On a Nonlinear Partial Integro-Differential Equation

2009/11/19 by Frédéric Abergel, Abergel, Frederic, Rémi Tachet +1 · 1 citation
Computer Science · Mathematics · #35A01 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0911.3664

openalex publication_date 2009/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. Local volatility models introduced by Dupire in 1994 are widely used to price and manage the risks of structured products. However, the inconsistencies observed between the dynamics of the smile in those models and in real markets motivate researches for stochastic volatility modelling. Combining both those ideas to form Local and Stochastic Volatility models is of interest for practitioners. In this paper, we study the calibration of the vanillas in those models. This problem can be written as a nonlinear and nonlocal partial differential equation, for which we prove short-time existence of solutions.

Cited by

• On the surjectivity of the conditional expectation given a real random variable

Related

• Calibration of Local-Stochastic and Path-Dependent Volatility Models to\n Vanilla and No-Touch Options
• Strong existence and uniqueness of a calibrated local stochastic volatility model
• USLV: Unspanned Stochastic Local Volatility Model
• Calibration of the Bass Local Volatility model
• Existence of a calibrated regime switching local volatility model and new fake Brownian motions