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Boris Günther, Thomas Kruse, Ludger Overbeck, Thorsten Schmidt · 2026-06-22
The paper extends affine processes—a mathematical class widely used in finance—to a 'path-dependent' setting where the model's coefficients depend on the whole history of the path, not just its current value. It derives analytic formulas (via generalized Riccati-type equations) for the Fourier–Laplace transform of these processes, proves characterization results, and applies them to path-dependent volatility models including a delayed version of the Heston model.
Why it matters: Affine and Heston-type models underpin much of options pricing and volatility modeling, so a tractable path-dependent generalization could allow models that capture memory effects in volatility while remaining computationally manageable. This is foundational math that could feed into more realistic derivative-pricing and volatility tools, though it is not itself a trading strategy.
⚠ This is a pure mathematical-finance theory paper with no empirical calibration or trading results, so practical benefit depends entirely on downstream implementation and testing.
We extend the classical theory of affine processes to a path-dependent setting by introducing path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform in terms of generalized Riccati-type equations. In the proposed framework, we define path-dependent affine processes through their exponential-affine Fourier--Laplace transform on the path space and establish a characterization theorem. Conversely, for path-dependent stochastic differential equations with affine path-dependent coefficients, we also provide explicit exponential-affine representations of the Fourier--Laplace functional in terms of those Riccati equations. Moreover, we derive a condition ensuring non-negativity of the path-dependent diffusion coefficient, guaranteeing well-posedness of the model. Finally, we apply these results to a path-dependent volatility model and a path-dependent extension of the Heston model, including a delayed Heston model as a special case.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.