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Title: Relation between volatility and returns through a quantile fuzzy regression model Authors:  Luciano Stefanini - University of Urbino (Italy)
Maria Letizia Guerra - (Italy) [presenting]
Abstract: The purpose is to analyze the nature of the relation of the pair volatility and return in the particular case of CBOE VIX and S&P 500 index. In particular, the S\&P500 returns time series is modelled through fuzzy-valued functions, whose level-cuts are interpreted in the framework of expectile and quantile fuzzy regressions which are built by defining fuzzy-valued expectile (L2-norm) and quantile (L1-norm) extensions of the F-transforms. Since in the whole time period the relationship between VIX and S\&P500 returns changes dramatically, we introduce the clustering of the data into subsets to significantly improve the quality of fitting; the clustering is applied only when a preliminary evaluation test based on Kendall and Spearman correlation verifies its e efficacy. A forecasting methodology is proposed based on specific forms of local trends such as parametric exponential functions which we prove to be more suitable and stable for extrapolation than polynomials. We show how it is possible to forecast S\&P500 returns to time $T+M$ ($M$-steps ahead) by having available volatility and returns observations till time $T$.