By assuming that the stochastic discount factor (SDF) [Formula: see text] is a proper but unspecified function of state variables [Formula: see text], we show that this function [Formula: see text] must solve a simple second-order linear differential equation specified by state variables’ risk-neutral dynamics. Therefore, this assumption determines the most general possible SDFs and associated preferences, that are consistent with the given risk-neutral state dynamics and interest rate. A consistent SDF then implies the corresponding state dynamics in the data-generating measure. Our approach offers novel flexibilities to extend several popular asset pricing frameworks: affine and quadratic interest rate models, as well as models built on linearity-generating processes. We illustrate the approach with an international asset pricing model in which (i) interest rate has an affine dynamic term structure and (ii) the forward premium puzzle is consistent with consumption-risk rationales; the two asset pricing features previously deemed conceptually incompatible.

中文翻译：

---

功能随机贴现因子

通过假设随机贴现因子 (SDF) [公式：见文本] 是状态变量 [公式：见文本] 的一个适当但未指定的函数，我们证明这个函数 [公式：见文本] 必须解决一个简单的二阶由状态变量的风险中性动态指定的线性微分方程。因此，该假设确定了与给定的风险中性状态动态和利率相一致的最一般可能的 SDF 和相关偏好。然后，一致的 SDF 意味着数据生成度量中的相应状态动态。我们的方法提供了新的灵活性来扩展几种流行的资产定价框架：仿射和二次利率模型，以及建立在线性生成过程上的模型。我们用国际资产定价模型来说明该方法，其中 (i) 利率具有仿射动态期限结构，并且 (ii) 远期溢价难题与消费风险原理一致；以前认为在概念上不兼容的两种资产定价特征。