Chapter 9

HJM Interest Rate Modeling with Three Risk Factors1

In Chapters 6 through 8, we provided worked examples of how to use the yield curve simulation framework of Heath, Jarrow, and Morton (HJM) using two different assumptions about the volatility of forward rates and one or two independent risk factors. The first volatility assumption was that volatility was dependent on the maturity of the forward rate and nothing else. The second volatility assumption was that the volatility of forward rates was dependent on both the level of rates and the maturity of the forward rates being modeled. The first two models were one-factor models, implying that random rate shifts are either all positive, all negative, or zero. Our third example postulated that the change in one-year spot rates, along with an unspecified second risk factor, was in fact driving interest rate movements. In this chapter, we generalize the model to include three risk factors in order to increase further the realism of the simulated yield curve. Consistent with the research of Dickler, Jarrow, and van Deventer cited in Chapters 6, 7, and 8, we explain why even three risk factors understate the number of risk factors driving the U.S. Treasury yield curve.


In Chapter 8, we explained that one-factor models imply no twists in the yield curve. Random yield shocks drive yields either all up or all down together. Yield curve twists, where some rates rise and ...

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