Interest rate modeling equilibrium

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. 

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. 

Francis A. Longstaff and Eduardo Schwartz, Interest Rate Volatility and the Term Structure A Two-Factor General Equilibrium Model, Journal of Finance 47 (1992), pp. 1259-1282 and Fletcher A. Longstaff and Eduardo Schwartz, A Two-Factor Interest Rate Model and Contingent Claim Valuation, Journal of Fixed Income 3 (1992), pp. 16-23. 

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... 

Returning to our earlier discussion we can generalize by stating that when a reaction is reversible and its rate is fast compared to the residence time of the aqueous system of interest ty2) then equilibrium models may be used to describe the state of that reaction. When a reaction is irreversible, or its rate comparable to or slower than the system residence time i 2 t/ ) a kinetic model is needed to describe the state of such a reaction. 

The most fundamental experimental determinations in model studies of proton transfer at weakly basic carbon are of the rate and equilibrium constants for carbon deprotonation to form an unstable carbanion (Eq. (1.1)). These parameters define the kinetic and thermodynamic barriers to proton transfer (Eq. (1.2) for Fig. 1.1). They are of interest to enzymologists because they specify the difficulty of the problem that must be solved in the evolution of proteins which catalyze proton transfer with second-order rate constants kcat/ m of 10 "-10 s that are typically ob-... 

Longstaff FA, Schwartz ES (1992) Interest Rate Volatility an the Term Structure A Two-Factor General Equilibrium Model. Journal of Finance 47 1259-1282. 

Model inputs Arbitrage models use the term structure of spot rate as an input, and this data is straightforward to obtain. Equilibrium models require a measure of the investor s market risk premium, which is rather more problematic. Practitioners analyse historical data on interest rate movements, which is considered less desirable. 

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. 

Attari, M., 1996. Discontinuous Interest Rate Processes An Equilibrium Model for Bond Option Prices Working Paper. University of Iowa, Iowa City, pp. 1-32. 

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 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. 

Geochemical models can be conceptualized in terms of certain false equilibrium states (Barton et al., 1963 Helgeson, 1968). A system is in metastable equilibrium when one or more reactions proceed toward equilibrium at rates that are vanishingly small on the time scale of interest. Metastable equilibria commonly figure in geochemical models. In calculating the equilibrium state of a natural water from a reliable chemical analysis, for example, we may find that the water is supersaturated with respect to one or more minerals. The calculation predicts that the water exists in a metastable state because the reactions to precipitate these minerals have not progressed to equilibrium. 

In this chapter we consider the problem of the kinetics of the heterogeneous reactions by which minerals dissolve and precipitate. This topic has received a considerable amount of attention in geochemistry, primarily because of the slow rates at which many minerals react and the resulting tendency of waters, especially at low temperature, to be out of equilibrium with the minerals they contact. We first discuss how rate laws for heterogeneous reactions can be integrated into reaction models and then calculate some simple kinetic reaction paths. In Chapter 26, we explore a number of examples in which we apply heterogeneous kinetics to problems of geochemical interest. 

In the thermodynamic models (FIAM and BLM), the internalisation flux is assumed to be rate-limiting, and the concentration of carriers or sensitive sites bound by the solute of interest negligible with respect to the total number of carriers (i.e. free carrier concentration constant). The fundamental equations describing the equilibrium models can be summarised as ... 

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

It is interesting to note that in chaotic regime, the flow rate outlet stream, which is manipulated by the control valve CVl (see Figure 12), and the reactor volume, are driven by the PI controller to the equilibrium point without chaotic oscillations. However, the other variables have a chaotic behavior as shown in Figure 18. So it is possible to obtain a reactor behavior, in which some variables are in steady state and the others are in regime of chaotic oscillations, due to the decoupling or serial connection phenomena. In this case the control system and the volumetric flow limitation of coolant flow rate through the control valve VC2, are the responsible of this behavior. Similar results can be obtained from model. 

Complementing the equilibrium measurements will be a series of time resolved studies. Dynamics experiments will measure solvent relaxation rates around chromophores adsorbed to different solid-liquid interfaces. Interfacial solvation dynamics will be compared to their bulk solution limits, and efforts to correlate the polar order found at liquid surfaces with interfacial mobility will be made. Experiments will test existing theories about surface solvation at hydrophobic and hydrophilic boundaries as well as recent models of dielectric friction at interfaces. Of particular interest is whether or not strong dipole-dipole forces at surfaces induce solid-like structure in an adjacent solvent. If so, then these interactions will have profound effects on interpretations of interfacial surface chemistry and relaxation. 

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