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Extending the universality of the Heath–Jarrow–Morton model

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  • Dwight Grant
  • Gautam Vora

Abstract

Heath, Jarrow, and Morton (HJM) developed an important model of the evolution of interest rates. A key assumption of the model is that interest rate changes are normally distributed in continuous time. Implementing the HJM‐method of evolution of interest rates in discrete time for more complex volatility functions remains a significant challenge. In this article, we present a relatively simple and flexible method of implementation, that extends the usefulness of the HJM model. The derivation assumes that the distribution of interest rates is stable, but not necessarily identical, for each discrete time period. This allows us to identify the drift‐adjustment terms necessary to build interest rate lattices and trees and Monte Carlo simulations that satisfy exactly the no‐arbitrage and volatility conditions, even complex ones, of the model. The much more difficult discrete‐time implementation methods suggested in the literature (Heath, Jarrow, and Morton (1991) [Heath, D., Jarrow, R. & Morton, A. (1991). Contingent claim valuation with a random evolution of interest rates. Review of Futures Markets, 54–76.] and Jarrow (1996) [Jarrow, R. (1996). Modeling fixed income securities and interest rate options. New York, NY: McGraw‐Hill Companies Inc.]) do not accomplish that. We illustrate our analytical implementation with three examples of volatility functions and demonstrate its superiority to other methods of implementation.

Suggested Citation

  • Dwight Grant & Gautam Vora, 2006. "Extending the universality of the Heath–Jarrow–Morton model," Review of Financial Economics, John Wiley & Sons, vol. 15(2), pages 129-157.
    • Handle: RePEc:wly:revfec:v:15:y:2006:i:2:p:129-157
    DOI: 10.1016/j.rfe.2005.04.003
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    References listed on IDEAS

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    1. Heath, David & Jarrow, Robert & Morton, Andrew, 1990. "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(4), pages 419-440, December.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    4. Dwight Grant & Gautam Vora, 2002. "The Hull and White Model of the Short Rate: An Alternative Analytical Representation," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 25(4), pages 463-476, December.
    5. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    6. Ho, Thomas S Y & Lee, Sang-bin, 1986. "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, American Finance Association, vol. 41(5), pages 1011-1029, December.
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