcomes. The findings provide valuable insights for policymakers aiming to enhance the design and efficiency of emissions trading systems through a deeper understanding of stakeholder responses across varying market conditions.

q-fin.GN↗

arXiv2025

Comparison of Tax and Cap-and-Trade Carbon Pricing Schemes

Carbon pricing has become a central pillar of modern climate policy, with carbon taxes and emissions trading systems (ETS) serving as the two dominant approaches. Although economic theory suggests these instruments are equivalent under idealized assumptions, their performance diverges in practice due to real-world market imperfections. A particularly less explored dimension of this divergence concerns the role of financial intermediaries in emissions trading markets. This paper develops a unified framework to compare the economic and environmental performance of tax- and market-based schemes, explicitly incorporating the involvement of financial intermediaries. By calibrating both instruments to deliver identical aggregate emission reduction targets, we assess their economic performance across alternative market structures. Our results suggest that although the two schemes are equivalent under perfect competition, the presence of intermediaries in ETS reduces both regulatory wealth and the aggregate wealth of economic agents relative to carbon taxation. These effects stem from intermediaries' influence on price formation and their appropriation of part of the revenue stream. The findings underscore the importance of accounting for intermediaries' behavior in the design of carbon markets and highlight the need for further empirical research on the evolving institutional structure of emissions trading systems.

econ.GN↗

arXiv2024

Provisions and Economic Capital for Credit Losses

Based on supermodularity ordering properties, we show that convex risk measures of credit losses are nondecreasing w.r.t. credit-credit and, in a wrong-way risk setup, credit-market, covariances of elliptically distributed latent factors. These results support the use of such setups for computing credit provisions and economic capital or for conducting stress test exercises and risk management analysis.

q-fin.RM↗

arXiv2024

Spanning Multi-Asset Payoffs With ReLUs

We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to numerically exploit. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges.

q-fin.RM↗

arXiv2024

Resolving a Clearing Member's Default, A Radner Equilibrium Approach

For vanilla derivatives that constitute the bulk of investment banks' hedging portfolios, central clearing through central counterparties (CCPs) has become hegemonic. A key mandate of a CCP is to provide an efficient and proper clearing member default resolution procedure. When a clearing member defaults, the CCP can hedge and auction or liquidate its positions. The counterparty credit risk cost of auctioning has been analyzed in terms of XVA metrics in Bastide, Cr{é}pey, Drapeau, and Tadese (2023). In this work we assess the costs of hedging or liquidating. This is done by comparing pre- and post-default market equilibria, using a Radner equilibrium approach for portfolio allocation and price discovery in each case. We show that the Radner equilibria uniquely exist and we provide both analytical and numerical solutions for the latter in elliptically distributed markets. Using such tools, a CCP could decide rationally on which market to hedge and auction or liquidate defaulted portfolios.

q-fin.RM↗

arXiv2024

Handling model risk with XVAs

In this paper we revisit Burnett (2021) \& Burnett and Williams (2021)'s notion of hedging valuation adjustment (HVA), originally intended to deal with dynamic hedging frictions such as transaction costs, in the direction of model risk. The corresponding HVA reconciles a global fair valuation model with the local models used by the different desks of the bank. Model risk and dynamic hedging frictions indeed deserve a reserve, but a risk-adjusted one, so not only an HVA, but also a contribution to the KVA of the bank. The orders of magnitude of the effects involved suggest that local models should not so much be managed via reserves, as excluded altogether.

q-fin.PR↗

arXiv2024

CVA Sensitivities, Hedging and Risk

We present a unified framework for computing CVA sensitivities, hedging the CVA, and assessing CVA risk, using probabilistic machine learning meant as refined regression tools on simulated data, validatable by low-cost companion Monte Carlo procedures. Various notions of sensitivities are introduced and benchmarked numerically. We identify the sensitivities representing the best practical trade-offs in downstream tasks including CVA hedging and risk assessment.

q-fin.CP↗

arXiv2024

Invariance Times Transfer Properties

Invariance times are stopping times $τ$ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $τ$ , are local martingales with respect to the original model filtration and probability measure. They arise naturally for modeling the default time of a dealer bank, in the mathematical finance context of counterparty credit risk. Assuming an invariance time endowed with an intensity and a positive Az{é}ma supermartingale, this work establishes a dictionary relating the semimartingale calculi in the original and reduced stochastic bases, regarding in particular conditional expectations, martingales, stochastic integrals, random measure stochastic integrals, martingale representation properties, semimartingale characteristics, Markov properties, transition semigroups and infinitesimal generators, and solutions of backward stochastic differential equations.

math.PR↗

arXiv2024

An Explicit Scheme for Pathwise XVA Computations

Motivated by the equations of cross valuation adjustments (XVAs) in the realistic case where capital is deemed fungible as a source of funding for variation margin, we introduce a simulation/regression scheme for a class of anticipated BSDEs, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessing the regression error of the scheme at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in a high-dimensional and hybrid market/default risks XVA use-case.

q-fin.RM↗

arXiv2022

Beyond Surrogate Modeling: Learning the Local Volatility Via Shape Constraints

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.

q-fin.MF↗

arXiv2022

Pathwise CVA Regressions With Oversimulated Defaults

We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes $(X, Y )$. Here an exogenous component $Y$ (Markov by itself) is time-consuming to simulate, while the endogenous component $X$ (jointly Markov with $Y$) is quick to simulate given $Y$, but is responsible for most of the variance of the simulated payoff. To address the related variance issue, we introduce a conditionally independent, hierarchical simulation scheme, where several paths of $X$ are simulated for each simulated path of $Y$. We analyze the statistical convergence of the regression learning scheme based on such block-dependent data. We derive heuristics on the number of paths of $Y$ and, for each of them, of $X$, that should be simulated. The resulting algorithm is implemented on a graphics processing unit (GPU) combining Python/CUDA and learning with PyTorch. A CVA case study with a nested Monte Carlo benchmark shows that the hierarchical simulation technique is key to the success of the learning approach.

q-fin.CP↗

arXiv2022

Anomaly Detection on Financial Time Series by Principal Component Analysis and Neural Networks

A major concern when dealing with financial time series involving a wide variety ofmarket risk factors is the presence of anomalies. These induce a miscalibration of the models used toquantify and manage risk, resulting in potential erroneous risk measures. We propose an approachthat aims to improve anomaly detection in financial time series, overcoming most of the inherentdifficulties. Valuable features are extracted from the time series by compressing and reconstructingthe data through principal component analysis. We then define an anomaly score using a feedforwardneural network. A time series is considered to be contaminated when its anomaly score exceeds agiven cutoff value. This cutoff value is not a hand-set parameter but rather is calibrated as a neuralnetwork parameter throughout the minimization of a customized loss function. The efficiency of theproposed approach compared to several well-known anomaly detection algorithms is numericallydemonstrated on both synthetic and real data sets, with high and stable performance being achievedwith the PCA NN approach. We show that value-at-risk estimation errors are reduced when theproposed anomaly detection model is used with a basic imputation approach to correct the anomaly.

q-fin.ST↗

arXiv2022

Derivatives Risks as Costs in a One-Period Network Model

We present a one-period XVA model encompassing bilateral and centrally cleared trading in a unified framework with explicit formulas for most quantities at hand. We illustrate possible uses of this framework for running stress test exercises on a financial network from a clearing member's perspective or for optimizing the porting of the portfolio of a defaulted clearing member.

q-fin.RM↗

arXiv2020

Deep Local Volatility

Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.

q-fin.CP↗