Abstract: 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.

Key words: pricing kernel, probability weight function, rank-dependent expected utility, maximum likelihood