HJM model

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The term HJM-Modell refers to the interest structure model by Heath, Jarrow and Morton , an arbitrage-free interest structure model , which basically models the entire term structure of the current interest rate and derives the entire interest structure from it. It was presented in the discrete-time form in 1990. In 1992, after about two years, the continuous version of the model appeared. The majority of the interest structure models can be interpreted as special cases of the HJM model.

As a one-factor model, it only takes into account the volatility of forward rate changes as a risk factor. However, the model can be expanded to include any number of risk factors. By adding a further risk factor, not only a shift in the yield curve but also its rotation can be described. For reasons of complexity, a discrete one-factor HJM model is often used to evaluate European swaptions . This results in a binomial evaluation model, since the upward and downward movements only depend on one risk factor.

Comparison with the Black 76 model

The great advantage of the classic Black 76 model is its simplicity in calculation and implementation. The biggest criticism of this model is the assumption that changes in interest rates are lognormally distributed . This assumption is very unrealistic due to the mean reversion properties of the interest rates. In addition, the sum of lognormally distributed random variables is not necessarily lognormally distributed, which is assumed in the Black 76 model. Furthermore, the volatilities of the forward interest rates are assumed to be constant.

As a result, there is a high probability that the application of this model will lead to significant mispricing. The HJM model does not have these weaknesses. It does not start from the log normal distribution and takes the mean reversion property into account. This model also takes into account that the individual interest rates are interdependent. The HJM model takes account of these dynamics in interest rates, so that a change in one interest rate also changes the overall interest structure.

However, the HJM model also has some disadvantages. On the one hand, there must be an initial interest structure of the forward interest rates f (0, t, t + 1). In practice, however, these cannot be observed directly. Hence, there is a problem of data availability. On the other hand, the model is difficult to implement because it has very complex structures. The modeling of the interest rate processes creates non-closed binomial trees that lead to a large number of nodes: In the HJM model, a total of 2 * t nodes must be recorded at time t . For example, the evaluation of a swaption with a total term of 21 years when modeling the term rate process f (20,20,21) leads to 2 ^ 21 = 2,097,152 nodes, which already generates considerable computational effort.

literature

  • John Hull: Options, Futures, and Other Derivatives . 6th edition. Pearson Studies, Munich 2006, ISBN 978-3-8273-7142-3 .
  • Markus Rudolf: Interest structure models . Physica-Verlag, Heidelberg 2000.
  • Markus Rudolf: Heath, Jarrow, Morton made easy: For the preference-free evaluation of swaptions . In: Financial Market and Portfolio Management . tape 12 , no. 2 , 1998, p. 170-1196 .
  • D. Heath, R. Jarrow, A. Morton: Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation . 1992 (English).
  • Klaus Sandmann: Introduction to the stochastics of the financial markets . Springer, 1999.