Skip to main content
SpringerLink
Log in

Discrete-Time Quadratic Hedging of Barrier Options in Exponential Lévy Model

  • Conference paper
  • First Online:
Book cover Advanced Modelling in Mathematical Finance

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 189))

Abstract

We examine optimal quadratic hedging of barrier options in a discretely sampled exponential Lévy model that has been realistically calibrated to reflect the leptokurtic nature of equity returns. Our main finding is that the impact of hedging errors on prices is several times higher than the impact of other pricing biases studied in the literature.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In numerical calculations with a fixed value of z we choose c so as to minimize the value of the integrand at \(\lambda =0\), see [36], Eq. (3).

  2. 2.

    Computation of the discretely monitored option price in Black–Scholes model follows the methodology of [15]. Effectively, the calculation is the same as for V in the empirical model, but the multinomial transition probabilities approximate the Black-Scholes risk-neutral distribution \(N\left( \left( r-\sigma ^{2}/2\right) \varDelta ,\sigma ^{2}\varDelta \right) \).

  3. 3.

    Objective probability distribution of log returns in the Black-Scholes model is \(N\left( \left( \mu -\sigma ^{2}/2\right) \varDelta ,\sigma ^{2}\varDelta \right) \).

  4. 4.

    The barrier with delta of \(10^{-100}\) is so high that the corresponding results are, for all intents and purposes, indistinguishable from a plain vanilla option.

References

  1. Andricopoulos, A.D., Widdicks, M., Duck, P.W., Newton, D.P.: Universal option valuation using quadrature methods. J. Financ. Econ. 67(3), 447–471 (2003)

    Article  Google Scholar 

  2. Bowie, J., Carr, P.: Static simplicity. Risk 7(9), 45–49 (1994)

    Google Scholar 

  3. Boyle, P.P., Emanuel, D.: Discretely adjusted option hedges. J. Financ. Econ. 8, 259–282 (1980)

    Article  Google Scholar 

  4. Boyle, P.P., Lau, S.: Bumping up against the barrier with the binomial method. J. Deriv. 1(4), 6–14 (1994)

    Article  Google Scholar 

  5. Broadie, M., Glasserman, P., Kou, S.: A continuity correction for discrete barrier options. Math. Financ. 7(4), 325–348 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Financ. 11(3), 285–314 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carr, P., Chou, A.: Breaking barriers. Risk 10(9), 139–146 (1997)

    Google Scholar 

  8. Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J. Financ. 53(3), 1165–1191 (1998)

    Article  Google Scholar 

  9. Černý, A.: Optimal continuous-time hedging with leptokurtic returns. Math. Financ. 17(2), 175–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Černý, A.: Mathematical Techniques in Finance: Tools for Incomplete Markets, 2nd edn. Princeton University Press (2009)

    Google Scholar 

  11. Černý, A., Denkl, S., Kallsen, J.: Hedging in Lévy models and time step equivalent of jumps (2013). ArXiv preprint http://arxiv.org/pdf/1309.7833

  12. Derman, E., Ergener, D., Kani, I.: Forever hedged. Risk 7(9), 139–145 (1994)

    Google Scholar 

  13. Derman, E., Ergener, D., Kani, I.: Static options replication. J. Deriv. 2, 78–95 (1995)

    Article  Google Scholar 

  14. Derman, E., Kani, I., Ergener, D., Bardhan, I.: Enhanced numerical methods for options with barriers. Finan. Anal. J. 51(6), 65–74 (1995)

    Article  Google Scholar 

  15. Duan, J.C., Dudley, E., Gauthier, G., Simonato, J.G.: Pricing discretely monitored barrier options by a Markov chain. J. Deriv. 10(4), 9–31 (2003)

    Article  Google Scholar 

  16. Dupont, D.Y.: Hedging barrier options: Current methods and alternatives. Economics Series 103, Institute for Advanced Studies (2001)

    Google Scholar 

  17. Eberlein, E., Özkan, F.: Time consistency of Lévy models. Quan. Financ. 3, 40–50 (2003)

    Article  Google Scholar 

  18. Eberlein, E., Prause, K.: The generalized hyperbolic model: Financial derivatives and risk measures. In: Geman, H., Madan, D., Pliska, S.R., Vorst, T. (eds.) Mathematical Finance: Bachelier Congress 2000, pp. 245–267. Springer (2002)

    Google Scholar 

  19. Fang, F., Oosterlee, C.W.: Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numer. Math. 114(1), 27–62 (2009). doi:10.1007/s00211-009-0252-4, http://dx.doi.org/10.1007/s00211-009-0252-4

    Google Scholar 

  20. Feng, L., Linetsky, V.: Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach. Math. Financ. 18(3), 337–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Figlewski, S., Gao, B.: The adaptive mesh model: A new approach to efficient option pricing. J. Financ. Econ. 53(3), 313–351 (1999)

    Article  Google Scholar 

  22. Fusai, G., Abrahams, I., Sgarra, C.: An exact analytical solution for discrete barrier options. Financ. Stochast. 10(1), 1–26 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fusai, G., Recchioni, M.: Analysis of quadrature methods for pricing discrete barrier options. J. Econ. Dyn. Control 31(3), 826–860 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gobet, E., Temam, E.: Discrete time hedging errors for options with irregular payoffs. Finan. Stochast. 5(3), 357–367 (2001). doi:10.1007/PL00013539, http://dx.doi.org/10.1007/PL00013539

    Google Scholar 

  25. Hörfelt, P.: Pricing discrete European barrier options using lattice random walks. Math. Financ. 13(4), 503–524 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. İlhan, A., Sircar, R.: Optimal static-dynamic hedges for barrier options. Math. Financ. 16(2), 359–385 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, second edn. Springer, Berlin (2003)

    Google Scholar 

  28. Kou, S.G.: On pricing of discrete barrier options. Statistica Sinica 13(4), 955–964 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Kuan, G., Webber, N.: Valuing discrete barrier options on a Dirichlet lattice. FORC Preprint 140/04, University of Warwick (2004)

    Google Scholar 

  30. Leisen, D.: Valuation of barrier options in a Black-Scholes setup with jump risk. Eur. Finance Rev. 3(3), 319–342 (1999)

    Article  MATH  Google Scholar 

  31. Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Finance Rev. 2(1), 79–105 (1998)

    Article  MATH  Google Scholar 

  32. Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  33. Reiner, E., Rubinstein, M.: Breaking down the barriers. Risk 4(8), 28–35 (1991)

    Google Scholar 

  34. Ritchken, P.H.: On pricing barrier options. J. Deriv. 3(2), 19–28 (1995)

    Article  Google Scholar 

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

    Google Scholar 

  36. Strawderman, R.L.: Computing tail probabilities by numerical Fourier inversion: The absolutely continuous case. Statistica Sinica 14(1), 175–201 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Toft, K.B.: On the mean-variance tradeoff in option replication with transactions costs. J. Finan. Quant. Anal. 31(2), 233–263 (1996)

    Article  MathSciNet  Google Scholar 

  38. Wei, J.: Valuation of discrete barrier options by interpolations. J. Deriv. 6(1), 51–73 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cass Business School, City University London, 106 Bunhill Row, London, EC1Y 8TZ, UK

    Aleš Černý

Authors
  1. Aleš Černý
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aleš Černý .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

    Jan Kallsen

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Antonis Papapantoleon

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Černý, A. (2016). Discrete-Time Quadratic Hedging of Barrier Options in Exponential Lévy Model. In: Kallsen, J., Papapantoleon, A. (eds) Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-45875-5_12

Download citation

Publish with us

Policies and ethics

Search

Navigation