Our first seminar of 2026 will be held on Monday, March 16, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time). The meeting will take place at room C116- Bloco C - CT – Instituto de Matemática – UFRJ. There will be no transmission online.
Speaker: Vinicius Viana Luiz Albani (IM-UFRJ)
Title: Joint Calibration of S&P 500 and Volatility Index Smiles by a Stochastic Volatility Framework
Abstract:
In recent years, the quantitative finance community has faced the challenge of developing models capable of jointly calibrating the observed call and put option prices on the S&P 500 and the VIX volatility index. Many existing approaches rely on non-Markovian dynamics, which often implies substantial theoretical complexity in the pricing framework and may lead to computationally demanding numerical implementations. In contrast, we propose a Markovian framework based on a stochastic volatility model with functional parameters. Within this setting, we derive a partial differential equation governing the VIX dynamics. Moreover, the associated forward Kolmogorov equation and its adjoint partial differential equation are employed to compute option prices on both the S&P 500 and the VIX.
This structure significantly simplifies the pricing and calibration procedures while preserving modeling flexibility. Nonetheless, important theoretical questions remain open, particularly concerning well-posedness and analytical properties of the proposed system. These issues will be discussed in the talk. The problem is based on tools from stochastic analysis, partial differential equations, and functional analysis. Numerical experiments based on real market data will be presented to demonstrate the practical viability of the methodology.
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