Heath–Jarrow–Morton framework

The term HJM model called the term structure model of Heath, Jarrow and Morton, a forward rate and arbitrage- free term structure model. It was introduced in the time-discrete form in 1990. In 1992, after about two years, the continuous version of the model. The majority of the yield models can be interpreted as special cases of the HJM model.

As a one-factor model, it takes into account only the volatility of the forward rate changes as a risk factor. The model can, however, extend to any number of risk factors. With the addition of a further risk factor, not only a shift in the yield curve, but also their rotation are described. For complexity reasons, a discrete one-factor HJM model for the evaluation of European swaptions is used often. This results in a binomial valuation model, since the dependent upward and downward movements of only one risk factor.

Comparison of HJM model and the Black- 76 model

The great advantage of the classical Black- 76 model is its simplicity in the calculation and implementation. The biggest criticism of this model is the assumption that changes in interest rates are log-normally distributed. This assumption is very far from reality due to the mean reversion properties of interest rates. In addition, the sum of the log normal distributed random variables is not necessarily log-normally distributed, what is assumed in the Black- 76 model. Furthermore, the assumed volatility of forward rates as constant.

Consequently, the application of this model with a large probability to significant mispricing. In contrast, the HJM model does not have these weaknesses. It's not out of the lognormal distribution and takes into account the mean reversion property. In addition, it can be assumed that the individual interest rates are interdependent, ie the change in an interest rate tends to be associated with the shift in the entire yield. This important fact is observed in the HJM model by the dynamics of interest rates is taken into account.

However, the HJM model has some drawbacks. For one must be an initial term structure of forward rates f ( 0, t, t 1) be given. In practice, however, these are not directly observable. Therefore, there is the problem of the data availability. Secondly, the model is difficult to implement, since it has very complex structures. Thus, in modeling the interest rate processes occur unclosed binomial trees, which lead to a large number of nodes. Consequently, must at time t in a one-factor HJM model, a total of 2 * t nodes are detected. For example, f ( 20,20,21 ), the evaluation of a swaption with a total term of 21 years in the modeling of the forward rate process to 2 ^ 21 = 2,097,152 nodes, this can possibly cause computational problems.