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.

Highlights

  • Stochastic analysis on manifolds is a vibrant and well-studied field dating back to the seminal work of Varadhan [30], followed by Elworthy [11], Hsu [19], Stroock [29], Grigoryan [13], Avramidi [2] and, in a financial context [1, 12, 16, 17]; of particular importance in these works is Brownian motion on a Riemannian manifold1

  • 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 to each particular model

  • The underlying manifold here is the state space of the process, which is in most cases a complete open manifold

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Summary

Introduction

Stochastic analysis on manifolds is a vibrant and well-studied field dating back to the seminal work of Varadhan [30], followed by Elworthy [11], Hsu [19], Stroock [29], Grigoryan [13], Avramidi [2] and, in a financial context [1, 12, 16, 17]; of particular importance in these works is Brownian motion on a Riemannian manifold. There, Hobson provides the following classification (and examples) of the large-time behaviour of sample paths of the X process for such models: (i) it can hit zero in finite time, (ii) it converges to a strictly positive limit, or (iii) it is always positive, but converges to zero as time tends to infinity Note that these cases are not necessarily exclusive from one another, and (i) and (ii) can both happen with positive probability; for a given model, it is in general difficult to estimate these probabilities precisely. We further present transformations of the hyperbolic Brownian motion under which this large-time property remains valid, and provide formulae for these probabilities; the resulting processes turn out to be precisely of the form (1.1).

SABR geometry and geometry preserving mappings
HO j φ0
Probability of hitting the boundary

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