Maximum Likelihood Estimation for Discrete Multivariate Vasicek Processes

Michael Pokojovy, Ebenezer Nkum, Thomas M. Fullerton Jr.

Abstract: Because of low correlation with other asset classes, bonds play major role in portfolio diversification efforts. As investment funds can create robust diversified portfolios with bonds, it is imperative that multiple bonds be analyzed simultaneously. We consider a multivariate extension of the original Vasicek model to multiple zero-coupon bonds. Due to the low-frequency nature of bonds and other debt securities, instead of working in continuous time, we apply the Euler-Maruyama discretization and study the resulting discrete multivariate Vasicek model. We adopt the maximum likelihood estimation (MLE) approach to estimate the parameters of the model, i.e., the long-term mean vector, reversion speed matrix and volatility matrix. Instead of reparametrizing the problem as a VAR(1) model and applying the classical ordinary least squares (OLS) approach for calibration, we rigorously derive a system of nonlinear estimating equations using multivariate vector and matrix calculus and propose a new statistical estimator based on solving this system with a Banach-type fixed point iteration. The performance of our new MLE vs the classical OLS estimator is thoroughly evaluated through a simulation study as well as backtesting analysis performed on 3-month US Treasury and AAA-rated Euro bond yield rates. In both cases, we conclude that our new estimator significantly outperforms the conventional OLS estimator. We also provide a set of Matlab® codes that may prove to be a useful tool for model calibration in connection with portfolio optimization and risk management in the bond market.