  We analyze approximation rates by deep ReLU networks of a class of
multi-variate solutions of Kolmogorov equations which arise in option pricing.
Key technical devices are deep ReLU architectures capable of efficiently
approximating tensor products. Combining this with results concerning the
approximation of well behaved (i.e. fulfilling some smoothness properties)
univariate functions, this provides insights into rates of deep ReLU
approximation of multi-variate functions with tensor structures. We apply this
in particular to the model problem given by the price of a European maximum
option on a basket of $d$ assets within the Black-Scholes model for European
maximum option pricing. We prove that the solution to the $d$-variate option
pricing problem can be approximated up to an $\varepsilon$-error by a deep ReLU
network with depth $\mathcal{O}\big(\ln(d)\ln(\varepsilon^{-1})+\ln(d)^2\big)$
and $\mathcal{O}\big(d^{2+\frac{1}{n}}\varepsilon^{-\frac{1}{n}}\big)$ non-zero
weights, where $n\in \mathbb{N}$ is arbitrary (with the constant implied in
$\mathcal{O}(\cdot)$ depending on $n$). The techniques developed in the
constructive proof are of independent interest in the analysis of the
expressive power of deep neural networks for solution manifolds of PDEs in high
dimension.
