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 Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure varepsilon error of DNN expressed option prices with DNNs of size that grows polynomially with respect to {mathcal{O}}(varepsilon ^{-1}), and with constants implied in {mathcal{O}}(, cdot , ) 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 Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide varepsilon expression error in DNN size {mathcal{O}}(|log (varepsilon )|^{a}) with exponent a proportional to d, but with constants implied in {mathcal{O}}(, cdot , ) growing exponentially with respect to d. Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy 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 Lévy triplet. We indicate several consequences and possible extensions of the presented results.
Highlights
Recent years have seen a dynamic development in applications of deep neural networks (DNNs for short) in expressing high-dimensional input–output relations
We study the expression rates of Deep neural networks (DNNs) for prices of options written on possibly large baskets of risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps
This article is concerned with establishing expression rate bounds of deep neural networks (DNNs) for prices of options written on possibly large baskets of risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps
Summary
Recent years have seen a dynamic development in applications of deep neural networks (DNNs for short) in expressing high-dimensional input–output relations. We prove two types of results on DNN expression rate bounds for European options in exponential Lévy models, with one probabilistic and one “deterministic” proof The former is based on concepts from statistical learning theory and provides for relevant payoffs The latter bound is based on parabolic smoothing of the Kolmogorov equation and allows us to prove exponential expressivity of prices for positive maturities, i.e., an expression error O(ε) with DNN sizes of O(| log ε|a) for some a > 0, albeit with constants implied in O( · ) possibly growing exponentially in d For the latter approach, a certain non-degeneracy is required for the symbol of the underlying Lévy process.
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