A new factor to explain implied volatility smirk

ABSTRACT

In this article, we find empirical evidence of a new smirk factor, obtained from the jump structure of the risk neutral distribution of the underlying Lévy process. As an application we show how to price a barrier style contract.

KEYWORDS:

• Skewness
• Lévy processes
• implied volatility smirk
• symmetry

JEL CLASSIFICATION:

• C52
• G10
• G13

Acknowledgements

I would like to thank an anonymous referee for his very useful comments. Also, I would like to thank Bruno Dupire, Liuren Wu, Roger Lee, and Ernst Eberlein for very useful comments and Fernando Mendo Lopez for research assistance. I also thank the comments of seminar participants at Stevanovich center University of Chicago. PIMSV–Universitat Bern. IMUB-Universitat de Barcelona. Universität Freiburg. CRM-Universitat Autonoma de Barcelona. V LubraFin. 7th Bachelier Finance Society Meeting and 10th World Congress of The Econometric Society for comments. Finally, I thank CNPq from Brazil for financial support. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ $\mathrm{\Pi }(\left\{0\right\})$ could be defined as 0. Here we follow Cont and Tankov (2004).

² Henceforth FW.

³ The intuition for this name came from the fact that when we plot implied volatility versus moneyness and $\beta$, we can obtain a figure with a torsion, as in Figure 1, build using Brazilian asset Petróbras (PETR4).

⁴ Taken from http://www.cboe.com/data/mktstat.aspx

⁵ It is also important to mention that this choice of $\beta$ allow us to guarantee ${\sigma }_{imp}>0$ for the sample set of moneyness.

⁶ Here we need the law of $S_T$ be absolutely continuous. The absolute continuity holds in all Lévy models of interest, see Theorem 27.4 in Sato (1999).

⁷ We simulate realizations of a NIG distribution with the estimated parameters.

⁸ see Table 15.2 in Cont and Tankov (2004).

Additional information

Funding

This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico.