Affine models with path-dependence under parameter uncertainty and their application in finance

TL;DR

This paper develops a framework for valuing path-dependent financial derivatives under model uncertainty using functional Itô calculus and neural networks, enabling efficient numerical solutions for complex payoffs like barrier options.

Contribution

It introduces a path-dependent extension of affine models under uncertainty and applies neural networks to approximate functional derivatives for valuation.

Findings

Neural network approximation enables tractable valuation under uncertainty.

Path-dependent models extend affine processes to complex payoffs.

Application to credit and counterparty risk modeling demonstrates practical relevance.

Abstract

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 such payoffs in an efficient manner, we approximate the functional derivatives with deep neural networks and show that the numerical valuation under parameter uncertainty is highly tractable. Finally, we consider the application to structural modelling of credit and counterparty risk, where both parameter uncertainty and path-dependence are crucial and the approach proposed here opens the door to efficient numerical methods in this field.