We develop a restriction that precludes implausibly high reward-for-risk in incomplete international economies to consider a theoretical problem that characterizes a lower bound on the covariance between stochastic discount factors (SDFs) subject to correct pricing. The problem is analytically solvable and synthesizes domestic and foreign SDFs into spanned and unspanned components. Our novelty is that exchange rate growth need not equal the ratio of SDFs and that the SDF correlations are plausibly lowered. Exploiting the realities of cross-country correlations of macroeconomic quantities, namely, consumption, wealth, dividend growths, and asset returns, our empirical investigation refutes the specification of complete markets. Received September 19, 2016; editorial decision August 31, 2017 by Editor Andrew Karolyi.

We document the phenomenon that average excess returns of out-of-the-money puts and calls on bond futures are negative, both unconditionally and conditionally on economic states. To explain these findings, we develop economically motivated restrictions in the context of a theory in which the pricing kernel is a general diffusion process with spanned and unspanned components. Our reconciliation is a framework that introduces market incompleteness and priced unspanned volatility risks, allowing for time-varying downside and upside futures risk premiums. The estimated model shows consistency with data on bond yields, yield volatilities, bond futures return volatilities, option prices, and option risk premiums.