Chapter 13

Special Cases of Heath, Jarrow, and Morton Interest Rate Modeling

In this chapter, we continue our review of the history of interest rate analytics, but this time we focus on term structure models. We start with the example of parallel yield curves shifts and duration as a naïve term structure model to illustrate the approach academics have typically taken to interest rate modeling:

Step 1: Make an assumption about how interest rates vary
Step 2: Impose no-arbitrage restrictions
Step 3: Derive what the shape of the yield curve must be

The approach taken by Heath, Jarrow, and Morton (HJM) was the exact opposite:

Step 1: Observe what the shape of the yield curve actually is
Step 2: Add assumptions about the volatility of rates
Step 3: Impose no-arbitrage conditions on yield curve movement from its current shape
Step 4: Derive the nature of yield curve movements from the existing yield curve shape

The power of the HJM approach was illustrated in Chapters 6 through 9, with four different sets of assumptions about interest rate volatility. In this chapter, we demonstrate that the early academic work on term structure models, in every case, is a special case of HJM. By special case, we mean a specific assumption in Step 2 of the general HJM process above.

In Chapter 12, we were introduced to Frederick Macaulay’s duration concept. At the heart of the duration concept is the implicit assumption that rates at all maturities move at the same time in the same direction by the ...

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