Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

33 Pages Posted: 10 Jan 2017

See all articles by Leif Döring

Leif Döring

University of Mannheim - School of Mathematics and Computer Science

Blanka Horvath

Imperial College London - Department of Mathematics

Josef Teichmann

ETH Zurich; Swiss Finance Institute

Date Written: January 8, 2017

Abstract

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

Keywords: SABR model, time change, asymptotics, semigroups, Feller property, Dirichlet forms

JEL Classification: C5, C6

Suggested Citation

Döring, Leif and Horvath, Blanka and Teichmann, Josef, Functional Analytic (Ir-)Regularity Properties of SABR-type Processes (January 8, 2017). Available at SSRN: https://ssrn.com/abstract=2895723 or http://dx.doi.org/10.2139/ssrn.2895723

Leif Döring

University of Mannheim - School of Mathematics and Computer Science ( email )

United States

Blanka Horvath (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
Imperial College
LONDON, SW7 2AZ
United Kingdom

HOME PAGE: http://www.imperial.ac.uk/people/b.horvath

Josef Teichmann

ETH Zurich ( email )

Rämistrasse 101
ZUE F7
Zürich, 8092
Switzerland

HOME PAGE: http://www.math.ethz.ch/~jteichma

Swiss Finance Institute ( email )

c/o University of Geneva
40 Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

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