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

The purpose of this paper is to examine the asymmetric relationship between price and implied volatility and the associated extreme quantile dependence using linear and non linear quantile regression approach. Our goal in this paper is to demonstrate that the relationship between the volatility and market return as quantified by Ordinary Least Square (OLS) regression is not uniform across the distribution of the volatility-price return pairs using quantile regressions. We examine the bivariate 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 and STOXX Volatility (VSTOXX) and STOXX. The assumption of a normal distribution in the return series is not appropriate when the distribution is skewed and hence OLS does not capture the complete picture of the relationship. Quantile regression on the other hand can be set up with various loss functions, both parametric and non-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 and non-linear nature of the relationship between price and volatility. In the empirical analysis we compare the results from linear quantile regression (LQR) and copula based non linear quantile regression known as copula quantile regression (CQR). The discussion of the properties of the volatility series and empirical findings in this paper have significance for portfolio optimization, hedging strategies, trading strategies and risk management in general.