The Stochastic Discount Factor: Extending the Volatility Bound and a New Approach to Portfolio Selection with Higher-Order Moments

The authors extend the well-known Hansen and Jagannathan (HJ) volatility bound. HJ characterize the lower bound on the volatility of any admissible stochastic discount factor (SDF) that prices correctly a set of primitive asset returns. The authors characterize this lower bound for any admissible SDF that prices correctly both primitive asset returns and quadratic payoffs of the same primitive assets. In particular, they aim at pricing derivatives whose payoffs are defined as non-linear functions of the underlying asset payoffs. The authors construct a new volatility surface frontier in a three-dimensional space by considering not only the expected asset payoffs and variances, but also asset skewness. The intuition behind the authors' portfolio selection is motivated by the duality between the HJ mean-variance frontier and the Markowitz mean-variance portfolio frontier. The authors' approach consists of minimizing the portfolio risk subject not only to portfolio cost and expected return, as usual, but also subject to an additional constraint that depends on the portfolio skewness. In this sense, the authors shed light on portfolio selection when asset returns exhibit skewness.