arXiv subjects

Youness Boutaib

Publications and source records attributed to Youness Boutaib.

7 recordsLinked to original sources

arXiv2026

Asymptotically-informed neural networks for Black-Scholes implied volatility computation

The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.

q-fin.CP↗

arXiv2026

Separation capacity of linear reservoirs with random connectivity matrix

A natural hypothesis for the success of reservoir computing in generic tasks is the ability of the untrained reservoir to map distinct input time series to separable reservoir states, a property we term separation capacity. In this work, we develop a rigorous mathematical framework for analysing the separation capacity of random linear reservoirs. We show that the expected separation induced by a random reservoir is completely characterised by the spectral properties of a positive semi-definite matrix naturally associated with the connectivity matrix, which we call the generalised matrix of moments. Taking Gaussian connectivity matrices as an example, we investigate how separation depends on the reservoir dimension $N$, the scaling of the connectivity matrix, and structural assumptions such as symmetry. In the symmetric case, we show that, although the classical scaling $N^{-1/2}$ yields the most balanced separation for large reservoirs, the quality of separation inevitably deteriorates as the length of the input time series increases. In contrast, for reservoirs with independent and identically distributed connectivity entries, we prove that the classical scaling $N^{-1/2}$ is asymptotically optimal from the perspective of separation and derive quantitative bounds describing the evolution of separation with the time horizon. Finally, numerical experiments suggest a strong practical connection between balanced separation profiles and downstream learning performance. Beyond providing a theoretical justification for several common reservoir design choices, our results introduce separation capacity as a tractable and informative framework for the analysis of random reservoirs.

stat.ML↗

arXiv2022

The accessibility problem for geometric rough differential equations

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields with the rough calculus on manifolds developed by Cass, Litterer and Lyons.

math.CA↗

arXiv2022

Path classification by stochastic linear recurrent neural networks

We investigate the functioning of a classifying biological neural network from the perspective of statistical learning theory, modelled, in a simplified setting, as a continuous-time stochastic recurrent neural network (RNN) with identity activation function. In the purely stochastic (robust) regime, we give a generalisation error bound that holds with high probability, thus showing that the empirical risk minimiser is the best-in-class hypothesis. We show that RNNs retain a partial signature of the paths they are fed as the unique information exploited for training and classification tasks. We argue that these RNNs are easy to train and robust and back these observations with numerical experiments on both synthetic and real data. We also exhibit a trade-off phenomenon between accuracy and robustness.

stat.ML↗

arXiv2019

A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in "Rough Paths on Manifolds" (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

math.CA↗

arXiv2019

On Lipschitz maps and the Hölder regularity of flows

This paper regroups some of the basic properties of Lipschitz maps and their flows. Many of the results presented here are classical in the case of smooth maps. We prove them here in the Lipschitz case for a better understanding of the Lipschitz geometry and for a quantification of the related properties, which would be of use to the development of numerical methods for rough paths for example. We also introduce the notion of almost Lipschitz maps, which provide a sharper control and description of flows of Lipschitz vector fields and local inverses of Lipschitz injective immersions

math.CA↗

arXiv2013

Dimension-free Euler estimates of rough differential equations

We give a dimension-free Euler estimation of solution of rough differential equations in term of the driving rough path. In the meanwhile, we prove that, the solution of rough differential equation is close to the exponential of a Lie series, with a concrete error bound.

math.CA↗