Pricing options under rough volatility with backward SPDEs
- Bayer, Christian
- Qiu, Jinniao
- Yao, Yao
2010 Mathematics Subject Classification
- 91G20 60H15 91G60
- Rough volatility, option pricing, stochastic partial differential equation, machine learning, stochastic Feynman-Kac formula
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 solutions 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 learning-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.