  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.
