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.
Highlights
Applying Heath-Jarrow-MortonStocks and bonds play a crucial role in making investment decisions
For a maturity T ≥ 0, let P(t, T ) denote the price of a zero-coupon bond at time t ∈ [0, T ]. It is known from the theory of interest rates that the instantaneous forward rate of a zero-coupon bond is given by
We provide a more powerful framework, referred to as the abstract HJM model, which incorporates arbitrage-free infinite-dimensional models
Summary
Stocks and bonds play a crucial role in making investment decisions. there is a significant difference between stock and bond markets. For a maturity T ≥ 0, let P(t, T ) denote the price of a zero-coupon bond at time t ∈ [0, T ] It is known from the theory of interest rates (viz. Definition 10 in Section 2) that the instantaneous forward rate of a zero-coupon bond is given by. In. Section 4, we summarize central properties of the abstract HJM model, in particular, we give sufficient conditions under which the abstract HJM model defines an arbitrage-free market, describe the long rates associated with the model, and give an example of a state space compatible with the instantaneous forward rate dynamics. Appendices A–C contain all seminal probabilistic and functional-analytic concepts and results [6,11,12,13,14,15] used in the article
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