Short-Rate Processes

The original interest rate models describe the dynamics of the short rate later ones—known as HJM, after Heath, Jarrow, and Morton, who created the first whole yield-curve model—focus on the forward rate. 

The short rate follows a stochastic process, or probability distribution. So, although the rate itself can assume a range of possible future values, the process by which it changes from value to value can be modeled. A one-factor model of interest rates specifies the stochastic process that describes the movement of the short rate. 

Changes or jumps in yield that follow a Weiner process are scaled by the volatility of the stochastic process that drives interest rates, which is denoted by 7. The stochastic process for change in yields is expressed by (4.2). 

The value of the volatility parameter is user-specified—that is, it is set at a value that the user feels most accurately describes the current interest rate environment. The value used is often the volatility implied by the market price of interest rate derivatives such as caps and floors. 

The zero-coupon bond yield has thus far been described as a stochastic process following a geometric Brownian motion that drifts with no discernible trend. This description is incomplete. It implies that the yield will either rise or fall continuously to infinity, which is clearly not true in practice. To be more realistic, the model needs to include a term capturing the fact that interest rates move up and down in a cycle. The short rate s expected direction of change is the second parameter in an interest rate model. This is denoted in some texts by a letter such as a or i , in others by /i. The short-rate process can therefore be described as function (4.3). 

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Santa-Clara and Sornette [67] argue that there are no empirical findings that would lead to a preference of a T-differential or non-differential type of RF. We show that the integrated RF dU t, T) enforces a well-defined short rate process, whereas the non-differential field dW t, T) fails. In the following, we restrict our analysis to these two t5 es of RF models, but keeping in mind that only the T-differential RF ensures a well defined short rate process. Their correlation functions fit with the requirements for a correct modeling of the forward rate curve, while the models remain tractable. 

Note that the extension of the HJM-framework to RF models implies that the short rate dynamics depends on the T-derivative of the RF dWp t,T). First of all, this means that admissible short rate dynamics can be derived only for T-differential Random Fields. In reverse this implies that the nondifferential RF dZ t,T) does not lead to a well-defined short rate process. Secondly, the mean reversion parameter itself evolves stochastically. 

To recap on the issues involved in fitting the extended Vasicek model or Hull-White model this describes the short-rate process as following the form... 

From a practical point of view, we can safely assume that the majority of stochastic processes representing prices of traded financial assets are adapted to the filtration F and that the short rate process r = r(r) >o is a predictable process, meaning that r(t) is Ff i measurable. This implies that B t) is also Ff i measurable and this condition is automatically satisfied for continuous or left-continuous processes. 

Many models proposed for the short rate process r = [r(t)]f>Q are particular cases of the general diffusion equation... 

Formulas for bond options were found by Cox, Ingersoll, and Ross using the CIR model (square root process) for short rates, and by Jam-shidian, Rabinovitch, and by Chaplin using the Vasicek model for the short rate process. 

When the short rate process r = r(t) follows the continuous time version of the Ho-Lee model given by equation (18.10), the price at time... 

A more general case is discussed by Shiryaev" for single-factor Gaussian models modelling the short interest rate. These are single-factor affine models where the short rate r is also a Gauss-Markov process. The equation for this short rate process is... 

Equation (4.3) describes a stochastic short-rate process modified to include the direction of change. To be more realistic, it should also include a term describing the tendency of interest rates to drift back to their long-run average level. This process is known as mean reversion and is perhaps best captured in the Hull-White model. Adding a general specification of mean reversion to (4.3) results in (4.4). 

The Vasicek model was the first term-structure model described in the academic literature, in Vasicek (1977). It is a yield-based, one-factor equilibrium model that assumes the short-rate process follows a normal distribution and incorporates mean reversion. The model is popular with many practitioners as well as academics because it is analytically tractable—that is, it is easily implemented to compute yield curves. Although it has a constant volatility element, the mean reversion feature removes the certainty of a negative interest rate over the long term. Nevertheless, some practitioners do not favor the model because it is not necessarily arbitrage-free with respect to the prices of actual bonds in the market. 

Given this description of the short-rate process, the price at time r of a zero-coupon bond with maturity Tmay be expressed as (4.10). 

Equilibrium interest rate models also exist. These make the same assumptions about the dynamics of the short rate as arbitrage models do, but they are not designed to match the current term structure. The prices of zero-coupon bonds derived using such models, therefore, do not match prices seen in the market. This means that the prices of bonds and interest rate derivatives are not given purely by the short-rate process. In brief, arbitrage models take as a given the current yield curve described by the... 

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