Instantaneous forward rates

For derived forward rates, the bond price functirm P t, T) is continuously differentiable with respect to t. Therefore, the model produces the following for the instantaneous forward rates ... 

A drawback of the model is that it requires the input of instantaneous forward rates, which cannot necessarily be observed directly in the market. Models have been developed that are in the HIM approach that take this factor into account, including those presented by Brace et al. (1997) and Jamshidian (1997). This family of models is known as the LIBOR market model or the BGM model, fri the BGM model, there is initially one factor, the forward rate fit) which is the rate applicable from time to time t +i at time t. The forward rate is described by Equation (4.33) ... 

Let f t,s) be the forward rate at time s > 0 calculated at time t < s. The instantaneous forward rate at time t to borrow at time T can be calculated from the bond prices using... 

If the period between T and the maturity of the longer-term bond is progressively reduced, the result is an instantaneous forward rate, which is given by formula (3.29). 

Figure 14.6. Simplified substitution rate matrix used in ML and distance phylogenetic analysis. The off-diagonal values a represent a product of an instantaneous rate of change, a relative rate between the different substitutions, and the frequency of the target base. In practice, the forward rates (upper triangular values) are presumed to equal the reverse rates (corresponding lower triangular values). The diagonal elements are nonzero, which effectively accounts for the possibility that more divergent sequences are more likely to share the same base by chance. In the simplest model of sequence evolution (the Jukes-Cantor model), all values of a are the same all substitution types and base frequencies are presumed equal.

We can define forward rates in terms of the short rate. Again for infinitesimal change in time from a forward date TitoT (for example, two bonds whose maturity dates are very close together), we can define a forward rate for instantaneous borrowing, given by... 

In a continuous time environment we do not assume discrete time intervals over which interest rates are applicable, rather a period of time in which a borrowing of funds would be repaid instantaneously. So we define the forward rate f(t, s) as the interest rate applicable for borrowing funds where the deal is struck at time f the actual loan is made at s (with s>t) and repayable almost instantly. In mathematics the period s — f is described as infinitesimally small. The spot interest rate is defined as the continuously compounded yield or interest rate r(f, T). In an environment of no arbitrage, the return generated by investing at the forward rate f(t, s) over the period s — t must be equal to that generated by investing initially at the spot rate r(f, T). So we may set... 

The derived forward rate is a decreasing function of the instantaneous standard deviation a, one of the model parameters. The partial derivative of the forward rate with respect to the standard deviation is given in Equation (3.30) ... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

In the previous chapter, and indeed in previous analysis, we have defined the forward rate as the interest rate applicable to a loan made at a future point in time and repayable instantaneously. We assume that the dynamics of the forward rate follow a Wiener process. The spot rate is the rate for borrowing undertaken now and maturing at T, and we know from previous analysis that it is the geometric average of the forward rates from 0 to T that is... 

Let us look now at the T-period forward rate again as a function of the range of spot rates from the time f today to point T in more detail than in Section 7.1. If P t, T) is the price today of a zero-coupon bond that has a redemption value of 1 at time T, then this price is given in terms of the instantaneous stmcture of forward rates by Equation (7.9) ... 

However, the price of the zero-coupon bond is also given in terms of the spot rate as the expression in (7.1), where E, is the expectation under the risk-free probability function. Therefore, forward rates are related to the expected level of the instantaneous spot rates, and if we differentiate the expression in (7.10), we obtain a result that states that the forward rate is a weighted average of the range of spot rates in the period t to T. This is given in Equation (7.11), which we encountered earUer as Equation (7.2) ... 

In stepping forward from t to a new point in time t, the instantaneous rate will change as the fluid s chemistry evolves. Rather than carrying the rate at t over the step, it is more accurate (e.g., Richtmyer, 1957 Peaceman, 1977) to take the average of the rates at t and t. In this case, the new bulk composition (at t) is given from its previous value (at t ) and Equations 16.7-16.9 by,... 

One way to ensure that back reactions are not important is to measure initial rates. The initial rate is the limit of the reaction rate as time reaches zero. With an initial rate method, one plots the concentration of a reactant or product over a short reaction time period during which the concentrations of the reactants change so little that the instantaneous rate is hardly affected. Thus,by measuring initial rates, one can assume that only the forward reaction in Eq. (35) predominates. This would simplify the rate law to that given in Eq. (36) which as written would be a second-order reaction, first-order in reactant A and first-order in reactant B. Equation (35), under these conditions, would represent a second-order irreversible elementary reaction. 

If the reaction between the absorbed species, and the nonvolatile reactant is reversible, the term instantaneous reaction is.synonymous with equilibrium reaction. Both forward and backward reactions in this case are so fast that, at all times, concentrations of the various reacting species in the liquid are in equilibrium. The absorption rate in this situation would be independent of the reaction and solely determined by the diffusion of various reacting species. 

Kobayashi (22) performed computer simulations via Eq. (4) as applied to his differential fixed-bed reactor. The model gas-phase reaction X Y is considered to pass in series through elementary steps to adsorbed X, an adsorbed intermediate in, adsorbed Y, to give finally Y. The forward and backward rate parameters were adjusted to simulate various mechanisms with their rate-determining steps. The shapes of the response curves for Y for typical mechanisms arc classified as instantaneous, monotonic, overshoot, S shaped, false start, and complex. This paper is a good so urce of ideas for the interpretation of transient responses. These ideas are illustrated by application to the oxidation of ethylene over a silver catalyst (23). The response curves last more than 100 min because the temperature is only 91°C and the bed contains 261 g of catalyst the flow rate is 160 ml/min. 

To analyse bond breakage under steady loading, we take advantage of the enormous gap in time scale between the ultrafast Brownian diffusion (r 10 — 10 s) and the time frame of laboratory experiments ( 10 s to min). This means that the slowly increasing force in laboratory experiments is essentially stationary on the scale of the ultrafast kinetics. Thus, dissociation rate merely becomes a function of the instantaneous force and the distribution of rupture times can be described in the limit of large statistics by a first-order (Markov) process with time-dependent rate constants. As force rises above the thermal force scale, i.e. rj-t> k T/x, the forward transition... 

Figure 5. Time-dependence of the transformation in 25-A CdSe crystals at 463 K at different pressmes. (A) The forward transition at A 5.2 GPa, 5.7 GPa, 0 6.9 GPa. (B) The reverse transition at 0 0.7 GPa, 1.0 GPa, A1.2 GPa. (1 GPa s 10,000 atm) The abscissa is the intensity of the four-coordinate electronic absorption feature. Fits are single-exponential decays. Rate constants were obtained from the slope of the fitted lines, and are equivalent to relaxation times (= In 2/k) in the forward transition of A 21 min, 3.6 min, 0 20 s. In the reverse transition 0 24 min, 0 3.8 min, and A 16 s. Each crystal in the ensemble transforms instantaneously relative to the experiment time such that the relaxation is a measure of the average time required to overcome the kinetic barrier. Once a nanocrystal transforms to the stable stmcture, it is statistically unlikely that it will fluctuate back unless the pressure is changed accordingly, because the transition is measured far from equilibrium. [Used by permission of the editor of Science, from Jacobs et al (2001), Fig. 2.]...

Assume that the first reaction (the protonation of the surface hydroxyl group) in the above mechanism occurs instantaneously so that this reaction is always at equilibrium. The rates of the forward and reverse reactions are equal at equilibrium, therefore... 

The actual rate of the forward reaction is determined by the sum of the instantaneous rates from reactant levels, /, to the final product levels, p that is,... 

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