Regularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options

TL;DR

This paper proves the regularity of probabilistic solutions to a degenerate parabolic PDE used in pricing Asian options, clarifying the mathematical properties of the solution in financial mathematics.

Contribution

It establishes the regularity of probabilistic solutions to Vecer's degenerate PDE, bridging the gap between probabilistic and classical solutions in option pricing models.

Findings

Probabilistic solutions are shown to be regular.

The PDE admits a classical solution under certain conditions.

Clarifies the mathematical foundation of Asian option pricing models.

Abstract

Vecer derived a degenerate parabolic equation with a boundary condition characterizing the price of Asian options with generally sampled average. It is well understood that there exists a unique probabilistic solution to such a problem but it remained unclear whether the probabilistic solution is a classical solution. We prove that the probabilistic solutions to Vecer's PDE are regular.