Interest rates dynamics, model

Brace A, Gatarek D, Musiela M (1997) The Market Model of Interest Rate Dynamics. Mathematical Finance 7 127-154. 

Brace, A., Gatarek, D., Musiela, M., 1997. The market model of interest-rate dynamics. Math. Financ. 7 (2), 127-155. 

Term-stmcture modeling is based on theories concerning the behavior of interest rates. Such models seek to identify elements or factors that may explain the dynamics of interest rates. These factots are random, or stochastic. That means their future levels cannot be predicted with certainty. Interest rate models therefore use statistical processes to describe the factors stochastic properties and so arrive at reasonably accurate representations of interest rate behavior. 

The approach described in Heath-Jarrow-Morton (1992) represents a radical departure from earlier interest rate models. The previous models take the short rate as the single or (in two- and multifactor models) key state variable in describing interest rate dynamics. The specification of the state variables is the fundamental issue in applying multifactor models. In the HJM model, the entire term structure and not just the short rate is taken to be the state variable. Chapter 3 explained that the term structure can be defined in terms of default-free zero-coupon bond prices or yields, spot rates, or forward rates. The HJM approach uses forward rates. 

The dynamic model was validated both in steady and unsteady-state conditions, which is quite interesting in case that a control based on feed-forward strategy is applied. The prediction of the final concentration of Orange II from the initial data would allow the system control to modify the flow rate of MnP, Orange II or H2O2, in order to adapt the conditions to the desired final value. 

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... 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

An interest-rate model provides a description of the dynamic process by which rates change over time, in terms of a statistical construct, as well as a means by which interest-rate derivatives such as options can be priced. It is often the practical implementation of the model that dictates which type is used, rather than mathematical neatness or more realistic assumptions. An excellent categorisation is given in James and Webber (2000), who list models as being one of the following types ... 

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 ... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

In response to these issues, interest-rate models have been developed that model the entire yield curve. In a whole yield curve, the dynamics of the entire term structure are modelled. The Ho-Lee model is a simple type of whole curve... 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

A no-arbitrage model that is implemented in a realistic approach matches precisely the term structure of interest rates that are implied by the current (or initial) observed market yields. It then derives a forward curve for the fumre that is dependent on the way it has modelled the dynamics of the interest-rate process. 

In selecting the model, a practitioner will select the market variables that are incorporated in the model these can be directly observed such as zero-coupon rates or forward rates, or swap rates, or they can be indeterminate such as the mean of the short rate. The practitioner will then decide the dynamics of these market or state variables, so, for example, the short rate may be assumed to be mean reverting. Finally, the model must be calibrated to market prices so, the model parameter values input must be those that produce market prices as accurately as possible. There are a number of ways that parameters can be estimated the most common techniques of calibrating time series data such as interest rate data are general method of moments and the maximum likelihood method. For information on these estimation methods, refer to the bibliography. 

Model consistency As we have noted elsewhere, using models requires their constant calibration and re-calibration over time. For instance, an arbitrage model makes a number of assumptions about the interest rate drift rate and volatility, and in some cases, the mean reversion of the dynamics of the rate process. Of course, these values will fluctuate constantly over time so that the estimate of these model parameters used one day will not remain the same over time. So, the model will be inconsistent over time and must be re-calibrated to the market Equilibrium models use parameters that are estimated from historical data, and so there is no unused daily change. Model parameters remain stable. Over time therefore these models remain consistent, at least with themselves. However, given the points we have noted above, market participants usually prefer to use arbitrage models and re-calibrate them frequently. 

From oiu understanding of derivatives, we know that option pricing models such as Black-Scholes assume that asset price returns follow a lognormal distribution. The dynamics of interest rates and the term structure is the subject of... 

Another approach, which was used by Cox, Ingersoll, and Ross to model the term structure in a general equilibrium environment, consists of a model of the dynamics of interest rates. This process provides a... 

The first term-structure models described in the academic literature explain interest rate behavior in terms of the dynamics of the short rate. This term refers to the interest rate for a period that is infinitesimally small. (Note that spot rate and zero-coupon rate are terms used often to... 

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. 

Market practitioners armed with a term-structure model next need to determine how this relates to the fluctuation of security prices that are related to interest rates. Most commonly, this means determining how the price T of a zero-coupon bond moves as the short rate r varies over time. The formula used for this determination is known as Itos lemma. It transforms the equation describing the dynamics of the bond price P into the stochastic process (4.5). 

Although published officially in 1985, the Cox-Ingersoll-Ross model was described in academic circles in 1977, or perhaps even earlier, which would make it the first interest rate model. Like Vasiceks it is a one-factor model that defines interest rate movements in terms of the dynamics of the short rate. It differs, however, in incorporating an additional feature, which relates the variation of the short rate to the level of interest rates. This feature precludes negative interest rates. It also reflects the fact that interest rate volatility rises when rates are high and correspondingly decreases when rates are low. The Cox-lngersoll-Ross model is expressed by equation (4.11). 

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