Barrier style contracts under Lévy processes once again

Abstract

In this paper we present new pricing formulas for some Barrier style contracts of European type when the underlying process is driven by an important class of Lévy processes, which includes CGMY model, generalized hyperbolic Model and Meixner Model, when no symmetry properties are assumed, complementing in this way previous findings in Fajardo (J Bank Financ 53:179–187, 2015). Also, we show how to implement our new formulas.

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

    \(\Pi (\{0\})\) could be defined as 0. Here I follow Cont and Tankov (2004).

  2. 2.

    Equivalently for all t, see Theorem 25.17 in Sato (1999).

  3. 3.

    More precisely, \(\frac{d\tilde{\mathbb Q}_t}{d{\mathbb Q}_t}=e^{X_t-(r-\delta )t},\;\;t\ge 0\).

  4. 4.

    The Lévy models of interest satisfy it.

  5. 5.

    A payoff function is a nonnegative Borel function on \(\mathbb R\).

  6. 6.

    Here we use the same denomination used in example 5.15 by Carr and Lee (2009).

  7. 7.

    This method is widely used to estimate Lévy processes, see for example Ramezani and Zeng (2007).

References

  1. Bates, D.: The skewness premium: Option pricing under asymmetric processes. Adv Futur Options Res 9, 51–82 (1997)

    Google Scholar 

  2. Carr, P., Crosby, J.: A class of levy process models with almost exact calibration of both barrier and vanilla fx options. Quant Financ 10(10), 1115–1136 (2010)

    Article  Google Scholar 

  3. Carr, P., Lee, R.: Put call symmetry: Extensions and applications. Math Financ 19(4), 523–560 (2009)

    Article  Google Scholar 

  4. Carr, P., Wu, L.: Stochastic skew in currency options. J Financ Econ 86(1), 213–247 (2007)

    Article  Google Scholar 

  5. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall /CRC Financial Mathematics Series, London (2004)

    Google Scholar 

  6. Corcuera, J., De Spiegeleer, J., Fajardo, J., Jonsson, H., Schoutens, W., Valdivia, A.: Close form pricing formulas for coupon cancellable cocos. J Bank Financ 42, 339–351 (2014)

    Article  Google Scholar 

  7. Eberlein, E., Glau, K., Papapantoleon, A.: Analysis of fourier transform valuation formulas and applications. Appl Math Financ 17(3), 211–240 (2010)

    Article  Google Scholar 

  8. Eberlein, E., Glau, K., Papapantoleon, A.: Analyticity of the wiener-hopf factors and valuation of exotic options in Lévy models. In: Nunno, G.D., Øksendal, B. (eds.) Advanced Mathematical Methods for Finance. Springer, Berlin (2011)

    Google Scholar 

  9. Eberlein, E., Prause, K.: The generalized hyperbolic model: Financial derivatives and risk measures. In: Geman, S.P.T.V.H., Madan, D. (eds.) Mathematical Finance-Bachelier Congress 2000. Springer, Berlin (2002)

    Google Scholar 

  10. Fajardo, J.: Barrier style contracts under Lévy processes: An alternative approach. J Bank Financ 53, 179–187 (2015)

    Article  Google Scholar 

  11. Fajardo, J.: A new factor to explain implied volatility smirk. Appl Econ 49(40), 4026–4034 (2017)

    Article  Google Scholar 

  12. Fajardo, J., Farias, A.: Generalized hyperbolic distributions and Brazilian data. Braz Rev Econ 24(2), 249–271 (2004)

    Google Scholar 

  13. Fajardo, J., Mordecki, E.: Symmetry and duality in Lévy markets. Quant Financ 6(3), 219–227 (2006)

    Article  Google Scholar 

  14. Fajardo, J., Mordecki, E.: Skewness premium with Lévy processes. Quant Financ 14(9), 1619–1626 (2014)

    Article  Google Scholar 

  15. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes. Springer, Berlin, Heidelberg (1987)

    Google Scholar 

  16. Lewis, A.L.: A simple option formula for general jump-diffusion and other exponential Lévy processes. Working paper. Envision Financial Systems and OptionCity.net Newport Beach, California, USA. http://www.optioncity.net (2001)

  17. Ramezani, C.A., Zeng, Y.: Maximum likelihood estimation of the double exponential jump-diffusion process. Ann Financ 3, 487–507 (2007)

    Article  Google Scholar 

  18. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  19. Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York (2003)

    Google Scholar 

  20. Schoutens, W., Cariboni, J.: Lévy Processes in Credit Risk. Wiley, New York (2009)

    Google Scholar 

Download references

Acknowledgements

I would like to thank the comments of an anonymous referee who helps to improve the present version of the paper. Also, I would like to thank seminar participants at EMAp/FGV, SBFin 2015, SAET 2016 and Barcelona Workshop on Mathematical Finance. Financial support from CNPq of Brazil is also acknowledge.

Author information

Affiliations

Authors

Corresponding author

Correspondence to José Fajardo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fajardo, J. Barrier style contracts under Lévy processes once again. Ann Finance 14, 93–103 (2018). https://doi.org/10.1007/s10436-017-0303-2

Download citation

Keywords

  • Skewness
  • Lévy processes
  • Absence of symmetry
  • Barrier contracts

JEL Classification

  • C52
  • G12