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Boris Günther, Thomas Kruse, Ludger Overbeck, Thorsten Schmidt · 2026-06-22
The paper extends the mathematical theory of 'affine processes' — a class of models widely used in finance for their tractability — to cases where the coefficients depend on the entire past path, not just the current state. It derives analytic formulas (via Riccati-type equations) for the Fourier–Laplace transform of these processes and applies them to path-dependent volatility models, including a path-dependent and delayed version of the Heston model.
Why it matters: Affine and Heston-type models underpin much of derivatives pricing and volatility modeling, and this work provides tractable formulas for versions where volatility depends on history — potentially useful for pricing options under more realistic, memory-driven volatility dynamics. It may help quants build and calibrate path-dependent volatility models with the analytic convenience previously limited to non-path-dependent ones.
⚠ This is a theoretical mathematical-finance paper with no empirical calibration or trading results, so practical value depends on implementation and real-market validation.
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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