How to make Dupire’s local volatility work with jumps

¶ The last author, Marc Yor, passed away shortly after final acceptance of this manuscript.

Abstract

There are several (mathematical) reasons why Dupire’s formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note, we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire’s local vol diffusion process recreates the correct option prices, even in manifest presence of jumps.

Keywords:

• Local volatility theory
• Volatility surfaces
• Computational finance
• Fokker–Planck equation

JEL Classification:

• C0
• C00

View correction statement:

Corrigendum

Acknowledgements

P. Friz acknowledges support from MATHEON and ERC grant Nr.258237. S. Gerhold acknowledges financial support from the Austrian Science Fund (FWF) under grant P 24880-N25. We thank Peter Laurence for sending us the unpublished preprint (Laurence 2009). Last but not least, we thank Rama Cont, Bruno Dupire, Sean Violante and the anonymous referees for helpful remarks.

Notes

¶ The last author, Marc Yor, passed away shortly after final acceptance of this manuscript.

ǁ Throughout, we assume a zero interest rate for simplicity. Extension to a constant rate is straightforward; see Remark 2.

† Doing so without introducing arbitrage is decisively non-trivial, see Gatheral and Jacquier (2011, 2012).

‡ The mathematics here is well understood (see Bichteler et al. (1987), Norris (1986, 1988)); a smooth density (and then call prices) are typically the result of a (hypo)elliptic diffusion part, possibly aided by infinite activity jumps.

This article was originally published with errors. This version has been amended. Please see Corrigendum (http://dx.doi.org/10.1080/14697688.2014.912872).