Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

Abstract

We study the mass at the origin in the uncorrelated stochastic alpha, beta, rho stochastic volatility model and derive several tractable expressions, in particular when time becomes small or large. As an application—in fact the original motivation for this paper—we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

Keywords:

• SABR model
• Asymptotic expansions
• Implied volatility

JEL Classification:

• 60H30
• 91G99
• 91G60

Acknowledgements

The authors would like to thank Rama Cont and Josef Teichmann for initiating the series of ETH-Imperial College workshops, where this project initiated. BH would like to thank Leif Döring and Leonid Mytnik for stimulating discussions on time change techniques. BH acknowledges financial support from the SNF Early Postdoc Mobility Grant 165248. AJ acknowledges financial support from the EPSRC First Grant EP/M008436/1 and EPSRC Platform Grant EP/I019111/1. The numerical implementations have been carried out on the collaborative platform Zanadu (www.zanadu.io).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† That is, rates ‘stick’ to zero for certain periods of time, see Antonov et al. (2015b) for more details.

† We thank the anonymous referee for making us aware of this improvement.

Additional information

Funding

This work was suuported by the SNF Early Postdoc Mobility [grant number 165248]; EPSRC [grant number EP/M008436/1], [grant number EP/I019111/1].