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Bi-cubic B-spline fitting-based local volatility model with mean reversion process

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Abstract

This paper studies the traditional local volatility model and proposes: A novel local volatility model with mean-reversion process. The larger is the gap between local volatility and its mean level, the higher will be the rate at which local volatility will revert to the mean. Then, a B-spline method with proper knot control is applied to interpolate the local volatility matrix. The bi-cubic B-spline is used to recover the local volatility surface from this local volatility matrix. Finally, empirical tests show that the proposed mean-reversion local volatility model offers better prediction performance than the traditional local volatility model.

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References

  1. Keynes M J, The General Theory of Employment, Interest and Money, Macmillan Cambridge University Press, New York, 1936.

    Google Scholar 

  2. Cecchetti S, Lam P S, and Mark N, Mean reversion in equilibrium asset prices, The American Economic Review, 1990, 80: 398–418.

    Google Scholar 

  3. Black F, Mean reversion and consumption smoothing, Review of Financial Studies, 1990, 3: 107–114.

    Article  Google Scholar 

  4. Bessembinder H, Coughenour J, Seguin P, and Smoller M M, Mean reversion in equilibrium asset prices: Evidence from the futures term structure, Journal of Finance, 1995, 50: 361–375.

    Article  Google Scholar 

  5. Engle R and Patton A, What good is a volatility model?, Quantitative Finance, 2001, 1: 237–245.

    Article  Google Scholar 

  6. Bouchouev I and Isakov V, The inverse problem of option pricing, Inverse Problems, 1997, 13: L11.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dupire B, Pricing with a smile, Risk, 1994, 7: 18–20.

    Google Scholar 

  8. Sepp A, Pricing barrier options under local volatility, Mail: artursepp@hotmail.com,Web: www.hot.ee/seppar, 2002.

    Google Scholar 

  9. Coleman T, Li Y, and Verma A, Reconstructing the unknown local volatility function, Journal of Computation Finance, 1999, 2: 77–100.

    Google Scholar 

  10. Crépey S, Calibration of the local volatility in a generalized black-scholes model using TIKHONOV regularization, SIAM Journal of Mathematical Analysis, 2003, 34: 1183–1206.

    Article  MATH  Google Scholar 

  11. Bernard D, Fleming J, and Whaley R, Implied volatility functions: Empirical tests, Journal of Finance, 1998, 53: 2059–2106.

    Article  Google Scholar 

  12. Bedendo M and Hodges S, The dynamics of the volatility skew: A kalman filter approach, Journal of Banking and Finance, 2009, 33: 1156–1165.

    Article  Google Scholar 

  13. Harker M, Least squares surface reconstruction from measured gradient fields, IEEE Conference on Computer Vision and Pattern Recognition, 2008, 1–7.

    Google Scholar 

  14. Catmuli E and Clark J, Recursively generated B-spline surfaces on arbitrary topological meshes, Journal of Computer-Aided Design, 1978, 10: 350–355.

    Article  Google Scholar 

  15. Schoenberg J I, Contributions to the problem of approximation of equidistant data by analytic functions, Quarterly of Applied Mathematics, 1946, 4: 45–99.

    MathSciNet  Google Scholar 

  16. Lee E T Y, A simplified B-spline computation routine, Computing, 1982, 29: 365–371.

    Article  MathSciNet  MATH  Google Scholar 

  17. Schoenberg J I, Cardinal interpolation and spline functions, Journal of Approximation Theory, 1969, 2: 167–206.

    Article  MathSciNet  MATH  Google Scholar 

  18. Carl D B, A Practical Guide to Splines, Springer-Verlag, New York, 1978.

    MATH  Google Scholar 

  19. Schumaker L, Spline Functions: Basic Theory, Wiley, New York, 1981.

    MATH  Google Scholar 

  20. Brunet P, Increasing the smoothness of bicubic spline surfaces, Computer Aided Geometric Design, 1985, 2: 157–164.

    Article  MathSciNet  MATH  Google Scholar 

  21. Vasicek O, An equilibrium characterisation of the term structure, Journal of Financial Economics, 1977, 5: 177–188.

    Article  Google Scholar 

  22. Yang H, Wang W, and Sun J, Control point adjustment for B-spline curve approximation, Computer-Aided Design, 2004, 36: 639–652.

    Article  Google Scholar 

  23. Park H, B-spline surface fitting based on adaptive knot placement using dominant columns, Computer-Aided Design, 2011, 43: 258–264.

    Article  Google Scholar 

  24. Przemyslaw K, Bicubic B-spline blending patches with optimized shape, Computer-Aided Design, 2011, 43: 133–144.

    Article  Google Scholar 

  25. Kineri Y, Wang M, Lin H, and Maekawa T, B-spline surface fitting by iterative geometric interpolation/approximation algorithms, Computer-Aided Design, 2012, 44: 697–708.

    Article  Google Scholar 

  26. Ncube M, Stochastic Models and Inferences for Commodity Futures Pricing, Florida State University, 2009.

    Google Scholar 

  27. Bollerslev T, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 1986, 31: 307–327.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hansen P and Lunde A, A forecast comparison of volatility models: Does anything beat a GARCH (1,1)?, Journal of Applied Econometrics, 2005, 20: 873–889.

    Article  MathSciNet  Google Scholar 

  29. Poon S H and Granger W J C, Forecasting volatility in financial markets: A review, Journal of Economic Literature, 2003, XLI: 478–539.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Management Sciences, City University of Hong Kong, Hong Kong, China

    Shifei Zhou, Hao Wang & Kin Keung Lai

  2. Department of Finance, Hong Kong University of Science and Technology, Hong Kong, China

    Jerome Yen

  3. International Business School, Shaanxi Normal University, Xi’an, 710061, China

    Kin Keung Lai

Authors
  1. Shifei Zhou
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  2. Hao Wang
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  3. Jerome Yen
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  4. Kin Keung Lai
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Corresponding author

Correspondence to Shifei Zhou.

Additional information

This paper was recommended for publication by Editor WANG Shouyang.

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Zhou, S., Wang, H., Yen, J. et al. Bi-cubic B-spline fitting-based local volatility model with mean reversion process. J Syst Sci Complex 29, 119–132 (2016). https://doi.org/10.1007/s11424-015-3066-8

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  • DOI: https://doi.org/10.1007/s11424-015-3066-8

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