Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

# Option implied moments obtained through fuzzy regression

• 4 Accesses

## Abstract

The aim of this paper is to investigate the potential of fuzzy regression methods for computing more reliable estimates of higher-order moments of the risk-neutral distribution. We improve upon the formula of Bakshi et al. (RFS 16(1):101–143, 2003), which is used for the computation of market volatility and skewness indices (such as the VIX and the SKEW indices traded on the Chicago Board Options Exchange), through the use of fuzzy regression methods. In particular, we use the possibilistic regression method of Tanaka, Uejima and Asai, the least squares fuzzy regression method of Savic and Pedrycz and the hybrid method of Ishibuchi and Nii. We compare the fuzzy moments with those obtained by the standard methodology, based on the Bakshi et al. (2003) formula, which relies on an ex-ante choice of the option prices to be used and cubic spline interpolation. We evaluate the quality of the obtained moments by assessing their forecasting power on future realized moments. We compare the competing forecasts by using both the Model Confidence Set and Mincer–Zarnowitz regressions. We find that the forecasts for skewness and kurtosis obtained using fuzzy regression methods are closer to the subsequently realized moments than those provided by the standard methodology. In particular, the lower bound of the fuzzy moments obtained using the Savic and Pedrycz method is the best ones. The results are important for investors and policy makers who can rely on fuzzy regression methods to get a more reliable forecast for skewness and kurtosis.

This is a preview of subscription content, log in to check access.

1. 1.

An option is a financial contract that gives the holder the right to buy (call option) or the right to sell (put option) an asset (underlying asset) at a given date (the maturity date) for a pre-specified price (the strike price). A call option gives the holder the right to buy the underlying asset, while a put option gives the holder the right to sell it.

2. 2.

An option is said to be at-the-money, in-the-money, or out-of-the-money if it generates a zero, positive, or negative payoff, respectively, if exercised immediately.

3. 3.

The smile depicts implied volatility (obtained by inverting the Black and Scholes 1973 formula) as a function of the strike price. Its shape resembles a smile (when implied volatility is higher for out-of-the-money options than it is for at-the-money options) or a smirk (when the implied volatility is higher for put prices and lower for call prices).

4. 4.

Call and put prices are obtained by using the Black–Scholes formula. It is important to note that the Black–Scholes formula is used only as a mirror to convert option prices into implied volatilities (in order to get the smile function to be interpolated) and implied volatilities into option prices (to plug into (3)–(6)).

5. 5.

The procedure described in Sects. 24 has been implemented and executed using MATLAB R2018B (9.5.0.944444). The average execution time is obtained on an Intel Core i5 2450M 2.50 GHz processor.

## References

1. Alfonso, G., López, Roldán, de Hierro, A. F., & Roldán, C. (2017). A fuzzy regression model based on finite fuzzy numbers and its application to real-world financial data. Journal of Computational and Applied Mathematics,318, 47–58.

2. Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies,16(1), 101–143.

3. Bernardi, M., & Catania, L. (2014). The model confidence set package for R. CEIS Working Paper No. 362.

4. Bhattacharyya, R., Hossain, S. A., & Kar, S. (2014). Fuzzy cross-entropy, mean, variance, skewness models for portfolio selection. Journal of King Saud University-Computer and Information Sciences,26(1), 79–87.

5. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy,81(3), 637–654.

6. Britten-Jones, M., & Neuberger, A. (2000). Option prices, implied price processes, and stochastic volatility. Journal of Finance,55(2), 839–866.

7. Capotorti, A., & Figà-Talamanca, G. (2013). On an implicit assessment of fuzzy volatility in the Black and Scholes environment. Fuzzy Sets and Systems,223, 59–71.

8. Carr, P., & Madan, D. (2005). A note on sufficient conditions for No Arbitrage. Finance Research Letters,2, 125–130.

9. CBOE (2010). The CBOE Skew Index. https://www.cboe.com/micro/skew/documents/skewwhitepaperjan2011.pdf. Accessed February 6, 2019.

10. Chen, W., Wang, Y., Zhang, J., & Lu, S. (2017). Uncertain portfolio selection with high-order moments. Journal of Intelligent and Fuzzy Systems,33, 1397–1411.

11. Conrad, J., Dittmar, R. F., & Ghysels, E. (2013). Ex Ante Skewness and Expected Stock Returns. Journal of Finance,68(1), 85–124.

12. De Andrés-Sánchez, J. (2017). An empirical assessment of fuzzy Black and Scholes pricing option model in Spanish stock option market. Journal of Intelligent & Fuzzy Systems,33(4), 2509–2521.

13. De Andrés-Sánchez, J. (2018). Pricing European options with triangular fuzzy parameters: assessing alternative triangular approximations in the spanish stock option market. International Journal of Fuzzy Systems,20(5), 1624–1643.

14. Deng, X., & Liu, Y. (2018). A high-moment trapezoidal fuzzy random portfolio model with background risk. Journal of Systems Science and Information,6(1), 1–28.

15. Elyasiani, E., Gambarelli, L., & Muzzioli, S. (2020). Moment risk premia and the cross-section of stock returns in the European stock market. Journal of Banking & Finance, 111, 105732. https://doi.org/10.1016/j.jbankfin.2019.105732.

16. Feng, Z. Y., Cheng, J. T. S., Liu, Y.-H., & Jiang, I. M. (2015). Options pricing with time changed Lévy processes under imprecise information. Fuzzy Optimization and Decision Making,14(1), 97–119.

17. Hansen, P. R., Lunde, A., & Nason, J. M. (2011). The model confidence set. Econometrica,79(2), 453–497.

18. He, Y.-L., Wang, X., & Huang, J. Z. (2016). Fuzzy nonlinear regression analysis using a random weight network. Information Sciences,364–365, 222–240.

19. Ishibuchi, H., & Nii, M. (2001). Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks. Fuzzy Sets and Systems,119(2), 273–290.

20. Jiang, G. J., & Tian, Y. S. (2005). The model-free implied volatility and its information content. Review of Financial Studies,18(4), 1305–1342.

21. Mincer, J., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In J. Zarnowitz (Ed.), Economic forecasts and expectations. New York: National Bureau of Economic Research.

22. Muzzioli, S. (2010). Option-based forecasts of volatility: an empirical study in the DAX-index options market. The European Journal of Finance,16(6), 561–586.

23. Muzzioli, S. (2013). The forecasting performance of corridor implied volatility in the Italian market. Computational Economics,41(3), 359–386.

24. Muzzioli, S., & De Baets, B. (2013). A comparative assessment of different fuzzy regression methods for volatility forecasting. Fuzzy Optimization and Decision Making,12(4), 433–450.

25. Muzzioli, S., & De Baets, B. (2017). Fuzzy approaches to option price modelling. IEEE Transactions on Fuzzy Systems,25(2), 392–401.

26. Muzzioli, S., Gambarelli, L., & De Baets, B. (2017). Towards a fuzzy volatility index for the Italian market. In Proceedings of the IEEE international conference on fuzzy systems (FUZZ-IEEE 2017) https://doi.org/10.1109/fuzz-ieee.2017.8015446.

27. Muzzioli, S., Gambarelli, L., & De Baets, B. (2018). Indices for financial market volatility obtained through fuzzy regression. International Journal of Information Technology & Decision Making,17(6), 1659–1691.

28. Muzzioli, S., Ruggeri, A., & De Baets, B. (2015). A comparison of fuzzy regression methods for the estimation of the implied volatility smile function. Fuzzy Sets and Systems,266, 131–143.

29. Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics,160(1), 246–256.

30. Savic, D. A., & Pedrycz, W. (1991). Evaluation of fuzzy linear regression models. Fuzzy Sets and Systems,39(1), 51–63.

31. Tanaka, H., Uejima, S., & Asai, K. (1982). Linear regression analysis with fuzzy model. IEEE Transactions on Systems, Man, and Cybernetics,12, 903–907.

32. Wang, X., He, J., & Li, S. (2014). Compound option pricing under fuzzy environment. Journal of Applied Mathematics, 2014(1), 1–9.

33. Wang, N., Zhang, W.-X., & Mei, C.-L. (2007). Fuzzy nonparametric regression based on local linear smoothing technique. Information Sciences,177(18), 3882–3900.

34. Yue, W., & Wang, Y. (2017). A new fuzzy multi-objective higher order moment portfolio selection model for diversified portfolios. Physica A: Statistical Mechanics and its Applications,465(C), 124–140.

35. Zhang, D., Deng, L. F., Cai, K. Y., & So, A. (2005). Fuzzy nonlinear regression with fuzzified radial basis function network. IEEE Transactions on Fuzzy Systems,13(6), 742–760.

36. Zhang, W.-G., Xiao, W.-L., Kong, W.-T., & Zhang, Y. (2015). Fuzzy pricing of geometric Asian options and its algorithm. Applied Soft Computing,28, 360–367.

## Funding

This study was funded by Università Degli Studi di Modena e Reggio Emila (FAR19 Risk assessment in the EU: new indices based on machine learning methods (REU), FAR2017 The role of Asymmetry and Kolmogorov equations in financial Risk Modelling (ARM), FAR15 A SKEWness index for Europe (EU-SKEW)) and Fondazione Cassa di Risparmio di Modena (Volatility and higher order moments: new measures and indices of financial connectedness (IMOM)).

## Author information

Correspondence to Silvia Muzzioli.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

The objective of this appendix is to contrast the estimate for skewness and kurtosis obtained using the central value of the three proposed fuzzy regression models with the one obtained using the standard methodology for different initial inputs of strike prices and market volatilities. More specifically, we assess whether the difference in the estimates obtained using the standard method (cubic spline) and the most possible value (h = 1) obtained using the three fuzzy regression models could be attributed to the irregularities that affect real market option prices. To perform the test, we compute skewness and kurtosis estimates for two different settings, graphically represented in Fig. 2:

1. (i)

artificially generated input data characterized by a low spread and a regular pattern (depicted on the left);

2. (ii)

a real initial grid of strike prices and implied volatilities recorded on November 21, 2012, characterized by a high spread and other irregularities (depicted on the right).

Looking at Fig. 2, on the left, it is clear that the volatility smile function obtained by means of the standard methodology (cubic spline, depicted in red) is very close to the one obtained using the most possible value of the SP fuzzy regression model. On the other hand, the two estimates differ considerably for the smile function depicted in Fig. 2, on the right. The corresponding estimates of skewness and kurtosis are reported in Table 11, where, for the sake of completeness, we present the results also for the TUA and the IN fuzzy regression models.

The estimates of skewness and kurtosis obtained using the standard methodology (Std. Meth.) are very close to the most possible values computed by means of the three fuzzy regression models for the generated initial input of data (Panel A). On the other hand, the standard methodology overestimates both skewness and kurtosis when the smile function is characterized by a high spread between call and put implied volatilities and other irregularities in the smile function (such as very high implied volatilities associated to deep out-of-the-money options). Moreover, the relative difference between the estimates obtained using the standard method and the most possible value of the three fuzzy regression models is higher for kurtosis than for skewness.

This result could explain the considerable difference (in terms of MSE) in the forecasting performance between the standard methodology on the one hand, and the central estimate of the fuzzy regression models on the other hand (see Table 4). When the smile function is affected by irregularities and is characterized by a high spread, the standard methodology produces an estimate that could be very different from the most possible value of the three regression models, in particular for kurtosis.

## Rights and permissions

Reprints and Permissions