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The Black-Scholes Model and Its Modifications

  • Jia-An Yan
Chapter
Part of the Universitext book series (UTX)

Abstract

In Chap.  3, we have studied discrete-time financial market models, which are suitable for qualitative research and statistical analysis. However, for theoretical research, continuous-time models are proved to be a convenient framework, because one can use stochastic analysis tools in studying of pricing and hedging of contingent claims and portfolio selection. Using stochastic analysis tools can often lead to explicit solutions or analytical expressions.

References

  1. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 635–654 (1973)MathSciNetCrossRefGoogle Scholar
  2. Bollerslev, T.: Generalized autoregressive conditional heteroscedasticity. J. Econ. 31, 307–327 (1986)MathSciNetCrossRefGoogle Scholar
  3. Bollerslev, T., Engle, R.F., Nelson, D.B.: ARCH models. In: Engle, R.F., McFadden, D.L. (eds.) The Handbook of Econometrics, vol. 4. Elsevier, Amsterdam (1994)Google Scholar
  4. Breeden, D., Litzenberger, R.: Prices of state-contingent claims implicit in options prices. J. Bus. 51, 621–651 (1978)CrossRefGoogle Scholar
  5. Cox, J.C.: Notes on option pricing I: constant elasticity of variance diffusions. Working paper, Stanford University (1975)Google Scholar
  6. Cox, J.C., Ross, S.A.: The valuation of options for alternative stochastic processes. J. Financ. Econ. 3, 145–166 (1976)CrossRefGoogle Scholar
  7. Dudley, R.M.: Wiener functionals as Itô integrals. Ann. Probab. 5, 140–141 (1977)CrossRefGoogle Scholar
  8. Dupire, B. (1997): Pricing and hedging with smiles. In: Dempster, A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities, pp. 103–111. Publications of the Newton Institute, Cambridge University Press, CambridgeGoogle Scholar
  9. Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007 (1982)MathSciNetCrossRefGoogle Scholar
  10. Frey, R.: Derivative asset analysis in models with level-dependent and stochastic volatility. CWI Quart. (Amsterdan) 10, 1–34 (1997)Google Scholar
  11. Geske, R.: The valuation of compound options. J. Financ. Econ. 7, 63–81 (1979)CrossRefGoogle Scholar
  12. Glasserman, P., Wu, Q.: Forward and future implied volatility. Int. J. Theor. Appl. Financ. 14(3), 407–432 (2011)MathSciNetCrossRefGoogle Scholar
  13. Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Magazine, pp. 84–108 (2002)Google Scholar
  14. Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)CrossRefGoogle Scholar
  15. Hofmann, N., Platen, E., Schweizer, M.: Option pricing under incompleteness and stochastic volatility. Math. Financ. 2, 153–187 (1992)CrossRefGoogle Scholar
  16. Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Financ. 42, 281–300 (1987)CrossRefGoogle Scholar
  17. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, Berlin/Heidelberg/New York (1998)zbMATHGoogle Scholar
  18. Kwok, Y.-K.: Mathematical Models of Financial Derivatives. Springer, Singapore/New York (1998)zbMATHGoogle Scholar
  19. Madan, D.B.: Purely discontinuous asset price process. In: Jouini, E. et al. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 105–153. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  20. Madan, D.B., Seneta, E.: The variance gamma (V. G. ) model for share market returns. J. Bus. 63, 511–524 (1990)CrossRefGoogle Scholar
  21. Madan, D.B., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Financ. Rev. 2, 79–105 (1998)CrossRefGoogle Scholar
  22. Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973b)MathSciNetCrossRefGoogle Scholar
  23. Nelson, D.B.: ARCH as diffusion appriximations, J. of Econ. 45, 7-38 (1990)CrossRefGoogle Scholar
  24. Wilmott, P., Dewynne, J., and Howison, S.: Option Pricing: Mathematical Models and Computations. Oxford Financial Press, and Oxford (1993)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Science Press 2018

Authors and Affiliations

  • Jia-An Yan
    • 1
  1. 1.Academy of Mathematics and System ScienceChineses Academy of SciencesBeijingChina

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