Return-Volatility Relationship: Insights from Linear and Non-Linear Quantile Regression

The purpose of this paper is to examine the asymmetric relationship betweenprice and implied volatility and the associated extreme quantile dependence usinglinear and non linear quantile regression approach. Our goal in this paper is todemonstrate that the relationship between the volatility and market return as quantifiedby Ordinary Least Square (OLS) regression is not uniform across the distributionof the volatility-price return pairs using quantile regressions. We examine thebivariate relationship of six volatility-return pairs, viz. CBOE-VIX and S&P-500,FTSE-100 Volatility and FTSE-100, NASDAQ-100 Volatility (VXN) and NASDAQ,DAX Volatility (VDAX) and DAX-30, CAC Volatility (VCAC) and CAC-40 andSTOXX Volatility (VSTOXX) and STOXX. The assumption of a normal distributionin the return series is not appropriate when the distribution is skewed and henceOLS does not capture the complete picture of the relationship. Quantile regressionon the other hand can be set up with various loss functions, both parametric andnon-parametric (linear case) and can be evaluated with skewed marginal based copulas(for the non linear case). Which is helpful in evaluating the non-normal andnon-linear nature of the relationship between price and volatility. In the empiricalanalysis we compare the results from linear quantile regression (LQR) and copulabased non linear quantile regression known as copula quantile regression (CQR).The discussion of the properties of the volatility series and empirical findings inthis paper have significance for portfolio optimization, hedging strategies, tradingstrategies and risk management in general.