Pricing Kernel extends concepts from economics and finance to include adjustments for risk. When pricing kernel is given, by non-arbitrage theory, all securities can be priced. Searching for a proper pricing kernel is one of the most important tasks for researchers in asset pricing. In this thesis, we attempt to search a proper pricing kernel in three different scenarios.In chapter 1, we attempt to find a robust pricing kernel for a stochastic volatility model with parameter uncertainty in an incomplete commodity market. Based on a class of stochastic volatility models in Trolle and Schwartz (2009), we investigate how the parameter uncertainty affects the risk premium and commodity contingent claim pricing. To answer this question, we follow a two-step procedure. Firstly, we propose a benchmark approach to find an optimal pricing kernel for the model without parameter uncertainty. Secondly, we uncover a robust pricing kernel via a robust approach for the model with parameter uncertainty. Thirdly, we apply the two pricing kernels into the commodity contingent claim pricing and quantify effect of parameter uncertainty on contingent claim securities. We find that the parameter uncertainty attributes a negative uncertainty risk premium. Moreover, the negative uncertainty risk premium yields a positive uncertainty volatility component in the implied volatilities in the option market.In chapter 2, we propose a multi-factor model with a quadratic pricing kernel, in which the underlying asset return is a linear function of multi-factors and the pricing kernel is a quadratic function of multi-factors. The model provides a potential unified framework to link cross sectional literatures, time-series literatures, option pricing literatures and term structure literatures. By examining option data from 2005 to 2008, this model dramatically improves the cross-sectional fitting of option data both in sample and out of sample than many standard GARCH volatility models such as Christoffersen, Heston and Nandi (2011). This model also offers explanations for several puzzles such as the U shape relationship between the pricing kernel and market index return, the implied volatility puzzle and fat tails of risk neutral return density function relative to the physical distribution.In chapter 3, we investigate whether idiosyncratic volatility risk premium is cross-sectional variant. We use stock historical moving average price as a proxy for retail ownership and examine whether idiosyncratic volatility is correlated with stock price level. Evidence from cross-sectional regressions and portfolio analysis both suggests that low-priced stocks (high retail ownership) have a significantly higher idiosyncratic volatility risk premium than high-priced stocks (low retail ownership). Especially, evidence in subsample tests suggests that lowest-priced stocks (highest retail ownership) have a significantly positive idiosyncratic risk premium while highest-priced stocks (lowest retail ownership) have an insignificant one, which is consistent with theoretical predictions of Merton (1986) and classical portfolio theory.