Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-17T05:58:46.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Option-Based Estimation of the Price of Coskewness and Cokurtosis Risk

Published online by Cambridge University Press:  10 November 2020

Peter Christoffersen ,
Mathieu Fournier ,
Kris Jacobs  and
Mehdi Karoui
Peter Christoffersen
Affiliation:
(deceased)
Mathieu Fournier
Affiliation:
HEC Montréal mathieu.fournier@hec.ca
Kris Jacobs*
Affiliation:
University of Houstonkjacobs@bauer.uh.edu
Mehdi Karoui
Affiliation:
Ontario Municipal Employees Retirement System (OMERS) mehdi.karoui@mail.mcgill.ca
*
kjacobs@bauer.uh.edu (corresponding author)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the prices of risk for factors that are nonlinear in the market return can be obtained using index option prices. The price of coskewness risk corresponds to the market variance risk premium, and the price of cokurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An analysis of factor models with coskewness risk indicates that the new estimates of the price of risk improve the models’ performance compared with regression-based estimates.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 56 , Issue 1 , February 2021 , pp. 65 - 91
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Michael G. Foster School of Business, University of Washington

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Peter Christoffersen, our dear friend and coauthor, tragically passed away prior to the publication of this article. We thank the Bank of Canada, the Global Risk Institute, and the Social Sciences and Humanities Research Council (SSHRC) for financial support. For helpful comments, we thank Tim Bollerslev, Robert Dittmar (the referee), Adlai Fisher, Bryan Kelly, Eric Renault, Michael Rockinger, Sang Byung Seo, Bas Werker, Xiaoyan Zhang, and seminar participants at the 2015 American Finance Association (AFA) Meeting and the 2014 Society for Financial Econometrics (SoFiE) meeting.

References

Ang, A.; Chen, J.; and Xing, Y.. “Downside Risk.” Review of Financial Studies, 19 (2006), 1191–1239.CrossRefGoogle Scholar
Ang, A.; Hodrick, R.; Xing, Y.; and Zhang, X.. “The Cross-Section of Volatility and Expected Returns.” Journal of Finance, 61 (2006), 259–299.CrossRefGoogle Scholar
Azalideh, S.; Brandt, M.; and Diebold, F.. “Range-Based Estimation of Stochastic Volatility Models.” Journal of Finance, 57 (2002), 1047–1091.Google Scholar
Bakshi, G.; Kapadia, N.; and Madan, D.. “Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options.” Review of Financial Studies, 16 (2003), 101–143.CrossRefGoogle Scholar
Bakshi, G., and Madan, D.. “Spanning and Derivative Security Valuation.” Journal of Financial Economics, 55 (2000), 205–238.CrossRefGoogle Scholar
Bakshi, G., and Madan, D.. “A Theory of Volatility Spreads.” Management Science, 52 (2006), 1945–1956.CrossRefGoogle Scholar
Bakshi, G.; Madan, D.; and Panayotov, G.. “Returns of Claims on the Upside and the Viability of U-Shaped Pricing Kernels.” Journal of Financial Economics, 97 (2010), 130–154.CrossRefGoogle Scholar
Bali, T. G., and Engle, R. F.. “The Intertemporal Capital Asset Pricing Model with Dynamic Conditional Correlations.” Journal of Monetary Economics, 57 (2010), 377–390.CrossRefGoogle Scholar
Bali, T. G., and Zhou, H.. “Risk, Uncertainty, and Expected Returns.” Journal of Financial and Quantitative Analysis, 51 (2016), 707–735.CrossRefGoogle Scholar
Bansal, R., and Viswanathan, S.. “No-Arbitrage and Arbitrage Pricing: A New Approach.” Journal of Finance, 48 (1993), 1231–1262.CrossRefGoogle Scholar
Barone-Adesi, G. “Arbitrage Equilibrium with Skewed Asset Returns.” Journal of Financial and Quantitative Analysis, 20 (1985), 299–313.CrossRefGoogle Scholar
Barras, L., and Malkhozov, A.. “Does Variance Risk Have Two Prices? Evidence from the Equity and Option Markets.” Journal of Financial Economics, 121 (2016), 79–92.CrossRefGoogle Scholar
Barro, R. “Rare Disasters and Asset Markets in the Twentieth Century.” Quarterly Journal of Economics, 121 (2006), 823–866.CrossRefGoogle Scholar
Bollerslev, T.; Marrone, J.; Xu, L.; and Zhou, H.. “Predicting Stock Returns with Variance Risk Premia: Statistical Inference and International Evidence.” Journal of Financial and Quantitative Analysis, 49 (2014), 633–661.CrossRefGoogle Scholar
Bollerslev, T.; Tauchen, G.; and Zhou, H.. “Expected Stock Returns and Variance Risk Premia.” Review of Financial Studies, 22 (2009), 4463–4492.CrossRefGoogle Scholar
Bollerslev, T., and Todorov, V.. “Tails, Fears, and Risk Premia.” Journal of Finance, 66 (2011), 2165–2211.CrossRefGoogle Scholar
Brandt, M., and Jones, C.. “Volatility Forecasting with Range-Based EGARCH Models.” Journal of Business and Economic Statistics, 24 (2006), 470–486.CrossRefGoogle Scholar
Campbell, J. Y. “Intertemporal Asset Pricing Without Consumption Data.” American Economic Review, 83 (1993), 487–515.Google Scholar
Campbell, J. Y. “Understanding Risk and Return.” Journal of Political Economy, 104 (1996), 298–345.CrossRefGoogle Scholar
Carr, P., and Wu, L.. “Variance Risk Premia.” Review of Financial Studies, 22 (2009), 1311–1341.CrossRefGoogle Scholar
Chabi-Yo, F. “Conditioning Information and Variance Bounds on Pricing Kernels with Higher-Order Moments: Theory and Evidence.” Review of Financial Studies, 21 (2008), 181–231.CrossRefGoogle Scholar
Chabi-Yo, F. “Pricing Kernels with Stochastic Skewness and Volatility Risk.” Management Science, 58 (2012), 624–640.CrossRefGoogle Scholar
Chabi-Yo, F. “Real-Time Distribution of Stochastic Discount Factors.” Working Paper, University of Massachusetts at Amherst (2018).CrossRefGoogle Scholar
Chabi-Yo, F.; Leisen, D.; and Renault, E.. “Aggregation of Preferences for Skewed Asset Returns.” Journal of Economic Theory, 154 (2014), 453–489.CrossRefGoogle Scholar
Chang, B.-Y.; Christoffersen, P.; and Jacobs, K.. “Market Skewness Risk and the Cross-Section of Stock Returns.” Journal of Financial Economics, 107 (2013), 46–68.CrossRefGoogle Scholar
Chaudhuri, R., and Schroder, M.. “Monotonicity of the Stochastic Discount Factor and Expected Option Returns.” Review of Financial Studies, 28 (2015), 1462–1505.CrossRefGoogle Scholar
Chen, H.; Joslin, S.; and Tran, N.-K.. “Rare Disasters and Risk Sharing with Heterogeneous Beliefs.” Review of Financial Studies, 25 (2012), 2189–2224.CrossRefGoogle Scholar
Chou, R. “Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range.” Journal of Money, Credit and Banking, 37 (2005), 561–582.CrossRefGoogle Scholar
Cochrane, J. Asset Pricing. Princeton, NJ: Princeton University Press (2005).Google Scholar
Conrad, J.; Dittmar, R.; and Ghysels, E.. “Ex Ante Skewness and Expected Stock Returns.” Journal of Finance, 68 (2013), 85–124.CrossRefGoogle Scholar
Constantinides, G., and Duffie, D.. “Asset Pricing with Heterogeneous Consumers.” Journal of Political Economy, 104 (1996), 219–240.CrossRefGoogle Scholar
Corsi, F. “A Simple Approximate Long Memory Model of Realized Volatility.” Journal of Financial Econometrics, 7 (2009), 174–196.CrossRefGoogle Scholar
Cremers, M.; Halling, M.; and Weinbaum, D.. “Aggregate Jump and Volatility Risk in the Cross-Section of Stock Returns.” Journal of Finance, 70 (2015), 577–614.CrossRefGoogle Scholar
Dittmar, R. F. “Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross Section of Equity Returns.” Journal of Finance, 57 (2002), 369–403.CrossRefGoogle Scholar
Drechsler, I., and Yaron, A.. “What’s Vol Got to Do with It.” Review of Financial Studies, 24 (2011), 1–45.CrossRefGoogle Scholar
Driessen, J.; Maenhout, P.; and Vilkov, G.. “The Price of Correlation Risk: Evidence from Equity Options.” Journal of Finance, 64 (2009), 1377–1406.CrossRefGoogle Scholar
Fama, E. F., and French, K. R.. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics, 33 (1993), 3–56.CrossRefGoogle Scholar
Fama, E. F., and MacBeth, J. D.. “Risk, Return and Equilibrium: Empirical Tests.” Journal of Political Economy, 81 (1973), 607–636.CrossRefGoogle Scholar
Gabaix, X. “Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance.” Quarterly Journal of Economics, 127 (2012), 645–700.CrossRefGoogle Scholar
Garlappi, L., and Skoulakis, G.. “Taylor Series Approximations to Expected Utility and Optimal Portfolio Choice.” Mathematics and Financial Economics, 5 (2011), 121–156.CrossRefGoogle Scholar
Ghysels, E. “On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt?” Journal of Finance, 53 (1998), 549–573.CrossRefGoogle Scholar
Ghysels, E.; Plazzi, A.; and Valkanov, R.. “Why Invest in Emerging Markets? The Role of Conditional Return Asymmetry.” Journal of Finance, 71 (2016), 2145–2192.CrossRefGoogle Scholar
Golec, J., and Tamarkin, M.. “Bettors Love Skewness, Not Risk, at the Horse Track.” Journal of Political Economy, 106 (1998), 205–225.CrossRefGoogle Scholar
Gospodinov, N.; Kan, R.; and Robotti, C.. “Misspecification-Robust Inference in Linear Asset Pricing Models with Irrelevant Risk Factors.” Review of Financial Studies, 27 (2014), 2139–2170.CrossRefGoogle Scholar
Gourio, F. “Disaster Risk and Business Cycles.” American Economic Review, 102 (2012), 2734–2766.CrossRefGoogle Scholar
Guidolin, M., and Timmermann, A.. “International Asset Allocation under Regime Switching, Skew, and Kurtosis Preferences.” Review of Financial Studies, 21 (2008), 889–935.CrossRefGoogle Scholar
Han, B., and Zhou, Y.. “Variance Risk Premiums and Cross-Section of Stock Returns.” Working Paper, University of Texas at Austin (2011).CrossRefGoogle Scholar
Harvey, C. R., and Siddique, A.. “Conditional Skewness in Asset Pricing.” Journal of Finance, 55 (2000), 1263–1295.CrossRefGoogle Scholar
Heston, S. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327–343.CrossRefGoogle Scholar
Jackwerth, J., and Rubinstein, M.. “Recovering Probability Distributions from Option Prices.” Journal of Finance, 51 (1996), 1611–1631.CrossRefGoogle Scholar
Jagannathan, R., and Wang, Z.. “An Asymptotic Theory for Estimating Beta-Pricing Models Using Cross-Sectional Regression.” Journal of Finance, 53 (1998), 1285–1309.CrossRefGoogle Scholar
Jondeau, E., and Rockinger, M.. “Conditional Volatility, Skewness, and Kurtosis: Existence, Persistence, and Comovements.” Journal of Economic Dynamics and Control, 27 (2003), 1699–1737.CrossRefGoogle Scholar
Julliard, C., and Ghosh, A.. “Can Rare Events Explain the Equity Premium Puzzle?” Review of Financial Studies, 25 (2012), 3037–3076.CrossRefGoogle Scholar
Kan, R., and Zhang, C.. “Two-Pass Tests of Asset Pricing Models with Useless Factors.” Journal of Finance, 54 (1999), 204–235.CrossRefGoogle Scholar
Kelly, B., and Jiang, H.. “Tail Risk and Asset Prices.” Review of Financial Studies, 27 (2014), 2841–2871.CrossRefGoogle Scholar
Kleibergen, F. “Tests of Risk Premia in Linear Factor Models.” Journal of Econometrics, 149 (2009), 149–173.CrossRefGoogle Scholar
Kraus, A., and Litzenberger, R. H.. “Skewness Preference and the Valuation of Risk Assets.” Journal of Finance, 31 (1976), 1085–1100.Google Scholar
Leland, H. “Beyond Mean-Variance: Performance Measurement of Portfolios Using Options or Dynamic Strategies.” Working Paper, University of California, Berkeley (1997).Google Scholar
Lim, K.-G. “A New Test of the Three-Moment Capital Asset Pricing Model.” Journal of Financial and Quantitative Analysis, 24 (1989), 205–216.CrossRefGoogle Scholar
Liu, J.; Pan, J.; and Wang, T.. “An Equilibrium Model of Rare-Event Premia and Its Implication for Option Smirks.” Review of Financial Studies, 18 (2005), 131–164.CrossRefGoogle Scholar
Maheu, J.; McCurdy, T.; and Zhao, X.. “Do Jumps Contribute to the Dynamics of the Equity Premium?” Journal of Financial Economics, 110 (2013), 457–477.CrossRefGoogle Scholar
Martin, I. “What Is the Expected Return on the Market?” Quarterly Journal of Economics, 132 (2017), 367–433.CrossRefGoogle Scholar
Mitton, T., and Vorkink, K.. “Equilibrium Underdiversification and the Preference for Skewness.” Review of Financial Studies, 20 (2007), 1255–1288.CrossRefGoogle Scholar
Patton, A., and Timmermann, A.. “Monotonicity in Asset Returns: New Tests with Applications to the Term Structure, the CAPM, and Portfolio Sorts.” Journal of Financial Economics, 98 (2010), 605–625.CrossRefGoogle Scholar
Rietz, T. “The Equity Risk Premium: A Solution.” Journal of Monetary Economics, 22 (1988), 117–131.CrossRefGoogle Scholar
Rogers, L., and Satchell, S.. “Estimating Variance from High, Low and Closing Prices.” Annals of Applied Probability, 1 (1991), 504–512.CrossRefGoogle Scholar
Rubinstein, M. “The Fundamental Theory of Parameter-Preference Security Valuation.” Journal of Financial and Quantitative Analysis, 8 (1973), 61–69.CrossRefGoogle Scholar
Schneider, P. “Generalized Risk Premia.” Journal of Financial Economics, 116 (2015), 487–504.CrossRefGoogle Scholar
Schneider, P.; Wagner, C.; and Zechner, J.. “Low-Risk Anomalies?” Journal of Finance 75 (2020), 2673–2718.CrossRefGoogle Scholar
Scott, R. C., and Horvath, P. A.. “On the Direction of Preference for Moments of Higher Order than the Variance.” Journal of Finance, 35 (1980), 915–919.CrossRefGoogle Scholar
Shanken, J. “On the Estimation of Beta Pricing Models.” Review of Financial Studies, 5 (1992), 1–33.CrossRefGoogle Scholar
Skoulakis, G. “On the Quality of Taylor Approximations to Expected Utility.” Applied Financial Economics, 22 (2012), 863–876.CrossRefGoogle Scholar
Wachter, J. “Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?” Journal of Finance, 68 (2013), 987–1035.CrossRefGoogle Scholar
Supplementary material: PDF

Christoffersen et al. supplementary material

Online Appendix

Download Christoffersen et al. supplementary material(PDF)
PDF 329.3 KB