Stochastic Discount Factor

A stochastic discount factor generalizes the concept of discounted present value. The component of a stochastic discount factor for a state is the value per unit of probability of a one-dollar payoff in that state. The inner product of a stochastic discount factor and the payoff vector for an asset is the asset price. In a primal/dual analysis in finance theory, commonly the choice variable in the dual defines a stochastic discount factor. The law of one price states that a portfolio having payoff zero must have cost zero. Invoking the Fredholm alternative establishes a basic result: there exists a stochastic discount factor for asset markets if and only if the law of one price holds. The fundamental theorem of finance asserts that there is no opportunity for arbitrage if and only if there exists a positive stochastic discount factor. There exists a cost/mean/variance efficient portfolio if and only if there is no risk-free portfolio providing an arbitrage opportunity. Applications of efficiency duality and the Stiemke alternative obtain these results and provide insight. The payoff space is a moment space valuable in finance theory. When the law of one price holds, the “pricing payoff” is the unique payoff that is a stochastic discount factor. “Cost and mean positivity” is the condition that the inner product of the pricing payoff and the mean vector in the payoff space is positive. The cost of the mean vector is positive, and the expected value of the pricing payoff is positive. One should see positivity as a requirement for asset-market equilibrium, a condition distinct from the requirement that there should be no opportunity for arbitrage.