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The Relation between Physical and Risk-neutral Cumulants


  • We are especially grateful to Sudipto Dasgupta (Editor-in-Chief), an Associate Editor and an anonymous referee, whose helpful comments substantially improved the paper. We also acknowledge helpful comments from Long Chen, Jinghong Shu, Zhiguang Wang, and seminar participants at the University of Macau, the University of International Business and Economics, Sun Yat-Sen University, 2009 Financial Management Association (FMA) Annual Meeting in Reno, and 2010 China International Conference in Finance (CICF2010) in Beijing. H. Zhao has been supported by the Fundamental Research Funds for the Central Universities (Project No. 1209022), Guangdong Province University Key Project of Humanities and Social Sciences (Project No. 11ZGXM63002), and State Key Program of National Natural Science of China (Project No. 71231008). J. E. Zhang has been supported by an establishment grant from University of Otago. E. C. Chang has been partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. HKU 7403/06H and HKU 7179/07E).

Huimin Zhao

Sun Yat-Sen Business School

Sun Yat-Sen University

135 Xingang West Road

Guangzhou 510275



Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. The empirical observation on skewness research suggests that derivative professionals may also desire to hedge beyond volatility risk and there exists the need to hedge higher-moment market risks, such as skewness and kurtosis risks. We study two derivative contracts – skewness swap and kurtosis swap – which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the both skewness and kurtosis risk premiums are significantly negative.