Generalized Affine Models

Abstract

Affine models are very popular in modeling financial time series as they allow for analytical calculation of prices of financial derivatives like treasury bonds and options. The main property of affine models is that the conditional cumulant function, defined as the logarithmic of the conditional characteristic function, is affine in the state variable. Consequently, an affine model is Markovian, like an autoregressive process, which is an empirical limitation. The paper generalizes affine models by adding in the current conditional cumulant function the lagged conditional cumulant function. Hence, generalized affine models are non-Markovian, such as ARMA and GARCH processes, allowing one to disentangle the short term and long-run dynamics of the process. Importantly, the new model keeps the tractability of prices of financial derivatives. This paper studies the statistical properties of the new model, derives its conditional and unconditional moments, as well as the conditional cumulant function of future aggregated values of the state variable, which is critical for pricing financial derivatives. It derives the analytical formulas of the term structure of interest rates and option prices. Different estimating methods are discussed including MLE, QML, GMM, and characteristic function based estimation methods. In a term structure of interest rate out-of-sample forecasting exercise, our results suggest that for a many horizons, a simple multivariate generalized affine model on observed yields predicts the whole term structure of the interest rate better than the VAR and the Nelson-Siegel’s model with AR(1) factor dynamic.

Keywords: Affine models, cumulant function, option pricing, term structure of interest rates

JEL Classification: C12, C13, C14, C22, C51, E43, G12