We forecast the physical probability distribution of one-month equity index returns using an initial density forecast, bid-ask prices of a cross-section of monthly index options, and systems of asset pricing conditions for incomplete options markets with frictions. The option pricing kernel is restricted to be a positive, monotonic, and/or convex function of index return, resembling the Intertemporal Marginal Rate of Substitution for standard utility functions. A physical density forecast is obtained by information projection of the initial forecast onto the set of distributions that are consistent with the prices and restrictions. The implied physical significantly improves upon the initial by using forward-looking information contained in the option prices. It also improves upon the implied risk-neutral, which confounds the physical density with the pricing kernel. The improvements in forecasting ability translate to an annual Information Ratio for Growth Optimal Portfolios of well above 0.60 during both calm markets and volatile markets. The convexity condition appears crucial: relaxing it often leads to pathological kernel shapes, the convergence of the implied to the initial, and a sharp deterioration of the forecasting ability. By contrast, the monotonicity condition seems less relevant, and the forecasting ability actually benefits from allowing U-shaped kernels, especially during volatile markets.