Dear colleagues, you are all invited to participate in the following seminar organized by QFinLab - Department of Mathematics, Politecnico di Milano. Wednesday, 1 October 2025, 12.15-13.15 Seminar room, third floor, building 14, Via Bonardi 9, Milano (Leonardo Campus) Anna Maria Gambaro (Università del Piemonte Orientale) Title: Functional PCA for Risk-Neutral densities in Bayes Hilbert space. Abstract: In this work, we investigate the main drivers of risk-neutral densities of quoted stocks, using the functional principal component analysis (FPCA). To this end, we first construct a historical series of risk-neutral densities corresponding to quoted option prices with fixed time to maturity, using exponential expansions of orthogonal polynomials. Then, we apply the centered log-ratio transformation (CLRT) to the extracted densities and we perform the FPCA in the Bayes–Hilbert space. The CLRT provides an isometric isomorphism between the Bayes space of square log-integrable densities and the classical Hilbert space of square-integrable functions. As a result, the projected data onto the principal component basis correspond to the CLRT-transformed densities, and the application of the inverse CLRT yields proper density functions. Furthermore, by modeling the historical series of FPCA loadings as a stochastic process, we exploit the FPCA representation for forecasting purposes. Finally, we discuss extensions of this framework to cross-asset analyses and to the modeling of option price surfaces. This is a joint work with A. Amici e G. Fusai. Attendance is also possible online (Microsoft Teams), clicking here.<https://teams.microsoft.com/l/meetup-join/19%3ameeting_YmFlY2M5OWEtYzg1OS00MGVlLTliNjUtNGM4NzRiMGM5YmMx%40thread.v2/0?context=%7b%22Tid%22%3a%220a17712b-6df3-425d-808e-309df28a5eeb%22%2c%22Oid%22%3a%22afb4ebfa-b975-4c1c-879c-a0dad0b4e60b%22%7d> All news can be found on the QFinLab webpage<https://www.qfinlab.polimi.it/seminars-and-meetings/>. The organizers: Michele Azzone and Alessandro Calvia.