Option pricing under path-dependent stock models

TL;DR

This paper develops a framework for pricing and hedging options in path-dependent stock models using path-dependent PDEs, extending classical methods to more complex, history-dependent scenarios.

Contribution

It introduces a path-dependent Feynman-Kac formula and derives differentiability and Greeks formulas for models with path-dependent coefficients.

Findings

Option prices are differentiable with respect to time and path.

Provides a path-dependent PDE representation of option prices.

Derives formulas for Greeks under path-dependent coefficient perturbations.

Abstract

This paper studies how to price and hedge options under stock models given as a path-dependent SDE solution. When the path-dependent SDE coefficients have Fr\'{e}chet derivatives, an option price is differentiable with respect to time and the path, and is given as a solution to the path-dependent PDE. This can be regarded as a path-dependent version of the Feynman-Kac formula. As a byproduct, we obtain the differentiability of path-dependent SDE solutions and the SDE representation of their derivatives. In addition, we provide formulas for Greeks with path-dependent coefficient perturbations. A stock model having coefficients with time integration forms of paths is covered as an example.