On the probability of hitting the boundary for Brownian motions on the SABR plane

TL;DR

This paper derives an explicit formula for the probability that certain geometric stochastic models, related to the SABR model in finance, hit their domain boundaries, by leveraging hyperbolic Brownian motion properties and transformations.

Contribution

It introduces a method to compute boundary hitting probabilities for SABR-related models using hyperbolic geometry and time-change techniques, providing explicit formulas.

Findings

Explicit boundary hitting probability formula for SABR models

Connection between hyperbolic Brownian motion and finance models

Use of geometry-preserving transformations and time-change methods

Abstract

Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.