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Double Barrier Options Under Lévy Processes

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Modern Operator Theory and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 170))

Abstract

In this paper the problem of determination of the no arbitrage price of double barrier options in the case of stock prices is modelled on Lévy processes is considered. Under the assumption of existence of the Equivalent Martingale Measure this problem is reduced to the convolution equation on a finite interval with symbol generated by the characteristic function of the Lévy process. We work out a theory of unique solvability of the getting equation and stability of the solution under relatively small perturbations.

Author acknowledges financial support by CONACYT project 046936-F.

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References

  1. Kunimoto N., Ikeda M. Pricing options with curved boundaries. Math. Finance 2, 275–298, 1992.

    Article  Google Scholar 

  2. Geman H. and Yor M. Pricing and Hedging double-barrier options: A probabilistic approach. Math. Finance 6:4, 365–378, 1996.

    Article  MATH  Google Scholar 

  3. Sidenius J. Double Barrier Options — Valuating by Path Counting. J. Computat. Finance, 1998, V1, No. 3.

    Google Scholar 

  4. Pelsser A. Pricing double barrier options using Laplace transform. Finance Stochastics 4:1, 95–104, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. Baldi P., Caramellino L., Iovino M.G. Pricing general Barrier Options using sharp larger deviations. Math. Finance 9:4, 291–322, 1999.

    Article  MathSciNet  Google Scholar 

  6. Hui C.H. One-touch double barrier binary option values. Appl. Financial Econ. 6, 343–346, 1996.

    Article  Google Scholar 

  7. Hui C.H. Time dependent barrier option values. J. Futures Markets, 17, 667–688, 1997.

    Article  Google Scholar 

  8. Wilmott P., Dewynne J., Howison S. Option pricing: Mathematical models and computations. Oxford: Oxford Financial Press, 1993.

    Google Scholar 

  9. Raible R. Lévy processes in Finance: Theory, Numerics, and Empirical Facts. Dissertation. Mathematische Fakultät, Universität Freiburg im Breisgau, 2000.

    Google Scholar 

  10. Barndorff-Nielsen O.E. Processes of normal inverse Gaussian Type. Finance and Stochastics 2, 41–68, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  11. Madan D.B., Carr P. and Chang E.C. The variance Gamma process and option pricing. European Finance Review 2, 79–105, 1998.

    Article  MATH  Google Scholar 

  12. Eberlein E. Application of generalized hyperbolic Lévy motions to Finance. In: Lévy processes: Theory and applications, O.E. Barndorff-Nielsen, T. Mikosh and S. Resnik (Eds.), Birkhäuser, 319–337, 2001.

    Google Scholar 

  13. Bouchaud J.-P. and Potters M. Theory of financial risk. Cambridge University Press, Cambridge.

    Google Scholar 

  14. Matacz A. Financial modelling and option theory with the truncated Lévy process. Intern. Journ. Theor. and Appl. Finance 3:1, 143–160, 2000.

    Article  MathSciNet  Google Scholar 

  15. Boyarchenko S.I., Levendorskii Sergei. Non-Gaussian Merton-Black-Scholes theory. Advanced Series on Statistical Science and Applied Probability 9, World-Scientific, Singapore, 2002.

    Google Scholar 

  16. Boyarchenko S.I., Levendorskii S.Z. On rational pricing of derivative securities for a family of non-Gaussian processes. Preprint 98/7, Institut für Mathematik, Universit ät Potsdam, Potsdam.

    Google Scholar 

  17. Boyarchenko S.I., Levendorskii S.Z. Option pricing and hedging under regular Lévy processes of exponential type. In: Trends in Mathematics. Mathematical Finance, M. Kohlman and S. Tang (Eds.), 121–130, 2001.

    Google Scholar 

  18. Levendorskii S.Z. and Zherder V.M. Fast option pricing under regular Lévy processes of exponential type. Submitted to Journal of Computational Finance, 2001.

    Google Scholar 

  19. Boyarchenko S.I., Levendorskii S.Z. Option pricing for truncated Lévy processes. Intern. Journ. Theor. and Appl. Finance 3:3, 549–552, 2000.

    Article  MATH  Google Scholar 

  20. Boyarchenko S.I., Levendorskii S.Z. Perpetual American options under Lévy processes. SIAM Journ. of Control and Optimization, Vol. 40 (2002) no. 6, pp. 1663–1696

    Article  MATH  MathSciNet  Google Scholar 

  21. Mordecki E. Optimal stopping and perpetual options for Lévy processes. Talk presented at the 1 World Congress of the Bachelier Finance Society, June 2000.

    Google Scholar 

  22. Boyarchenko S.I., and Levendorskii S.Z. Barrier options and touch-and-out options under regular Lévy processes of exponential type. Annals of Applied Probability, Vol. 12 (2002), no. 4, pp. 1261–1298.

    Article  MATH  MathSciNet  Google Scholar 

  23. Cont R., Voltchkova E. Integro-differential equations for Options prices in exponential Lévy models. Finance Stochastics. 9, 299–325 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  24. Grudsky S.M. Convolution equations on a finite interval with a small parameter multiplying the growing part of the symbol. Soviet Math. (Iz. Vuz) 34,7, 7–18, 1990.

    MathSciNet  Google Scholar 

  25. Böttcher A., Grudsky S.M. On the condition numbers of large semi-definite Toeplitz matrices. Linear Algebra and its Applications 279,1–3, 285–301, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  26. Grudsky S.M., Mikhalkovich S.S. Semisectoriality and condition numbers of convolution operators on the large finite intervals. Integro-differential operators, Proceedings of different universities, Rostov-on-Don, 5, 78–87, 2001.

    Google Scholar 

  27. Grudsky S.M., Mikhalkovich S.S., and E. Ramíirez de Arellano. The Wiener-Hopf integral equation on a finite interval: asymptotic solution for large intervals with an application to acoustics. Proceedings of International Workshop on Linear Algebra, Numerical Functional Analysis and Wavelet Analysis, Allied Publisher Private Limite, India, 2003, 89–116.

    Google Scholar 

  28. Bertoin J. Lévy processes. Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

  29. Shiryaev A.N. Essentials of stochastic Finance. Facts, models, theory. World Scientific, Singapore Jersey London Hong Kong, 1999.

    Google Scholar 

  30. Karatzas I. and Shreve S.E. Methods of mathematical Finance. Springer-Verlag, Berlin Heidelberg New York, 1998.

    MATH  Google Scholar 

  31. Delbaen F. and Schachermayer W. A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  32. Eberlein E. and Jacod J. On the range of options prices. Finance and Stochastics 1, 131–140, 1997.

    Article  MATH  Google Scholar 

  33. Fölemer H. and Schweizer M. Hedging of contingent claims under incomplete information. In: Applied Stochastic Analysis, M.H.A. Davies and R.J. Elliot (eds.), New York, Gordon and Bleach, 389–414, 1991.

    Google Scholar 

  34. Keller U. Realistic modelling of financial derivatives. Dissertation. Mathematische Fakultät, Universität Freiburg im Breisgau, 1997.

    Google Scholar 

  35. Kallsen J. Optimal portfolios for exponential Lévy processes. Mathematical Methods of Operations Research 51:3, 357–374, 2000.

    Article  MathSciNet  Google Scholar 

  36. Madan D.B. and Milne F. Option prising with VG martingale components. Mathem. Finance 1, 39–55, 1991.

    Article  MATH  Google Scholar 

  37. Eberlein E., Keller U. and Prause K. New insights into smile, mispricing and value at risk: The hyperbolic model. Journ. of Business 71, 371–406, 1998.

    Article  Google Scholar 

  38. Eskin G.I. Boundary problems for elliptic pseudo-differential equations. Nauka, Moscow, 1973 (Transl. of Mathematical Monographs, 52, Providence, Rhode Island: Amer. Math. Soc., 1980).

    Google Scholar 

  39. Noble B. Methods based on the Wiener-Hopf technique for the solution of partial differential equations. International Series of Monographs on Pure and Applied Mathematics 7, Pergamon Press, New York-London-Paris-Los Angeles, 1958.

    Google Scholar 

  40. Gohberg I. and Krupnik N.Ya. Introduction to the Theory of One-dimensional Singular integral Operators. “Shtiintsa”, Kishinev, 1973 (Transl. One-dimensional Linear Singular Integral Equations. Vol. I. Introduction, Vol II General Theory and Applications, Translated from the 1979 German Translation. Operator theory: Advances and Applications, 53 and 54. Birkhäuser Verlag, Basel, 1992).

    Google Scholar 

  41. Böttcher A. and Silbermann B. Analysis of Toeplitz Operators. Springer-Verlag, Berlin, 1990.

    MATH  Google Scholar 

  42. Sarason D. Toeplitz operators with semi-almost-periodic symbols. Duke Math. J. 44,2, 357–364, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  43. Akhiezer N.I. Lectures on Approximation Theory. Second, revised and enlarged edition, “Nauka”, Moscow, 1965 (Transl. of first edition: Theory of Approximation, Frederick Ungar Publishing Co., New York, 1956).

    Google Scholar 

  44. Dybin V.B., Grudsky S.M. Introduction to the theory of Toeplitz operators with infinite index. Birkhäuser Verlag, Basel-Boston-Berlin, Operator Theory: Advances and Applications, 2002.

    Google Scholar 

  45. Böttcher A., Karlovich Yu., I Spitkovsky I.M. Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Application. Vol. 131, Birkhäuser Verlag, 2002.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Matemáticas, CINVESTAV del I.P.N., México, D.F., México

    Sergei M. Grudsky

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  1. Sergei M. Grudsky
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Editor information

Editors and Affiliations

  1. Mechanical-Mathematics Department, Rostov State University, Zorge Str. 5, Rostov-on-Don, 344104, Russia

    Yakob M. Erusalimsky

  2. School of Mathematical Sciences, Raymond and Beverly Sackler, Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Israel Gohberg

  3. Departamento de Matemáticas, CINVESTAV, Apartado Postal 14-740, 07000, Mexico, D.F., Mexico

    Sergei M. Grudsky  & Nikolai Vasilevski  & 

  4. Instituto Politecnico Nacional, ESIME Zacatenco, Avenida IPN, Mexico, D. F., 07738, Mexico

    Vladimir Rabinovich

Additional information

To I.B. Simonenko on the occasion of his 70th birthday

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Grudsky, S.M. (2006). Double Barrier Options Under Lévy Processes. In: Erusalimsky, Y.M., Gohberg, I., Grudsky, S.M., Rabinovich, V., Vasilevski, N. (eds) Modern Operator Theory and Applications. Operator Theory: Advances and Applications, vol 170. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7737-3_8

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