An Empirical Implementation of the Shadow Riskless Rate

We address the problem of asset pricing in a market where there are no risky assets. Previous work developed a theoretical model for a shadow riskless rate (SRR) for such a market, based on the drift component of the state-price deflator for that asset universe. Assuming that asset prices are modeled by correlated geometric Brownian motion, in this work, we develop a computational approach to estimate the SRR from empirical datasets. The approach employs principal component analysis to model the effects of individual Brownian motions, singular value decomposition to capture abrupt changes in the condition number of the linear system whose solution provides the SRR values, and regularization to control the rate of change of the condition number. Among other uses such as option pricing and developing a term structure of interest rates, the SRR can be used as an investment discriminator between different asset classes. We apply this computational procedure to markets consisting of various groups of stocks, encompassing different asset types and numbers. The theoretical and computational analysis provides the drift as well as the total volatility of the state-price deflator. We investigate the time trajectory of these two descriptive components of the state-price deflator for the empirical datasets.