The GBM process is widely used for stock prices and currencies. Fixed-income products are another matter.

Bond prices display long-term reversion to the face value (unless there is default). Such process is inconsistent with the GBM process, which displays no such mean reversion. The volatility of bond prices also changes in a predictable fashion, as duration shrinks to zero. Similarly, commodities often display mean reversion.

These features can be taken into account by modelling bond yields directly in a first step. In the next step, bond prices are constructed from the value of yields and a pricing function. The dynamics of interest rates rf can be modeled by

where Azf is the usual Wiener process. Here, we assume that 0 < k < 1,0 > 0, a >0. If there is only one stochastic variable in the fixed income market Az, the model is called a one-factor model.

This Markov process has a number of interesting features. First, it displays mean reversion to a long-run value of 0. The parameter k governs the speed of mean reversion. When the current interest rate is high, i.e. rf > 0, the model creates a negative drift k(0 - rf) toward 0. Conversely, low current rates create with a positive drift.

The second feature is the volatility process. This class of model includes the Va-sicek model when y = 0. Changes in yields are normally distributed because 5r is a linear function of Az. This model is particularly convenient because it leads to closed-form solutions for many fixed-income products. The problem, however, is that it could allow negative interest rates because the volatility of the change in rates does not depend on the level.

Equation (4.8) is more general because it includes a power of the yield in the variance function. With y = 1, the model is the lognormal model.1 This implies that the

1This model is used by RiskMetrics for interest rates.

rate of change in the yield has a fixed variance. Thus, as with the GBM model, smaller yields lead to smaller movements, which makes it unlikely the yield will drop below zero. With y = 0.5, this is the Cox, Ingersoll, and Ross (CIR) model. Ultimately, the choice of the exponent y is an empirical issue. Recent research has shown that y = 0.5 provides a good fit to the data.

This class of models is known as equilibrium models. They start with some assumptions about economic variables and imply a process for the short-term interest rate r. These models generate a predicted term structure, whose shape depends on the model parameters and the initial short rate. The problem with these models is that they are not flexible enough to provide a good fit to today's term structure. This can be viewed as unsatisfactory, especially by most practitioners who argue that they cannot rely on a model that cannot even be trusted to price today's bonds.

In contrast, no-arbitrage models are designed to be consistent with today's term structure. In this class of models, the term structure is an input into the parameter estimation. The earliest model of this type was the Ho and Lee model where Q(t) is a function of time chosen so that the model fits the initial term structure. This was extended to incorporate mean reversion in the Hull and White model

Finally, the Heath, Jarrow, and Morton model goes one step further and allows the volatility to be a function of time.

The downside of these no-arbitrage models, however, is that they impose no consistency between parameters estimated over different dates. They are also more sensitive to outliers, or data errors in bond prices used to fit the term structure.

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