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Option implied moments obtained through fuzzy regression

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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.

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Fig. 1


  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 ( The average execution time is obtained on an Intel Core i5 2450M 2.50 GHz processor.


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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)).

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Correspondence to Silvia Muzzioli.

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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:

Fig. 2

Estimated volatility smile function obtained from two different inputs of strike prices and implied volatilities: i) artificially generated input data characterized by a low spread and a regular pattern (on the left); ii) a real initial grid of strike prices and implied volatilities recorded on November 21, 2012, characterized by a high spread and a high implied volatility associated to deep out-of-the-money options (on the right). We report the option strike prices on the x-axis and the obtained implied volatility levels on the y-axis. Black dots and triangles indicate the initial input of put and call options, respectively. For each of the two figures we depict the estimate obtained by means of the standard methodology (cubic spline, red line), and three estimates obtained using the Savic and Pedrycz fuzzy regression method, taken as an example: the central estimate (in green) and the lower and upper bounds with h = 0

  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.

Table 11 Estimates of skewness and kurtosis obtained using the volatility smiles in Fig. 2 as inputs

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.

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Muzzioli, S., Gambarelli, L. & De Baets, B. Option implied moments obtained through fuzzy regression. Fuzzy Optim Decis Making (2020). https://doi.org/10.1007/s10700-020-09316-x

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  • Forecasting
  • Fuzzy regression
  • Skewness
  • Kurtosis
  • Italian market