Applying Heath-Jarrow-Morton Model to Forecasting the US Treasury Daily Yield Curve Rates

Abstract

The Heath-Jarrow-Morton (HJM) model is a powerful instrument for describing the stochastic evolution of interest rate curves under no-arbitrage assumption. An important feature of the HJM approach is the fact that the drifts can be expressed as functions of respective volatilities and the underlying correlation structure. Aimed at researchers and practitioners, the purpose of this article is to present a self-contained, but concise review of the abstract HJM framework founded upon the theory of interest and stochastic partial differential equations in infinite dimensions. To illustrate the predictive power of this theory, we apply it to modeling and forecasting the US Treasury daily yield curve rates. We fit a non-parametric model to real data available from the US Department of the Treasury and illustrate its statistical performance in forecasting future yield curve rates.

Multifactorial Heath-Jarrow-Morton model using principal component analysis

“…It describes the behavior of the future price at time t of a zero-coupon bond, denoted as 𝐵(𝑡, 𝑇), which pays one dollar at maturity 𝑇 as in [25]- [27]. The model is calibrated to the observed yield curve [28]- [31]. To estimate the prices of zero-coupon bonds at different maturities, the model begins with an exogenous specification of the stochastic dynamics of the forward rate and subsequently determines endogenously, in a risk-neutral world, the zero-coupon bond.…”

“…Hence, the model is implemented like in [4], [9], as these require the initial yield curve provided by the market at a previous date. Likewise, the instantaneous forward rate trend is calibrated so that the volatility-standardized risk premium is zero [31], [32]. However, it has notable differences from these models: a) The valuation process starts with an exogenous specification of the forward rate.…”

“…Taking the partial derivatives of the integrals on the right-hand side using Leibniz's rule: ) 𝑑𝑡 + 𝜑(𝑡)𝑑𝑊 𝑡 (8) As shown in (8) describes the behavior of the short rate where the trend of 𝑟 𝑡 is the slope of the initial forward rate. Due to the trend integrals in (8), the evolution of the short rate does not present the Markovian property [31]. Meanwhile, the multifactorial generalization of the HJM model has been developed as ( 9)…”