Citation: Di Persio, Luca, Emanuele Lavagnoli, and Marco Patacca. 2022. Calibrating FBSDEs Driven Models in Finance via NNs. Risks 10: 227. https://doi.org/10.3390/ risks10120227 Academic Editor: Mogens Steffensen Received: 8 October 2022 Accepted: 23 November 2022 Published: 30 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). risks Article Calibrating FBSDEs Driven Models in Finance via NNs Luca Di Persio 1,† , Emanuele Lavagnoli 1,† and Marco Patacca 2, * ,† 1 Department of Computer Science, University of Verona, via Ca’ Vignal 2, 37129 Verona, Italy 2 Department of Economics and Finance, University of Rome Tor Vergata, via Columbia 2, 00133 Rome, Italy * Correspondence: marco.patacca@uniroma2.it † These authors contributed equally to this work. Abstract: The curse of dimensionality problem refers to a set of troubles arising when dealing with huge amount of data as happens, e.g., applying standard numerical methods to solve partial differential equations related to financial modeling. To overcome the latter issue, we propose a Deep Learning approach to efficiently approximate nonlinear functions characterizing financial models in a high dimension. In particular, we consider solving the Black–Scholes–Barenblatt non-linear stochastic differential equation via a forward-backward neural network, also calibrating the related stochastic volatility model when dealing with European options. The obtained results exhibit accurate approximations of the implied volatility surface. Specifically, our method seems to significantly reduce the neural network’s training time and the approximation error on the test set. Keywords: Black–Scholes–Barenblatt; neural networks; stochastic volatility models JEL Classification: C45; C63 1. Introduction The curse of dimensionality problem refers to a set of troubles that arise when dealing with big data. Classical methods for solving partial differential equations suffer from this prob- lem, especially in the financial field (Bayer and Stemper 2018; Wang 2006; Weinan et al. 2019). For example, we can think of the non-linear Black–Scholes equation for pricing financial derivatives in which the number of considered underlying assets gives the dimensionality of the Partial Differential Equation (PDE). The latter scenario, from a purely mathematical point of view, typically translates in the need of approximating nonlinear functions in a high dimension. An effective solution can be obtained exploiting a Deep Learning approach, namely using a particular type of Machine Learning (set of) solutions. For the sake of completeness, let us recall that Machine Learning (ML) is the branch of computer science in which mathematical and statistical techniques are used to give computer systems the ability to learn through data. There are two main branches of ML algorithms: Supervised Learning and Unsupervised Learning. The former use labeled datasets while the latter use unlabeled data. The increase of available data in recent years and the need to analyze them, finds in Machine Learning the ideal tool. Indeed, the peculiar features and the change of programming paradigm of the latter, make it the best candidate to solve problems with huge number of data in different fields. For example, computer vision, natural lan- guage processing, time series analysis, differential equation. Many papers have been pub- lished with the aim of improving theoretical and empirical knowledge; see, among others, Germain et al. (2021); Han (2016) and (Carleo and Troyer 2017). This fame inspires specula- tions that Deep Learning might hold the key to solving the curse of dimensionality problem. Neural networks (NNs) are the backbone of deep learning algorithms and their use to solve real-world problems. Indeed, even if their conceptual development can be dated back to several years ago, only during last few years, particularly since powerful hardware became economical affordable, they proved to be concretely and efficiently applied to Risks 2022, 10, 227. https://doi.org/10.3390/risks10120227 https://www.mdpi.com/journal/risks