The paper presents a discrete state-space approximation method for solving nonlinear rational expectations models. The method uses numerical quadrature to approximate integral operators that arise in stochastic intertemporal models. It is particularly useful for approximating asset pricing models and has potential applications in other problems. The method is applied to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The method involves calibrating a Markov chain with a discrete state space whose probability distribution closely approximates the distribution of a given time series. The quality of the approximation improves as the discrete state space becomes finer. The technique is useful for taking a discrete approximation to the conditional density of strictly exogenous variables of a model. The conditional density can be specified a priori or estimated from data using parametric or nonparametric procedures. The method is particularly well-suited for asset pricing models without endogenous state variables. Once the state space is made discrete, the solution of the expectational equations involves matrix inversion. This maps the "difficult" problem of solving expectational equations into linear "approximating" problems requiring only matrix inversion. The paper extends this approach by identifying and making explicit the mapping from the continuous problem to the discrete linear problem and by providing an efficient method based on numerical quadrature for calibrating the discrete state-space economy. The method is applied to a representative agent/exchange economy asset pricing model in the style of Lucas (1978) and Mehra and Prescott (1985). The model considers a single asset with stochastic dividend stream and derives the asset pricing equation based on the agent's intertemporal utility maximization problem. The model assumes that consumption growth and dividend growth are functions of a finite-memory stationary stochastic process. The paper also discusses the convergence properties of the Markov chain model and the approximate solution to asset pricing equations. It shows that the method converges to the exact solution as the number of quadrature points increases. The method is tested on various models, including AR(1), AR(2), and VAR(2) processes, and is found to provide accurate approximations. The method is also applied to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The results show that the risk premium is positively related to the conditional variance of equity returns.The paper presents a discrete state-space approximation method for solving nonlinear rational expectations models. The method uses numerical quadrature to approximate integral operators that arise in stochastic intertemporal models. It is particularly useful for approximating asset pricing models and has potential applications in other problems. The method is applied to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The method involves calibrating a Markov chain with a discrete state space whose probability distribution closely approximates the distribution of a given time series. The quality of the approximation improves as the discrete state space becomes finer. The technique is useful for taking a discrete approximation to the conditional density of strictly exogenous variables of a model. The conditional density can be specified a priori or estimated from data using parametric or nonparametric procedures. The method is particularly well-suited for asset pricing models without endogenous state variables. Once the state space is made discrete, the solution of the expectational equations involves matrix inversion. This maps the "difficult" problem of solving expectational equations into linear "approximating" problems requiring only matrix inversion. The paper extends this approach by identifying and making explicit the mapping from the continuous problem to the discrete linear problem and by providing an efficient method based on numerical quadrature for calibrating the discrete state-space economy. The method is applied to a representative agent/exchange economy asset pricing model in the style of Lucas (1978) and Mehra and Prescott (1985). The model considers a single asset with stochastic dividend stream and derives the asset pricing equation based on the agent's intertemporal utility maximization problem. The model assumes that consumption growth and dividend growth are functions of a finite-memory stationary stochastic process. The paper also discusses the convergence properties of the Markov chain model and the approximate solution to asset pricing equations. It shows that the method converges to the exact solution as the number of quadrature points increases. The method is tested on various models, including AR(1), AR(2), and VAR(2) processes, and is found to provide accurate approximations. The method is also applied to study the relationship between the risk premium and the conditional variability of equity returns under an ARCH endowment process. The results show that the risk premium is positively related to the conditional variance of equity returns.