Diffusion transformations, Black-Scholes equation and optimal stopping

TL;DR

This paper introduces new path transformations for one-dimensional diffusions that change their long-term behavior, enabling explicit solutions for exit times, addressing non-uniqueness in Black-Scholes equations with bubbles, and solving optimal stopping problems.

Contribution

It develops a novel class of diffusion path transformations that unify the analysis of exit times, stochastic solutions of Cauchy problems, and optimal stopping in financial models.

Findings

Derived a formula for diffusion exit time distributions.

Resolved non-uniqueness issues in Black-Scholes equations with bubbles.

Provided explicit solutions for optimal stopping problems.

Abstract

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf \cite{KR} as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an {\em alternative} Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of {\em bubbles}. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.