Affine processes occupy a central role in mathematical finance and other applied fields due to their tractable structure. This article develops probabilistic representation formulas and integration-by-parts identities for expectations of affine processes. A defining feature of the resulting expressions is that they can be written as expectations of affine processes with suitably shifted parameters. The derivations rely on Fourier analysis and characteristic functions, rather than pathwise differentiability, thereby enabling the treatment of affine diffusion models across a wide range of parameter regimes. These identities are well suited for stable Monte Carlo evaluation, as derivatives are reduced to standard expectations under parameter shifts. To illustrate the approach, the formulas are applied to the Cox–Ingersoll–Ross interest rate model, where differentiation with respect to the initial value corresponds to a Greek (delta) in option pricing. The framework provides a unified and robust tool for settings in which classical Malliavin calculus is difficult to apply.

Free probability theory was established by Dan Voiculescu in the 1980s to resolve an open problem in operator algebra theory. The random-matrix theory has undergone active development in recent years, attracting considerable attention for its connections to diverse fields, such as quantum information theory and deep learning. Concurrently, various extensions and deformations have been proposed as part of foundational research for Voiculescu’s free probability theory. In 2006, Ben Arous and Voiculescu proposed a theoretical framework for analyzing the spectral maximum of freely independent, noncommutative random variables. This framework is considered the free probability analog of classical extreme value statistics, which has evolved into the free extreme value theory. In this study, we review advances in this theory, incorporating the author’s most recent findings.