Classical and deep pricing for Path-dependent options in non-linear generalized affine models

In this work we consider one-dimensional generalized affine processes under the paradigm of Knightian uncertainty (so-called non-linear generalized affine models). This extends and generalizes previous results in Fadina et al. (2019) and L\"utkebohmert et al. (2022). In particular, we study the case when the payoff is allowed to depend on the path, like it is the case for barrier options or Asian options. To this end, we develop the path-dependent setting for the value function which we do by relying on functional It\^o-calculus. We establish a dynamic programming principle which then leads to a functional non-linear Kolmogorov equation describing the evolution of the value function. While for Asian options, the valuation can be traced back to PDE methods, this is no longer possible for more complicated payoffs like barrier options. To handle these in an efficient manner, we approximate the functional derivatives with deep neural networks and show that numerical valuation under parameter uncertainty is highly tractable.