Proof of non-convergence of the short-maturity expansion for the SABR model

Alan Lewis and Dan Pirjol

Quantitative Finance, 2022, vol. 22, issue 9, 1747-1757

Abstract: We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ( $ \beta =1 $ β=1) SABR model. In this model, the option time-value can be represented as an integral of the form $ V(T) = \int _{0}^\infty e^{-\frac {u^2}{2T}} g(u)\,{\rm d}u $ V(T)=∫0∞e−u22Tg(u)du with $ g(u) $ g(u) a ‘payoff function’ which is given by an integral over the McKean kernel $ \mathcal {G}(t,s) $ G(t,s). We study the analyticity properties of the function $ g(u) $ g(u) in the complex u-plane and show that it is holomorphic in the strip $ |\Im (u) | 0). In a certain limit which can be defined either as the large volatility limit $ \sigma _0\to \infty $ σ0→∞ at fixed $ \omega =1 $ ω=1, or the small vol-of-vol limit $ \omega \to 0 $ ω→0 limit at fixed $ \omega \sigma _0 $ ωσ0, the short maturity T-expansion for the implied volatility has a finite convergence radius $ T_c = \frac {1.32}{\omega \sigma _0} $ Tc=1.32ωσ0.

Date: 2022
References: Add references at CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://hdl.handle.net/10.1080/14697688.2022.2071759 (text/html)
Access to full text is restricted to subscribers.

Related works:
Working Paper: Proof of non-convergence of the short-maturity expansion for the SABR model (2021)
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:taf:quantf:v:22:y:2022:i:9:p:1747-1757

Ordering information: This journal article can be ordered from
http://www.tandfonline.com/pricing/journal/RQUF20

DOI: 10.1080/14697688.2022.2071759

Access Statistics for this article

Quantitative Finance is currently edited by Michael Dempster and Jim Gatheral

More articles in Quantitative Finance from Taylor & Francis Journals
Bibliographic data for series maintained by Chris Longhurst ().