On a Nonlinear Partial Integro-Differential Equation

TL;DR

This paper addresses the calibration of local and stochastic volatility models in finance, formulating it as a nonlinear, nonlocal partial differential equation and proving short-time existence of solutions.

Contribution

It introduces a mathematical framework for calibrating complex volatility models by analyzing a nonlinear, nonlocal PDE and establishing short-time existence results.

Findings

Proves short-time existence of solutions to the PDE

Provides a mathematical foundation for model calibration

Addresses inconsistencies in volatility modeling

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.