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Evidence shows that the stochastic discount factor (SDF) is not always a downward-sloping function of S&P 500 returns when estimated using options data. In contrast, our results suggest that SDFs as functions of individual stock returns are generally downward sloping. A simple jump-diffusion model can reconcile these empirical findings. The same model also implies a steeper implied-volatility curve for the index than for the typical stock, a well-known empirical fact from the options literature. Both the SDF and volatility-curve results can be explained by a common source of jump risk among stocks, together with diversification of Brownian risk in the index. We also devise novel empirical tests of SDF monotonicity based on average returns of option trading strategies, thus avoiding the estimation of the return density functions.
