Chapter 7

HJM Interest Rate Modeling with Rate and Maturity-Dependent Volatility

In Chapter 6, we provided a worked example of interest rate modeling and valuation using the Heath, Jarrow, and Morton (HJM) framework and the assumption that forward rate volatility was dependent on years to maturity using actual U.S. Treasury yields prevailing on March 31, 2011, and historical volatility from 1962 to 2011. In this chapter, we increase the realism of the forward rate volatility assumption. Actual data makes it very clear that forward rate volatility depends not just on the time to maturity of the forward rate but also on the level of spot interest rates. This chapter shows how to incorporate that assumption into the analysis for enhanced accuracy and realism. This is a much more general assumption than that used by Ho and Lee (1986) for constant volatility or by Vasicek (1977) and Hull and White (1990b) for declining volatility.

We again use data from the Federal Reserve statistical release H15 published on April 1, 2011, for yields prevailing on March 31, 2011. U.S. Treasury yield curve data was smoothed using Kamakura Risk Manager version 7.3 to create zero-coupon bonds via the maximum smoothness forward rate technique of Adams and van Deventer discussed in Chapter 5.


Following Jarrow (2002), we make the same modeling assumptions for our worked example here as in Chapter 6:

  • Zero-coupon bond prices for the U.S. Treasury curve on March ...

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