  We derive quantitative error bounds for deep neural networks (DNNs)
approximating option prices on a $d$-dimensional risky asset as functions of
the underlying model parameters, payoff parameters and initial conditions. We
cover a general class of stochastic volatility models of Markovian nature as
well as the rough Bergomi model. In particular, under suitable assumptions we
show that option prices can be learned by DNNs up to an arbitrary small error
$\varepsilon \in (0,1/2)$ while the network size grows only sub-polynomially in
the asset vector dimension $d$ and the reciprocal $\varepsilon^{-1}$ of the
accuracy. Hence, the approximation does not suffer from the curse of
dimensionality. As quantitative approximation results for DNNs applicable in
our setting are formulated for functions on compact domains, we first consider
the case of the asset price restricted to a compact set, then we extend these
results to the general case by using convergence arguments for the option
prices.
