Multi-Lag Term Structure Models with Stochastic Risk Premia

The purpose of this paper is to propose discrete-time term structure models where the historical dynamics of the factor (x_t) is given, in the univariate case, by a Gaussian AR(p) process, and, in the multivariate case, by a Gaussian n-dimensional VAR(p) process. The factor (x_t) is considered as a latent or an observable variable and, in the second case, (x_t) is given by the short rate (in the scalar setting) or by a vector of several yields (in the multivariate setting). We consider an exponential-affine stochastic discount factor (SDF) with a stochastic factor risk correction coefficient defined, at time t, as an affine function of X_t = (x_t, ..., x_{t-p+1})' and, consequently, the yield-to-maturity formula at time t is an affine function of the p most recent lagged values of x_{t+1}. We study the Gaussian AR(p) and the Gaussian VAR(p) Factor-Based Term Structure Models. We investigate, under the risk-neutral and the S-forward probability, the Moving Average (or discrete-time Heath, Jarrow and Morton) representation of the yield and short-term forward rate processes. This representation gives the possibility to exactly replicate the currently-observed yield curve. We also study the problem of matching the theoretical and currently-observed market term structure by means of the Extended AR(p) approach