A large class of asset pricing models predicts that securities which have high payoffs when market returns are low tend to be more valuable than those with high payoffs when market returns are high. More generally, we expect the projection of the stochastic discount factor on the market portfolio-that is, the discounted pricing kernel evaluated at the market portfolio-to be a monotonically decreasing function of the market portfolio. Numerous recent empirical studies appear to contradict this prediction. The non-monotonicity of empirical pricing kernel estimates has become known as the pricing kernel puzzle. In this paper we propose and apply a formal statistical test of pricing kernel monotonicity. We apply the test using 17 years of data from the market for European put and call options written on the S&P 500 index. Statistically significant violations of pricing kernel monotonicity occur in a substantial proportion of months, suggesting that observed non-monotonicities are unlikely to be the product of statistical noise.
