Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential L\'evy models

TL;DR

This paper demonstrates that deep neural networks can efficiently approximate option prices in high-dimensional exponential Le9vy models, overcoming the curse of dimensionality under certain conditions, and provides alternative architectures with different error-growth trade-offs.

Contribution

It establishes conditions under which DNNs can approximate high-dimensional option prices with polynomial or logarithmic size, overcoming the curse of dimensionality in jump models.

Findings

DNNs can approximate option prices with polynomial size in psilon under certain conditions.

Alternative DNN architectures achieve logarithmic size in psilon but with exponential constants in dimension.

Under stronger conditions, DNN approximation rates are dimension-free and depend on sparsity.

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 L\'evy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the L\'evy 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 with respect $d$, thereby overcoming the curse of dimensionality 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 integrodifferential equations for certain multivariate L\'evy processes to present alternative architectures of ReLU DNNs that provide $\varepsilon$ expression error in DNN size $\mathcal{O}(|\log(\varepsilon)|^a)$ with exponent $a \sim d$, however, with constants implied in $\mathcal{O}(\cdot)$ growing exponentially with respect to $d$. Under stronger, dimension-uniform non-degeneracy conditions on the L\'evy symbol, we obtain algebraic expression rates of option prices in exponential L\'evy 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 L\'evy triplet. We indicate several consequences and possible extensions of the present results.