WIAS Preprint No. 2745, (2020)

Pricing options under rough volatility with backward SPDEs


  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • 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.

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