Multi-Lag Term Structure Models with Stochastic Risk Premia

In this paper we propose a family of discrete-time term structure models where we specify a Gaussian autoregressiveof order p > 1 historical and risk-neutral dynamics for the factor (xt), considered as a latent or observable variable: inthe second case the factor is a vector of several yields. We present the Gaussian AR(p) Factor-Based Term StructureModel in which the stochastic discount factor (SDF) for the period (t, t + 1) is specified as an exponential-affinefunction of xt+1, the factor risk-correction coefficient is stochastic, and the associated yield-to-maturity formulaat time t is an affine function of Xt = (xt, . . . , xt-p+1)0. We propose the Moving Average (or Heath, Jarrow andMorton) characterization of the yield and short-term forward rate processes, under the risk-neutral and the S-forwardprobability : this representation gives the possibility to exactly replicate the currently-observed yield curve. We alsostudy the problem of matching the theoretical and the currently-observed market term structure by means of theExtended AR(p) approach. We present the Gaussian VAR(p) Factor-Based Term Structure Model, generalizing thepreviously mentioned results to the multivariate framework.