  In this paper, we study the option pricing problems for rough volatility
models. As the framework is non-Markovian, the value function for a European
option is not deterministic; rather, it is random and satisfies a backward
stochastic partial differential equation (BSPDE). The existence and uniqueness
of weak solution is proved for general nonlinear BSPDEs with unbounded random
leading coefficients whose connections with certain forward-backward stochastic
differential equations are derived as well. These BSPDEs are then used to
approximate American option prices. A deep leaning-based method is also
investigated for the numerical approximations to such BSPDEs and associated
non-Markovian pricing problems. Finally, the examples of rough Bergomi type are
numerically computed for both European and American options.
