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

  • Aleš ČernýEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (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.

Keywords

Barrier option Quadratic hedging Lévy model 

References

  1. 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)CrossRefGoogle Scholar
  2. 2.
    Bowie, J., Carr, P.: Static simplicity. Risk 7(9), 45–49 (1994)Google Scholar
  3. 3.
    Boyle, P.P., Emanuel, D.: Discretely adjusted option hedges. J. Financ. Econ. 8, 259–282 (1980)CrossRefGoogle Scholar
  4. 4.
    Boyle, P.P., Lau, S.: Bumping up against the barrier with the binomial method. J. Deriv. 1(4), 6–14 (1994)CrossRefGoogle Scholar
  5. 5.
    Broadie, M., Glasserman, P., Kou, S.: A continuity correction for discrete barrier options. Math. Financ. 7(4), 325–348 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brown, H., Hobson, D., Rogers, L.C.G.: Robust hedging of barrier options. Math. Financ. 11(3), 285–314 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carr, P., Chou, A.: Breaking barriers. Risk 10(9), 139–146 (1997)Google Scholar
  8. 8.
    Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J. Financ. 53(3), 1165–1191 (1998)CrossRefGoogle Scholar
  9. 9.
    Černý, A.: Optimal continuous-time hedging with leptokurtic returns. Math. Financ. 17(2), 175–203 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Černý, A.: Mathematical Techniques in Finance: Tools for Incomplete Markets, 2nd edn. Princeton University Press (2009)Google Scholar
  11. 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. 12.
    Derman, E., Ergener, D., Kani, I.: Forever hedged. Risk 7(9), 139–145 (1994)Google Scholar
  13. 13.
    Derman, E., Ergener, D., Kani, I.: Static options replication. J. Deriv. 2, 78–95 (1995)CrossRefGoogle Scholar
  14. 14.
    Derman, E., Kani, I., Ergener, D., Bardhan, I.: Enhanced numerical methods for options with barriers. Finan. Anal. J. 51(6), 65–74 (1995)CrossRefGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 16.
    Dupont, D.Y.: Hedging barrier options: Current methods and alternatives. Economics Series 103, Institute for Advanced Studies (2001)Google Scholar
  17. 17.
    Eberlein, E., Özkan, F.: Time consistency of Lévy models. Quan. Financ. 3, 40–50 (2003)CrossRefGoogle Scholar
  18. 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. 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
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Figlewski, S., Gao, B.: The adaptive mesh model: A new approach to efficient option pricing. J. Financ. Econ. 53(3), 313–351 (1999)CrossRefGoogle Scholar
  22. 22.
    Fusai, G., Abrahams, I., Sgarra, C.: An exact analytical solution for discrete barrier options. Financ. Stochast. 10(1), 1–26 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fusai, G., Recchioni, M.: Analysis of quadrature methods for pricing discrete barrier options. J. Econ. Dyn. Control 31(3), 826–860 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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
  25. 25.
    Hörfelt, P.: Pricing discrete European barrier options using lattice random walks. Math. Financ. 13(4), 503–524 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    İlhan, A., Sircar, R.: Optimal static-dynamic hedges for barrier options. Math. Financ. 16(2), 359–385 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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. 28.
    Kou, S.G.: On pricing of discrete barrier options. Statistica Sinica 13(4), 955–964 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kuan, G., Webber, N.: Valuing discrete barrier options on a Dirichlet lattice. FORC Preprint 140/04, University of Warwick (2004)Google Scholar
  30. 30.
    Leisen, D.: Valuation of barrier options in a Black-Scholes setup with jump risk. Eur. Finance Rev. 3(3), 319–342 (1999)CrossRefzbMATHGoogle Scholar
  31. 31.
    Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Finance Rev. 2(1), 79–105 (1998)CrossRefzbMATHGoogle Scholar
  32. 32.
    Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Reiner, E., Rubinstein, M.: Breaking down the barriers. Risk 4(8), 28–35 (1991)Google Scholar
  34. 34.
    Ritchken, P.H.: On pricing barrier options. J. Deriv. 3(2), 19–28 (1995)CrossRefGoogle Scholar
  35. 35.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999)Google Scholar
  36. 36.
    Strawderman, R.L.: Computing tail probabilities by numerical Fourier inversion: The absolutely continuous case. Statistica Sinica 14(1), 175–201 (2004)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Toft, K.B.: On the mean-variance tradeoff in option replication with transactions costs. J. Finan. Quant. Anal. 31(2), 233–263 (1996)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wei, J.: Valuation of discrete barrier options by interpolations. J. Deriv. 6(1), 51–73 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Cass Business SchoolCity University LondonLondonUK

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