FUNCTIONAL ANALYTIC (IR-)REGULARITY PROPERTIES OF SABR-TYPE PROCESSES

Leif Döring (), Blanka Horvath and Josef Teichmann ()
Additional contact information
Leif Döring: Department of Mathematics, University of Mannheim, Germany
Blanka Horvath: #x2020;Department of Mathematics, Imperial College London, UK
Josef Teichmann: #x2021;Department of Mathematics, ETH Zürich, Switzerland

International Journal of Theoretical and Applied Finance (IJTAF), 2017, vol. 20, issue 03, 1-48

Abstract: The stochastic alpha, beta, rho (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 the markets, 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 a 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 behavior of SABR-type processes beyond the heat-kernel framework.

Keywords: SABR model; option pricing; time-change; asymptotics; semigroups; Feller property; Dirichlet forms (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0219024917500133
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:ijtafx:v:20:y:2017:i:03:n:s0219024917500133

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0219024917500133

Access Statistics for this article

International Journal of Theoretical and Applied Finance (IJTAF) is currently edited by L P Hughston

More articles in International Journal of Theoretical and Applied Finance (IJTAF) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().