Deep ReLU network expression rates for option prices in high-dimensional, exponential Levy models

Gonon, Lukas und Schwab, Christoph (2021): Deep ReLU network expression rates for option prices in high-dimensional, exponential Levy models. In: Finance and Stochastics, Bd. 25, Nr. 4: S. 615-657

Abstract

We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets whose log-returns are modelled by a multivariate Levy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Levy process X that ensure epsilon error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(epsilon(-1)), and with constants implied in O(center dot) which grow polynomially in d, thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Levy processes to present alternative architectures of ReLU (rectified linear unit) DNNs that provide e expression error in DNN size O(vertical bar log(epsilon)vertical bar(a)) with exponent a proportional to d, but with constants implied in O(center dot) growing exponentially with respect to d. Under stronger, dimension-uniform non-degeneracy conditions on the Levy symbol, we obtain algebraic expression rates of option prices in exponential Levy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Levy triplet. We indicate several consequences and possible extensions of the presented results.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 0949-2984
• Sprache: Englisch