The Heath, Jarrow, Morton Model


  • Oldrich Alfons Vasicek


Equilibrium models of the term structure of interest rates, such as Vasicek (1977) and Cox et al. (1985), hereafter CIR, determine the equilibrium yield curve by modelling the dynamics of the short-term interest rate, specifying the market price of risk, and solving the resulting partial differential equation for bond prices. Several multi-factor extensions of the Vasicek and CIR framework have been advanced in the recent term structure literature using as additional factors different variables, such as the volatility of interest rates (see, e.g. Longstaff and Schwartz, 1992; Dai and Singleton, 2000), the slope of the term structure (Brennan and Schwartz, 1979; Schaefer and Schwartz, 1984), monetary policy rates (Bakshi and Chen, 1996), and inflation (Pennacchi, 1991; Sun, 1992). Since a no-arbitrage condition must hold in equilibrium, this brief article starts from the stated law of motion for bond prices to tersely show how their implied instantaneous forward rates have an evolution under the pricing measure that is fully characterized by the forward rate volatilities. Thus, the outcome of the article is the fundamental equation of the classic model contributed by Heath et al. (1992), hereafter HJM, which sets off with the study of the forward rates' no-arbitrage dynamics. By doing so, it shows that, despite its different angle and its apparent complex structure, the HJM model is fully consistent and has a clear link with standard equilibrium set-ups like those of the Vasicek and CIR type. This note was written in 1994.