Interest-rate processes

The price processes of shares and bonds, as well as interest rate processes, are stochastic processes. That is, they exhibit a random change over time. For the purposes of modelling, the change in asset prices is divided into two components. These are the drift of the process, which is a deterministic element, also called the mean, and the random component known as the noise, also called the volatility of the process. 

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 Merton s (1971) model, the interest-rate process is assumed to be a generalised Weiner process, described by Equation (3.31) ... 

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. 

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

In this section we consider the pricing of a European option on the money fund. (This is the same as a bank account when the initial value B(0) = 1.) Thus, the payoff of a European call option with exercise price K is max[B(T) - K,0], The continuous version of the Ho-Lee model is assumed for the short interest rate process. The risk-neutral valuation methodology provides the solution as ... 

For entangled iV-state transfer processes, the hierarchy Green s functions formalism involves tensors each having elements. As far as the kinetics regime is concerned, however, many interesting rate processes proceed practically in a step-wise manner. In this case the kinetics rate matrix could be determined by the individual rates between two states. Remarkably, the hierarchy Green s functions for the modified ZE/HEOM is analytically solvable for two-state systems. The resulting analytical expression of rate resolution, K/,j[s), for individual elementary a) b rate process can therefore be used to construct the NxN kinetics rate matrix. 

A rise in volatility generates a range of possible future paths around the expected path. The actual expected path that corresponds to a zero-coupon bond price incorporating zero OAS is a function of the dispersion of the rai e of alternative paths around it. This dispersion is the result of the dynamics of the interest-rate process, so this process must be specified for the current term structure. We can illustrate this with a simple binomial model example. Consider again the spot rate structure in Table 12.1. Assume that there are only two possible future interest rate scenarios, outcome 1 and outcome 2, both of equal probability. The dynamics of the short-term interest rate are described by a constant drift rate a, together with a volatility rate a. These two parameters describe the evolution of the short-term interest rate. If outcome 1 occurs, the one-period interest rate one period from now will be... 

The cost of capital may also be considered as the interest rate at which money can be invested instead of putting it at risk in a manufacturing process. Let us consider the process data listed in Table 9-4 and plotted in Fig. 9-10. If the cost oi capital is 10 percent, then the appropriate discounted-cash-flow curve in Fig. 9-10 is abcdef. Up to point e, or 8.49 years, the capital is at risk. Point e is the discounted breakeven point (DEEP). At this point, the manufacturing process... 

Doping of solid reactant involves the introduction of a controlled amount of an impurity into solid solution in the host lattice. Such impurities can be selected to cause the generation or destruction of those electronic or structural defects which participate in the rate process of interest. Thus, the influence of the additive on kinetic behaviour can provide evidence concerning the mechanism of reaction [46,47]. Even if the... 

Kinetic observations for decomposition of some representative transition metal sulphides are summarized in Table 13. Several instances of an advancing interface [contracting volume, eqn. (7), n = 3] rate process have been identified and the rate may be diminished by the presence of sulphur. Diffusion control is, however, believed to be important in the reactions of two lower sulphides (Ni0.9sS. [687] and Cu1-8S [688]). These solids have attracted particular interest since both are commercially valuable ores and pyrolysis constitutes one possible initial step in metal extraction. 

Some writers claim that an advantage of the Rate of Return method is that no interest rate needs to be chosen. It has already been illustrated that this is not true. The difference in the two measures is when, not whether, a reasonable interest rate is chosen. With the Rate of Return method it does not need to be chosen until after the rate of return is obtained. If it is too high and two processes are both found acceptable, then the net present value must be calculated to determine which is best. In doing this the interest rates for proceeds and outlays may be different. 

Most chemical companies aim at making about a 10% profit on all sales. This means that the pretax profits must be around 20%. Since new products involve a large amount of risk and cost uncertainty, a new process is generally not considered unless at least 30% average profit is expected before paying federal corporate income taxes. From this a reasonable rate of return or interest rate may be obtained if some other information on the plants is available. This is done in the following example. 

Ordinary anodes for an electrochemical process last 2 years and then have to be replaced at a cost of 20,000. An alternative choice is to buy impregnated anodes that last 6 years and cost 56,000 (see Figure E3.5). If the annual interest rate is 6 percent per year, which alternative would be the cheapest ... 

Two advantages of this approach would be the minimized capital and operating costs, along with minimal space requirements. The best process (for iron removal) would probably be chemical precipitation with lime through a rapid mix, flocculation, and DAF step. DAF size requirements would be in the range of a 300- to 350-ft2 unit. Total costs for operation and maintenance of the preliminary treatment facility using an interest rate of 10% over 10 years are estimated at 0.44/1000 gal. A summary of the capital and operating cost estimates is presented in Table 8.2. 

One of the exciting new directions is the control of activated rate processes using external fields. Addition of an external field opens the way for a wide variety of new phenomena such as stochastic resonance, resonance activation, directed transport, control of the hopping distribution in surface diffusion and more. Even the addition of a constant force to the problem leads to interesting additional phenomena such as the locked to running transition, which remains a topic of ongoing research. " Quantum mechanics in the presence of external fields may differ significantly from the classical. 

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