Stochastic Black-Scholes equation for option pricing under rough volatility

Rough volatility is a pretty new paradigm in finance. We shall talk about 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) or so-called stochastic Black-Scholes equation.The wellposedness of such kind of BSPDEs and associated Feynman-Kac representationswill be discussed.These BSPDEs are then used to approximate American option prices. Moreover, a deep leaning-based method is also proposed and investigatedfor the numerical approximations tosuch BSPDEs and associated non-Markovian pricing problems. Two numerical examples will be presented for both European and American options. This talk is based on joint work with Christian Bayer and Yao Yao.

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