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.

翻译：我们推导了深度神经网络（DNNs）逼近$d$维风险资产期权价格作为底层模型参数、支付参数和初始条件函数的定量误差界。我们涵盖了马尔可夫性质的一般随机波动率模型以及粗糙Bergomi模型。特别地，在适当假设下，我们证明DNNs能以任意小的误差$\varepsilon \in (0,1/2)$学习期权价格，而网络规模仅随资产向量维度$d$和精度倒数$\varepsilon^{-1}$呈次多项式增长。因此，该逼近不受维度灾难影响。由于适用于我们设定下的DNN定量逼近结果是针对紧致域上的函数提出的，我们首先考虑资产价格限制在紧致集上的情况，然后通过期权价格收敛论证将这些结果推广到一般情形。