Sensitivity analysis of path-dependent options in an incomplete market with pathwise functional Ito calculus

TL;DR

This paper develops a functional Itô calculus framework for path-dependent options in incomplete markets, deriving formulas for option sensitivities and demonstrating their application through numerical methods in the Black-Scholes model.

Contribution

It introduces a pathwise functional Itô calculus approach for analyzing path-dependent options, including new formulas for Greeks and their numerical computation.

Findings

Derived functional Itô and Feynman-Kac formulas for path-dependent processes

Expressed Greeks as expectations suitable for Monte Carlo simulation

Applied methods to digital options in the Black-Scholes model

Abstract

Functional It^o calculus is based on an extension of the classical It^o calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on Follmer's deterministic proof of the It^o formula, see [3], and a notion of pathwise functional derivatives introduced by [5]. There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see [6, 9, 8]. Using functional It^o calculus and the notion of quadratic variation, we derive the functional It^o formula along with the Feynman-Kac formula for functional processes. Furthermore, we express the Greeks for path-dependent options as expectations, which can be efficiently computed numerically using Monte Carlo simulations. We illustrate these results by applying the formulae to digital options within the Black-Scholes model framework.