Computational Economics

, Volume 22, Issue 2–3, pp 113–138 | Cite as

An Implementation of Bouchouev's Method for a Short Time Calibration of Option Pricing Models

  • Carl Chiarella
  • Mark Craddock
  • Nadima El-Hassan


We analyse the Bouchouev integral equation for the deterministic volatility function in the Black–Scholes option pricing model. We areable to reduce Bouchouev's original triple integral equation to a single integral equation and describe its numerical solution. Moreover we show empirically that the most complex term in the equation may often be safely ignored for the purposes of numerical calculations. We present a selection of numerical examples indicating the range of time values for which we would expect the equation to be valid.

inverse problems calibration integral equations fundamental solutions of PDE 


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  1. 1.
    Abramowitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, 10th edn. Dover, New York.Google Scholar
  2. 2.
    Bouchouev, I. and Isakov, V. (1997). The inverse problem of option pricing. Inverse Problems, 13(1), Lll–L17.Google Scholar
  3. 3.
    Bouchouev, I. and Isakov, V. (1999). Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Problems, 15(1), R95–R116.Google Scholar
  4. 4.
    Bouchouev, I. (1997). Inverse parabolic problems with applications to option pricing. Ph.D. thesis, Department of Mathematics and Statistics, Wichita State University.Google Scholar
  5. 5.
    Chiarella, C. Craddock, M. and El-Hassan, N. (January 2002). The calibration of stock option pricing models using inverse problem methodology. Research Report, School of Finance and Economics, University of Technology, Sydney.Google Scholar
  6. 6.
    Derman, E. and Kani, I. (1994). Riding on a smile. Risk, 7(2), 32–39.Google Scholar
  7. 7.
    Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18–20.Google Scholar
  8. 8.
    Erdelyi, A. (1954). Tables of Integral Transforms. Bateman Manuscript Project, Vol. 1. McGraw Hill.Google Scholar
  9. 9.
    Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, N.J..Google Scholar
  10. 10.
    Levi, E.E. (1907). ItSulle equazioni lineari totalmente ellittiche alle derivate parzialli. Rendiconti del Circolo Matematico di Palermo, 24, 275–317.Google Scholar
  11. 11.
    Lagnado, R. and Osher, S. (1997). A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem. Journal of Computational Finance, 1(1), 13–25.Google Scholar
  12. 12.
    Porter, D. and Stirling, D.S.G. (1990). Integral equations. A practical treatment from spectral theory to applications, 1st edn. Cambridge Texts in Applied Mathematics, 5. Cambridge University Press, London.Google Scholar
  13. 13.
    Robinson, D. (1993). Elliptic Operators on Lie Groups. Cambridge University Press, Cambridge.Google Scholar
  14. 14.
    Rubinstein, M. (1994). Implied binomial trees. Journal of Finance, 69(1), 771–818.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Carl Chiarella
    • 1
  • Mark Craddock
    • 1
  • Nadima El-Hassan
    • 1
  1. 1.School of Finance and EconomicsUniversity of Technology SydneyBroadwayAustralia

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