Sharp Estimates for Geman-Yor Processes and applications to Arithmetic Average Asian options

TL;DR

This paper establishes sharp bounds for the fundamental solutions of PDEs linked to Geman-Yor processes, enhancing the understanding of Asian option pricing models in finance.

Contribution

It provides new pointwise bounds for PDEs associated with Geman-Yor processes, combining invariant Harnack inequalities and optimal control techniques.

Findings

Derived explicit lower bounds for fundamental solutions.

Established upper bounds via Hamilton-Jacobi-Bellman equations.

Applied results to improve Asian option pricing models.

Abstract

We prove the existence and pointwise lower and upper bounds for the fundamental solution of the degenerate second order partial differential equation related to Geman-Yor stochastic processes, that arise in models for option pricing theory in finance. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the optimal cost satisfies a specific Hamilton-Jacobi-Bellman equation.