The investor behavior has always been in focus in the literature on financial economics. Naturally, it involves the pricing kernel, which also known as the stochastic discount factor. In standard economic theory, the pricing kernel is a monotonically decreasing function of the market return, corresponds to a concave utility function and investor risk aversion. However, there has been a lot of discussion about the reliability of this theory. Many recent empirical studies based on index option data have provide evidence of non-monotonically decreasing pricing kernel. The non-monotonicity of empirical pricing kernel estimates has become known as the “pricing kernel puzzle” or “risk aversion puzzle”. Numerous attempts have been undertaken to explain the reason for the “pricing kernel puzzle” from different perspectives, including investor's heterogeneous beliefs, misspecification of the underlying state space, ambiguity aversion, rank-dependent expected utility, incomplete market, statistical artifact, investor's sentiment, etc. In this paper we consider a pricing kernel based on the rank-dependent expected utility model with a probability weighting function. The rank-dependent expected utility model was first introduced by Quiggin (1982), and further developed by Yaari (1987) and Allais (1988). We show that this model is consistent with several features of the empirical pricing kernel estimated from index options and that the data imply the shape of probability weights with the emphasis on tail events.Methods: In the last decades, there is a large literature on the estimation of the pricing kernel. A number of earlier papers estimate the pricing kernel using aggregate consumption data, problems with imprecise measurement of aggregate consumption can weaken the empirical results of these papers. Recently, many authors have used the historical underlying asset and option prices data to estimate the pricing kernel. This approach avoids the use of aggregate consumption data and can obtain more reliable results. Based on the option and underlying asset prices data, this paper derives the empirical pricing kernel by estimating the objective and risk-neutral densities based on the discrete-time EGARCH model and continuous-time GARCH diffusion model, respectively. Furthermore, the probability weighting functions are constructed based on the rank-dependent expected utility model and provides an explanation for the “pricing kernel puzzle”.Results: The empirical results based on the Hong Kong Hang Seng index (HSI) and index warrant prices data show that: (1) The estimated empirical pricing kernel is non-monotone and exhibits a hump, which known as the “pricing kernel puzzle”;(2) The estimated probability weights have the S shape, which overweights the probabilities in the middle and high of the distribution and underweights the tail events;(3) The “pricing kernel puzzle” can be explained by the rank-dependent expected utility model with standard utility and the S-shaped probability weighting function.Conclusions: The empirical pricing kernel from option and underlying asset prices are estimated and probability weighting functions are constructed based on the rank-dependent expected utility model, and provides an explanation for the “pricing kernel puzzle”. The results show that the estimated empirical pricing kernel is non-monotonic, i.e. “pricing kernel puzzle”. Under the standard CRRA utility function, the constructed probability weighting functions have S shape, indicating that investor underweights the tail events (the probabilities in the tails) and overweights the probabilities in the middle and high of the distribution. Thus, the nonmonotonic pricing kernel (“pricing kernel puzzle”) is explained by the rank-dependent expected utility with concave utility and the S-shaped probability weighting function. This type of weighting function represents investor's excessive optimism and overconfidence, which is also consistent with the findings of Barone-Adesi (2014) in the sense of sentiment theory. The probability weighting is an important and empirically relevant element for understanding asset prices, which can be applied to a wide range of problems in finance related to investment decision-making, option pricing, risk management and fund rating. Some evidences for the assumptions of investor behavior are provided in our findings.
[1] Brown D P,Jackwerth J C. The pricing kernel puzzle: Reconciling index option data and economic theory//Batten J, Wagner N,Thornton R,et al. Derivatives securities pricing and modelling, England: Emerald Group Publishing Limited, 2012, 155-183.
[2] At-Sahalia Y, Lo A W. Nonparametric risk management and implied risk aversion[J]. Journal of Econometrics, 2000, 94(1): 9-51.
[3] Jackwerth J C. Recovering risk aversion from option prices and realized returns[J]. Review of Financial Studies, 2000, 13(2): 433-451.
[4] Rosenberg J, Engle R F. Empirical pricing kernels[J]. Journal of Financial Economics, 2002, 64(3): 341-372.
[5] Golubev Y, Härdle W, Timofeev R. Testing monotonicity of pricing kernels. Working paper, Humboldt-Universität zu Berlin, 2009.
[6] Beare B K, Schmidt L D. An empirical test of pricing kernel monotonicity. Working paper, University of California, 2013.
[7] Ziegler A. Why does implied risk aversion smile?[J]. Review of Financial Studies, 2007, 20(3): 859-904.
[8] Bakshi G, Madan D. Investor heterogeneity and the non-monotonicity of the aggregate marginal rate of substitution in the market index. Working paper, University of Maryland, 2008.
[9] Detlefsen K, Härdle W, Moro R. Empirical pricing kernels and investor preferences. Working paper, Universite de Provence, 2007.
[10] Härdle W, Krätschmer V, Moro R. A microeconomic explanation of the EPK paradox. Working paper, Humboldt-Universität zu Berlin, 2009.
[11] Chabi-Yo F, Garcia R, Renault E. State dependence can explain the risk aversion puzzle[J]. Review of Financial Studies, 2008, 21(2): 973-1011.
[12] Chabi-Yo F. Pricing kernels with stochastic skewness and volatility risk[J]. Management Science, 2012, 58(3): 624-640.
[13] Christoffersen P, Heston S, Jacobs K. Capturing option anomalies with a variance-dependent pricing kernel. Working paper, University of Toronto, 2012.
[14] Gollier C. Portfolio choices and asset prices: The comparative statics of ambiguity aversion[J]. Review of Economic Studies, 2011, 78(4): 1329-1344.
[15] Polkovnichenko V, Zhao Feng. Probability weighting functions implied in options prices[J]. Journal of Financial Economics, 2013, 107(3): 580-609.
[16] Hens T,Reichlin C. Three solutions to the pricing kernel puzzle[J]. Review of Finance, 2013, 17(3): 1065-1098.
[17] Linn M,Shive S, Shumway T. Pricing kernel monotonicity and conditional information. Working paper, University of Michigan, 2014.
[18] Barone-Adesi G, Mancini L, Shefrin H. Sentiment, risk aversion, and time preference. Working paper, University of Lugano, 2014.
[19] Quiggin J. A theory of anticipated utility[J]. Journal of Economic and Behavioral Organization, 1982, 3(4): 323-343.
[20] Yaari M E. The dual theory of choice under risk[J]. Econometrica, 1987, 55(1): 95-115.
[21] Allais M. The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function[M]//Munier B R. Risk, decision and rationality. Dordrecht, the Netherlands:Springer, 1988, 233-289.
[22] Hansen L P, Singleton K J. Generalized instrumental variables estimation of nonlinear rational expectations models[J].Econometrica, 1982, 50(5): 1269-1286.
[23] Hansen L P, Singleton K J. Stochastic consumption, risk aversion, and the temporal behavior of asset returns[J]. Journal of Political Economy, 1983, 91(2): 249-265.
[24] Hansen L P,Jagannathan R. Implications of security market data for models of dynamic economies[J]. Journal of Political Economy, 1991, 99(2): 225-262.
[25] Chapman D. Approximating the asset pricing kernel[J]. Journal of Finance, 1997, 52(4): 1383-1410.
[26] Chernov M. Empirical reverse engineering of the pricing kernel[J]. Journal of Econometrics, 2003, 116(1): 329-364.
[27] Song Zhaogang,Xiu Dacheng. A tale of two option markets: Pricing kernels and volatility risk. Working paper, Federal Reserve Board, 2013.
[28] Barone-Adesi G, Engle R F, Mancini L. A GARCH option pricing model with filtered historical simulation[J]. Review of Financial Studies, 2008, 21(3): 1223-1258.
[29] Andersen T G,Bollerslev T, Diebold F, et al. The distribution of stock return volatility[J]. Journal of Financial Economics, 2001, 61(1): 43-76.
[30] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P 500: Evidence from realized volatility, daily returns, and option prices[J]. Review of Financial Studies, 2010, 23(8): 3141-3189.
[31] Chourdakis K, Dotsis G. Maximum likelihood estimation of non-affine volatility processes[J]. Journal of Empirical Finance, 2011, 18(3): 533-545.
[32] Wu Xinyu, Ma Chaoqun, Wang Shouyang. Warrant pricing under GARCH diffusion model[J]. EconomicModelling, 2012, 29(6): 2237-2244.
[33] Kahneman D, Tversky A. Prospect theory: An analysis of decision under risk[J]. Econometrica, 1979, 47(2): 263-291.
[34] Tversky A, Kahneman D. Advances in prospect theory: Cumulative representation of uncertainty[J]. Journal of Risk and Uncertainty, 1992, 5(4): 297-323.
[35] Prelec D. The probability weighting function[J]. Econometrica, 1998, 66(3): 497-527.
[36] Kliger D, Levy O. Theories of choice under risk: Insights from financial markets[J]. Journal of Economic Behavior and Organization, 2009, 71(2): 330-346.
[37] Dierkes M. Option-implied risk attitude under rank-dependent utility. Working paper, University of Münster, Münster, 2009.
[38] 董大勇, 史本山, 曾召友. 展望理论的权重函数与证券收益率分布[J]. 中国管理科学, 2005, 13(1): 24-29.
[39] 董大勇, 金炜东. 收益率分布主观模型及其实证分析[J]. 中国管理科学, 2007, 15(1): 112-120.
[40] 李昊. 基于概率权重函数和随机占优准则的开放式基金评级[J]. 中国管理科学, 2013, 21(1): 23-30.
[41] 张婷婷, 文凤华, 戴志锋, 等. 概率权重函数与股市收益率分布[J]. 系统工程, 2013, 31(11): 18-26.
[42] Cochrane J H. Asset Pricing[M].New Jersey: Princeton University Press, 2001.
[43] Bollerslev T, Wooldridge J M. Quasi-maximum likelihood estimation and inference in dynamic models with time varying covariances[J]. Economics Letters, 1992, 11(2): 143-172.
[44] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. Review of Financial Studies, 1993, 6(2): 327-343.
[45] Richard J F, Zhang Wei. Efficient high-dimensional importance sampling[J]. Journal of Econometrics, 2007, 127(2): 1385-1411.
[46] 刘杨树, 郑振龙, 张晓南. 风险中性高阶矩: 特征、风险与应用[J]. 系统工程理论与实践, 2012, 32(3): 647-655.
[47] Camerer C F, Ho T. Violations of the betweenness axiom and nonlinearity in probability[J]. Journal of Risk and Uncertainty, 1994, 8(2): 167-196.
