N°25-87: Tradable Recovery

We recover the projection of the stochastic discount factor (SDF) onto S&P 500 index returns as the Sharpe-optimal claim generated by a tradable portfolio of the underlying and listed options. Our nonparametric approach leverages the classical duality between the minimum-variance SDF and the maximum-Sharpe-ratio portfolio, provides finite-sample guarantees under weak temporal dependence, and supports conservative formal tests of monotonicity and curvature. The valuation of return states varies systematically with volatility and horizon: departures from standard SDF restrictions concentrate in specific return regions under low and high volatility, and intensify at shorter maturities.