AFFINE MODELS WITH PATH-DEPENDENCE UNDER PARAMETER UNCERTAINTY AND THEIR APPLICATION IN FINANCE

Benedikt Geuchen (), Katharina Oberpriller () and Thorsten Schmidt
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Benedikt Geuchen: 56 Rue de la Rochette 77000 Melun, France
Katharina Oberpriller: Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-Maximilians Universität, Theresienstr. 39, 80333 Munich, Germany
Thorsten Schmidt: Albert-Ludwigs University of Freiburg, Ernst-Zermelo-Str. 1 79104 Freiburg, Germany

International Journal of Theoretical and Applied Finance (IJTAF), 2024, vol. 27, issue 02, 1-36

Abstract: In this paper, we consider one-dimensional generalized affine processes under the paradigm of Knightian uncertainty (the so-called nonlinear generalized affine models). This extends and generalizes previous results in Fadina et al. (2019) and Lütkebohmert 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 relying on functional Itô calculus. We establish a dynamic programming principle which then leads to a functional nonlinear 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.

Keywords: Affine processes; Knightian uncertainty; VasiÄ ek model; Cox–Ingersoll–Ross model; nonlinear affine process; Kolmogorov equations; fully nonlinear PDE; functional Itô calculus; deep-learning; Merton model; structural models; credit risk (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S021902492450016X

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