arXiv subjects

Hervé Andrès

Publications and source records attributed to Hervé Andrès.

5 recordsLinked to original sources

arXiv2026

Approximation of stochastic insurer balance-sheet results using signatures of economic scenarios

In the insurance industry, Asset and Liability Management (ALM) models are key tools for numerous applications, including Solvency Capital Requirement (SCR) computation and asset allocation optimization. However, their use often entails a significant computational cost, especially when a large number of sensitivities or stressed balance-sheet evaluations must be performed. In this work, we propose an approximation framework for the outputs of an ALM model, such as the Value In Force or the Best Estimate, based on the theory of path signatures. More precisely, the proposed approach consists of approximating ALM outputs by a linear combination of signature terms derived from input economic scenarios. We show that the resulting surrogate is easy to calibrate, essentially through regularized linear regression, and exhibits strong predictive performance while drastically reducing computational costs. We further investigate its robustness under changes in the distribution of economic scenarios by considering variations in the parameters of the underlying model of risk factors while the surrogate model is kept fixed. These results make the proposed approach particularly suitable for large-scale sensitivity analyses and fast balance-sheet evaluations in practical actuarial applications.

q-fin.RM↗

arXiv2026

Linear independence properties of the signature components of time-augmented stochastic processes

Adding the time as a component of a stochastic process before computing its signature terminal value ensures injectivity and supports universal approximation results, but it induces linear dependence among the components of the signature terminal value. For any natural number $N$, the terminal values of the signature components associated with words of length not greater than $N$ are the image of the terminal values of the signature components associated with words of length $N$ by some universal linear map. We generalize this result by exhibiting other subfamilies of components -- represented by subfamilies of words -- with the same representation property. When considering the signature of the solution to a stochastic differential equation with a uniformly elliptic diffusion coefficient, we show that any such subfamily of components is linearly independent for the almost-sure equality and therefore provides a basis of the linear span of all components associated with words of length not greater than $N$. The linear independence of these subfamilies is preserved for the affine interpolation of this solution on a grid with a sufficiently small time step. We characterize bases of components with minimal computation cost. Finally, we remark that the subfamilies of words obtained above share a similar representation property when applied to the time-augmented EFM signature. For a Brownian semimartingale with a non-degenerate diffusion coefficient, we show that any such subfamily of components of its time-augmented EFM signature is almost-surely linearly independent for the $dt$-a.e. equality.

math.PR↗

arXiv2025

The implied volatility surface (also) is path-dependent

We propose a new model for the forecasting of both the implied volatility surfaces and the underlying asset price. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S\&P 500), we first study how vanilla options implied volatility can be predicted using the past trajectory of the underlying asset price. Our empirical study reveals that a large part of the movements of the at-the-money-forward implied volatility for up to two years time-to-maturities can be explained using the past returns and their squares. Moreover, we show that this feedback effect gets weaker when the time-to-maturity increases. Building on this new stylized fact, we fit to historical data a parsimonious version of the SSVI parameterization (Gatheral and Jacquier, 2014) of the implied volatility surface relying on only four parameters and show that the two parameters ruling the at-the-money-forward implied volatility as a function of the time-to-maturity exhibit a path-dependent behavior with respect to the underlying asset price. Finally, we propose a model for the joint dynamics of the implied volatility surface and the underlying asset price. The latter is modelled using a variant of the path-dependent volatility model of Guyon and Lekeufack and the former is obtained by adding a feedback effect of the underlying asset price onto the two parameters ruling the at-the-money-forward implied volatility in the parsimonious SSVI parameterization and by specifying Ornstein-Uhlenbeck processes for the residuals of these two parameters and Jacobi processes for the two other parameters. Thanks to this model, we are able to simulate highly realistic paths of implied volatility surfaces that are free from static arbitrage.

q-fin.CP↗

arXiv2025

Existence, uniqueness and positivity of solutions to the Guyon-Lekeufack path-dependent volatility model with general kernels

We show the existence and uniqueness of a continuous solution to a path-dependent volatility model introduced by Guyon and Lekeufack (2023) to model the price of an equity index and its spot volatility. The considered model for the trend and activity features can be written as a Stochastic Volterra Equation (SVE) with non-convolutional and non-bounded kernels as well as non-Lipschitz coefficients. We first prove the existence and uniqueness of a solution to the SVE under integrability and regularity assumptions on the two kernels and under a condition on the second kernel weighting the past squared returns which ensures that the activity feature is bounded from below by a positive constant. Then, assuming in addition that the kernel weighting the past returns is of exponential type and that an inequality relating the logarithmic derivatives of the two kernels with respect to their second variables is satisfied, we show the positivity of the volatility process which is obtained as a non-linear function of the SVE's solution. We show numerically that the choice of an exponential kernel for the kernel weighting the past returns has little impact on the quality of model calibration compared to other choices and the inequality involving the logarithmic derivatives is satisfied by the calibrated kernels. These results extend those of Nutz and Valdevenito (2023).

q-fin.CP↗

arXiv2024

Signature-based validation of real-world economic scenarios

Motivated by insurance applications, we propose a new approach for the validation of real-world economic scenarios. This approach is based on the statistical test developed by Chevyrev and Oberhauser (2022) and relies on the notions of signature and maximum mean distance. This test allows to check whether two samples of stochastic processes paths come from the same distribution. Our contribution is to apply this test to a variety of stochastic processes exhibiting different pathwise properties (H{ö}lder regularity, autocorrelation, regime switches) and which are relevant for the modelling of stock prices and stock volatility as well as of inflation in view of actuarial applications.

q-fin.ST↗