The fundamental insight of asset pricing is that an asset's value is determined by both its distribution of payoffs across economic states and state prices. SDFs summarize these state prices. In this section, we first define SDFs in complete and incomplete markets. We then define exchange rates and currency risk premia in terms of domestic and foreign SDFs.
SDFs are key components of the Euler equations that govern asset prices and returns. These Euler equations can be derived under three different assumptions: complete markets, the law of one price, or the existence of investors' preferences. We rapidly review these three assumptions here. Cochrane (2001) presents a more detailed exposition. Key references are Breeden (1979), Lucas (1978), and Harrison and Kreps (1979).
Assume that the investor derives some utility u from consumption C now and in the next period. This setup can be easily generalized to many periods. Let us find the price Pt at time t of a payoff Xt+1 at time t + 1. Let Q be the original consumption level absent at any asset purchase and let ξ be the amount of the asset the investor chooses to buy. The constant subjective discount factor is β. The maximization problem of this investor is
Substituting the constraints into the objective and setting the ...