Fuzzy Optimization and Decision Making

, Volume 12, Issue 4, pp 433–450 | Cite as

A comparative assessment of different fuzzy regression methods for volatility forecasting



The aim of this paper is to compare different fuzzy regression methods in the assessment of the information content on future realised volatility of option-based volatility forecasts. These methods offer a suitable tool to handle both imprecision of measurements and fuzziness of the relationship among variables. Therefore, they are particularly useful for volatility forecasting, since the variable of interest (realised volatility) is unobservable and a proxy for it is used. Moreover, measurement errors in both realised volatility and volatility forecasts may affect the regression results. We compare both the possibilistic regression method of Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) and the least squares fuzzy regression method of Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991). In our case study, based on intra-daily data of the DAX-index options market, both methods have proved to have advantages and disadvantages. Overall, among the two methods, we prefer the Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991) method, since it contains as special case (the central line) the ordinary least squares regression, is robust to the analysis of the variables in logarithmic terms or in levels, and provides sharper results than the Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) method.


Fuzzy regression methods Linear programming Least squares Volatility forecasting 



The authors gratefully acknowledge financial support from Fondazione Cassa di Risparmio di Modena for the project “Volatility modelling and forecasting with option prices: the proposal of a volatility index for the Italian market” and MIUR.


  1. Britten-Jones, M., & Neuberger, A. (2000). Option prices, implied price processes, and stochastic volatility. Journal of Finance, 55(2), 839–866.CrossRefGoogle Scholar
  2. Celminš, A. (1987). Least squares model fitting to fuzzy vector data. Fuzzy Sets and Systems, 22, 245–269.MathSciNetCrossRefGoogle Scholar
  3. Chang, Y.-H. O. (2001). Hybrid fuzzy least-squares regression analysis and its reliability measures. Fuzzy Sets and Systems, 119, 225–246.MathSciNetCrossRefMATHGoogle Scholar
  4. Chang, Y.-H. O., & Ayyub, B. M. (2001). Fuzzy regression methods—a comparative assessment. Fuzzy Sets and Systems, 119, 187–203.MathSciNetCrossRefGoogle Scholar
  5. Diamond, P. (1988). Fuzzy least squares. Information Sciences, 46, 141–157.MathSciNetCrossRefMATHGoogle Scholar
  6. D’Urso, P., & Gastaldi, T. (2000). A least-squares approach to fuzzy linear regression analysis. Computational Statistics & Data Analysis, 34, 427–440.CrossRefMATHGoogle Scholar
  7. Ishibuchi, H., & Nii, M. (2001). Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks. Fuzzy Sets and Systems, 119, 273–290.MathSciNetCrossRefMATHGoogle Scholar
  8. Jiang, G. J., & Tian, Y. S. (2005). The model-free implied volatility and its information content. Review of Financial Studies, 18(4), 1305–1342.CrossRefGoogle Scholar
  9. Kahraman, C., Beşkese, A., & Bozbura, F. T. (2006). Fuzzy regression approaches and applications. Studies in Fuzziness and Soft Computing, 201, 589–615.CrossRefGoogle Scholar
  10. Kao, C., & Chyu, C.-L. (2002). A fuzzy regression model with better explanatory power. Fuzzy Sets and Systems, 126, 401–409.MathSciNetCrossRefMATHGoogle Scholar
  11. Kim, K. J., Moskowitz, H., & Koksalan, M. (1996). Fuzzy versus statistical linear regression. European Journal of Operational Research, 92, 417–434.CrossRefMATHGoogle Scholar
  12. Moskowitz, H., & Kim, K. J. (1993). On assessing the H value in fuzzy linear regression. Fuzzy Sets and Systems, 58, 303–327.MathSciNetCrossRefMATHGoogle Scholar
  13. Muzzioli, S. (2010). Option-based forecasts of volatility: an empirical study in the DAX-index options market. European Journal of Finance, 16(6), 561–586.CrossRefGoogle Scholar
  14. Nasrabadi, M. M., Nasrabadi, E., & Nasrabadi, A. R. (2005). Fuzzy linear regression analysis: A multi objective programming approach. Applied Mathematics and Computation, 163, 245–251.MathSciNetCrossRefMATHGoogle Scholar
  15. Nather, W. (2006). Regression with fuzzy random data. Computational Statistics and Data Analysis, 51, 235–252.MathSciNetCrossRefGoogle Scholar
  16. Omrani, H., Aabdollahzadeh, S., & Alinaghian, M. (2011). A simple and efficient goal programming model for computing of fuzzy linear regression parameters with considering outliers. Journal of Uncertain Systems, 5(1), 62–71.Google Scholar
  17. Peters, G. (1994). Fuzzy linear regression with fuzzy intervals. Fuzzy Sets and Systems, 63, 45–55.MathSciNetCrossRefMATHGoogle Scholar
  18. Poon, S., & Granger, C. W. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature, 41, 478–539.CrossRefGoogle Scholar
  19. Sánchez, J. A., & Gómez, A. T. (2003). Estimating a term structure of interest rates for fuzzy financial pricing using fuzzy regression methods. Fuzzy Sets and Systems, 139, 313–331.MathSciNetCrossRefMATHGoogle Scholar
  20. Sánchez, J. A., & Gómez, A. T. (2004). Estimating a fuzzy term structure of interest rates using fuzzy regression techniques. European Journal of Operational Research, 154, 804–818.CrossRefMATHGoogle Scholar
  21. Savic, D. A., & Pedrycz, W. (1991). Evaluation of fuzzy linear regression models. Fuzzy Sets and Systems, 39, 51–63.MathSciNetCrossRefMATHGoogle Scholar
  22. Tanaka, H. (1987). Fuzzy data analysis by possibilistic linear models. Fuzzy Sets and Systems, 24, 363–375.MathSciNetCrossRefMATHGoogle Scholar
  23. Tanaka, H., Uejima, S., & Asai, K. (1982). Linear regression analysis with fuzzy model. IEEE Transactions on Systems, Man and Cybernetics, 12, 903–907.CrossRefMATHGoogle Scholar
  24. Tanaka, H., & Watada, J. (1988). Possibilistic linear systems and their application to the linear regression model. Fuzzy Sets and Systems, 27, 275–289.MathSciNetCrossRefMATHGoogle Scholar
  25. Tseng, F.-M., Tzeng, G.-H., Yu, H.-C., & Yuan, B.-J. C. (2001). Fuzzy ARIMA model for forecasting the foreign exchange market. Fuzzy Sets and Systems, 118, 9–19.MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Economics and CEFINUniversity of Modena and Reggio EmiliaModenaItaly
  2. 2.KERMIT, Department of Mathematical Modelling, Statistics and BioinformaticsGhent UniversityGhentBelgium

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