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

25 Pages Posted: 18 Apr 2013

See all articles by David E. Allen

David E. Allen

School of Mathematics and Statistics, The University of Sydney; Financial Research Network (FIRN); Department of Finance; School of Business and Law, Edith Cowan University

Abhay Kumar Singh

Edith Cowan University

Robert J. Powell

Edith Cowan University - School of Accounting, Finance and Economics; Financial Research Network (FIRN)

Michael McAleer

Erasmus University Rotterdam - Erasmus School of Economics, Econometric Institute; Tinbergen Institute; University of Tokyo - Centre for International Research on the Japanese Economy (CIRJE), Faculty of Economics

James W. Taylor

University of Oxford - Said Business School

Lyn C. Thomas

University of Southampton - School of Management

Date Written: April 18, 2013

Abstract

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.

Keywords: Return-Volatility relationship, quantile regression, copula, copula quantile regression, volatility index, tail dependence

Suggested Citation

Allen, David Edmund and Singh, Abhay Kumar and Powell, Robert J. and McAleer, Michael and Taylor, James W. and Thomas, Lyn C., Return-Volatility Relationship: Insights from Linear and Non-Linear Quantile Regression (April 18, 2013). Available at SSRN: https://ssrn.com/abstract=2253685 or http://dx.doi.org/10.2139/ssrn.2253685

David Edmund Allen

School of Mathematics and Statistics, The University of Sydney ( email )

School of Mathematics and Statistics F07
University of Sydney
Sydney, New South Wales 2006
Australia

HOME PAGE: http://www.maths.usyd.edu.au

Financial Research Network (FIRN)

C/- University of Queensland Business School
St Lucia, 4071 Brisbane
Queensland
Australia

HOME PAGE: http://www.firn.org.au

Department of Finance ( email )

Taiwan
Taiwan

School of Business and Law, Edith Cowan University

100 Joondalup Drive
Joondalup, WA 6027
Australia

HOME PAGE: http://www.dallenwapty.com

Abhay Kumar Singh (Contact Author)

Edith Cowan University ( email )

Joondalup Drive
Perth
Joondalup, WA 6027
Australia

Robert J. Powell

Edith Cowan University - School of Accounting, Finance and Economics ( email )

Joondalup Campus
Perth
Joondalup 6027, WA
Australia

Financial Research Network (FIRN)

C/- University of Queensland Business School
St Lucia, 4071 Brisbane
Queensland
Australia

HOME PAGE: http://www.firn.org.au

Michael McAleer

Erasmus University Rotterdam - Erasmus School of Economics, Econometric Institute ( email )

Rotterdam
Netherlands

Tinbergen Institute

Rotterdam
Netherlands

University of Tokyo - Centre for International Research on the Japanese Economy (CIRJE), Faculty of Economics

Tokyo
Japan

James W. Taylor

University of Oxford - Said Business School ( email )

Park End Street
Oxford, OX1 1HP
Great Britain

Lyn C. Thomas

University of Southampton - School of Management ( email )

Highfield
Southampton S017 1BJ, Hampshire SO17 1BJ
United Kingdom
(023) 8059 7718 (Phone)
(023) 8059 3844 (Fax)

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