Short rate models

In chapter (6), we extend the traditional HJM approach, by assuming that the sources of uncertainty are driven by Random Fields. For that reason, we introduce a non-differentiable Random Field (RF) and an equivalent T-differentiable counterpart. Given the particular Random Field, we derive the corresponding short rate model and show in contrast to Santa-Clara and Sor-nette [67] and Goldstein [33] that only a T-differentiable RF leads to admissible well-defined short rate dynamics". Santa-Clara and Sornette [67] argue that there is no empirical evidence for a T-differentiable RF. We conclude that the existence of some pre-defined short rate dynamics enforces the usage of a r-differentiable RF. Furthermore, we compute bond option prices when... 

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

First, following Goldstein [33] we start from anon-differential RF dW(t,T) and show that this type of Random Fields leads to an inconsistency with the properties of an appropriate defined short rate model. Then, we overcome this drawback hy defining a T-differential RF dU t,T), which can be derived by an integration of the Field dW(t,T) over the term T. 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

So, now we have determined that a short-rate model is related to the dynamics of bond yields and therefore may be used to derive a complete term structure. We also said that in the same way the model can be used to value bonds of any maturity. The original models were one-factor models, which describe the process for the short-rate r in terms of one source of uncertainty. This is used to capture the short-rate in the following form ... 

In this chapter we consider multi-factor and whole yield curve models. As we noted in the previous chapter, short-rate models have certain drawbacks, which, though not necessarily limiting their usefulness, do leave room for further development. The drawback is that as the single short-rate is used to derive the complete term structure, in practice, this can be unsuitable for the calculation of bond yields. When this happens, it becomes difficult to visualise the actual dynamics of the yield curve, and the model no longer fits observed changes in the curve. This means that the accuracy of the model cannot be observed. Another drawback is that in certain equilibrium model cases, the model cannot be fitted precisely to the observed yield curve, as they have constant parameters. In these cases, calibration of the model is on a goodness of fit or best fit approach. 

Short-rate models are composed of two components. The first attempts to capture the average rate of change, also called the drift, of the short-term rate at each instant, while the second component measures this drift as a function of the volatility of the short-term rate. This is given by ... 

This model is similar to Dothan model being a log-normal short rate model. This model, however, does not lead to explicit formulae for pure discount bonds or for options contingent on them. In addition this is an... 

No explicit mathematical model of the method was presented. However, a short descriptive model was outlined For each run, the average ignition and combustion rate (expressed as weight of fuel ignited or burnt per unit bed area and unit time) were calculated by determining the time taken for the ignition front to pass down through the bed and the completion of burn-out, respectively. No discussion is presented about limitations and assumptions of the method. 

Stubington et al [6] presented a short descriptive model of how the ignition front rate and the combustion rate are determined. No mathematical models to calculate the ignition rate and the combustion rate are shown. However. As far as this author can understand, the calculation results are time average values, that is, no time instant values are obtained by the method used by Stubington. No uncertainty analysis was presented and no verification method was used. The methods used are unclearly defined. Consequently, the results would be difficult to reproduce. Nevertheless, the study includes interesting result. 

A short descriptive model showed how the combustion heat rate was calculated. No discussion is presented about limitations and assumptions of the method. 

Aho [8] presented a short descriptive model of the relationship between combustion heat rate and mass loss rate of bed. He applies the same method as Koistinen et al [7]. It is clear that the method used presents time average values of combustion heat rate. No verification method is applied and no uncertainty analysis is carried out. The... 

The choice of experimental reactor is important to the success of the kinetic modeling effort. The short bench-scale reaction tubes sometimes used for studies of this sort give little or no insight into best mathematical form of the kinetic model, conduct the reaction over varying temperatures and partial pressures along the tube, and inevitably operate at velocities that are a small fraction of those to be encountered in the plant-scale reactor. Rate models from laboratory reactors of this sort rarely scale-up well. The laboratory differential reactor suffers from velocity problems but does at least conduct the reaction at known and relatively constant temperature and partial pressures. However, one usually runs into accuracy problems because calculated reaction rates are based upon the small observed differences in concentration between the reactor inlet and outlet. 

The rate model represented by equation 6.9 is particularly interesting since it will be shown shortly that a similar function arises from a consideration of formal diffusion theory. Therefore, providing it is established by experiment that the pseudo rate constant is truly constant over the range of experimental boundary conditions employed a, b, Xg) it remains perfectly valid to equate its value to appropriate mass transfer parameters required by diffusion theory. 

We also postulated this simplified volatilty function in chapter (5), where we derived the mean-reverting short rate dynamics for a iV-factor term structure model. Nevertheless, the ODE can also be solved postulating a more general volatiUy function. 

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. 

None the less, following Kennedy [50], [51], Goldstein [33], Longstaff, Santa-Clara and Schwartz [57], Collin-Dufresne and Goldstein [20] and Santa-Clara and Somette [67] we compute bond option prices assuming the non-differential RF dZ t,T), as well as the T-differential counterpart dU t,T), keeping in mind that only the T-differential RF model leads to a well-defined short rate dynamics. 

In other words, assuming a complete market stochastic volatility model implies that the short rate is modeled directly, while the traded asset (bond) has to be derived. Therefore, only the direct modeling of the bond price dynamics, together with stochastic volatility leads to an incomplete market model analog to the stochastic volatility models of equity markets". ... 

Starting from the dynamics of the short rates, extensive work has been done in implementing jumps in interest rates models (see e.g. Ahn and Thompson (1988) andChako and Das [15]). However only a few authors implemented jumps in a HJM-framework (see. e.g. Shirakawa [70] and Glasser-man and Kou [32]). Further work could be done in implementing jumps in the aforementioned framework combined with USV and correlated sources of uncertainty. Another area of research could result from combining a HJM-Uke multiple RF-framework with the class of USV models given by... 

Of course, interest rates are not constant but Equation (3.1) is valuable as it is used later in constructing a model. By using Equation (3.1), we are able to produce a yield curve, given a set of zero-coupon bond prices. For modelling purposes, we require a definition of the short rate, or the current interest rate for borrowing a sum of money that is paid back a very short period later (in fact, almost instantaneously). This is the rate payable at time t for repayment at time t+M where Af is an incremental passage of time. This is given by... 

Term structure models are essentially models of the interest-rate process. The problem being posed is, what behaviour is exhibited by interest rates, and by the short-term interest rate in particular An excellent description of the three most common processes that are used to describe the dynamics of the short-rate is given in Phoa (1998), who describes ... 

Therefore, once we have a full description of the random behaviour of the short-rate r, we can calculate the price and yield of any zero-coupon bond at any time, by calculating this expected value. The implication is clear specifying the process r t) determines the behaviour of the entire term structure so, if we wish to build a term structure model, we need only (under these assumptions) specify the process for r t). 

In the Vasicek (1977) model, the instantaneous short-rate r is assumed to follow a stochastic process known as the Omstein-Uhlenbeck process, a form of Gaussian process, described by Equation (3.24) ... 

This model incorporates mean reversion, which is not an imrealistic feature. Mean reversion is the process that describes that when the short-rate r is high, it will tend to be pulled back towards the long-term average level when the rate is low, it will have an upward drift towards the average level. In Vasicek s model, the short-rate is pulled to a mean level 6 at a rate of a. The mean reversion is governed by the stochastic term odW which is normally distributed. Using Equation (3.24), Vasicek shows that the price at time t of a zero-coupon bond of maturity T is given by ... 

In describing the dynamics of the yield curve, the Vasicek model only captures changes in the short-rate r, and not the long-run average rate b. 

From the previous section, we see that under a model that assumes the short-rate to follow a normal distribution, there can arise instances of negative forward rates. The Cox et al. (1985) is a one-factor model and as originally presented removed the possibility of negative rates. Under the CIR model, the dynamics of the short-rate are described by Equation (3.38) ... 

Some texts have suggested that equilibrium models can be converted into arbitrage-free models by making the short-rate drift rate time dependent. However, this may change the whole nature of the model, presenting problems in calibration. 

The Ho-Lee (1986) model was one of the first arbitrage-free models and was presented using a binomial lattice approach, with two parameters the standard deviation of the short-rate and the riskpremium of the short-rate. We summarise it here. Following Ho and Lee, let ( ) be the equilibrium price of a zero-coupon bond maturing at time T under state i. That is F( ) is a discount... 

The forward rate function/at time t is not static and is a function of the short-rate r at time f, the time t and the time to maturity T. The Hull-White model can be calibrated in terms of the forward rate /. That is, at time t the information (parameters) required to implement this are the short-rate r(t), the standard deviation a of the short-rate, the forward rate / and the standard deviations Bj t, T)a t) of the forward rates at time t. If the forward rates are known in a form that allows their first differential to be calculated with respect to t, using the other information, it is possible to calculate the function Bj, the derivative of this function and thereby the value for a f), using the relationship in Equation (3.56) ... 

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