Binomial model

They then derive their own binomial model relationships using Horwitz s data with variable apparent sample size. 

Chen, C. L. and Swallow, W. H. (1990). Using group testing to estimate a proportion, and to test the binomial model. Biometrics, 46, 1035-1046. 

If the classifiers in the ensemble are correlated, then we can use the beta-binomial model (Williams, 1975). This model allows only positive correlafion p in order to satisfy Var(p) > 0. Prentice (1988) showed that the beta-binomial model... 

Table 6.3 illustrates the prediction accuracy obtained by ensemble majority voting. When p = 0, the standard binomial probability in Eq. (6.1) is used for n<25 and the normal approximation is used for a larger n. The beta-binomial model is used when the correlation is positive, and the extended beta-binomial model is used when the correlation is negative. The table illustrates that negatively correlated classifiers improve the prediction accuracy more rapidly than the independent classifiers. 

Not available using the extended beta-binomial model by Prentice (1988). Greater than or equal to 0.995. 

If the two values differ and volatility and drift are already calibrated, the interest rates used in the binomial model are adjusted with a further spread. The spread is found with an iterative procedure ... 

In period 1 there are two possible levels for the interest rate at period 2 there are four possible levels. After N periods, there will be 2 possible values for the interest rate. Calculating the current price of a 10-year callable bond that pays semiannual coupons involves generating more than one million possible values for the last period s set of nodes. For a 20-year bond, the number jumps to one trillion. (Note that the binomial models actually used in analyses have much shorter periods than six months, increasing the number of nodes.)... 

The binomial model evaluates a bond s return by measuring the extent to which it exceeds those determined by the risk-free short rates in the tree. The spread between these returns is the bond s incremental return at a specified price. Determining the spread involves the following steps ... 

Effective duration may be calculated using the binomial model and equation (11.13), as follows ... 

If the prices used for P f, P, and Po are calculated assuming that the bond s remaining cash flows will not change when market rates do, the convexity computed is for an option-free bond. For bonds with embedded options, the prices used in the equation should be derived using a binomial model, in which the cash flows do change with interest rates. The result is effective or option-adjusted convexity. 

Effective duration is essentially approximate duration where P and P are obtained using a valuation model—such as a static cash flow model, a binomial model, or a simulation model—that incorporates the eflFect of a change in interest rates on the expected cash flows. The values of P and T, depend on the assumed prepayment rate. Generally analysts assume a higher prepayment rate when the interest rate is at the lower level of the two rates—interest and prepayment. 

Reliability can be calculated by several methods. One method used in other industries (such as the automotive industry) is to test 12 samples to the proposed life of the device. If all 12 samples survive, then the sample size is adequate to demonstrate a reasonable risk of failure using a binomial model. To determine failure modes, the stress or strain on the device can be increased by 10 percent for 10 percent of the proposed life of the device. If there are no failures at 110 percent of fhe proposed life, the stress or strain range is increased another 10 percent. This stair-step method continues until a failure mode is demonstrated. 

Ennis, D.M. and Bi, J. (1998). The beta-binomial model accounting for inter-trial variation in replicated difference and preference tests. J. Sensory Stud., 13, 389-412. 

Pham, T.V., Piersma, S.R., Warmoes, M., and Jimenez, C.R. (2010) On the beta-binomial model for analysis of spectral count data in label-free tandem mass spectrometry-based proteomics. Bioinformatics. 26 (3), 363-369. 

The literature offers a wide variety of count data models to regress a count variable on a set of independent variables (see for example Cameron and Trivedi, 1998). A well-known example is the Poisson model. The applicability of this model however is limited due to the fact that it imposes equi-dispersion, which means that the conditional mean and variance are equal. This is a rather strong assumption. A more flexible count data model is obtained when unobserved heterogeneity is introduced in the Poisson intensity parameter A. The most commonly used extension to the Poisson model is the negative binomial model (negbin). 

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... 

Fair Value of a Convertible Bond The Binomial Model... 

The fair price of a convertible bond is the one that provides no opportunity for arbitrage profit that is, it precludes a trading strategy of running simultaneous but opposite positions in the convertible and the underlying equity in order to realize a profit. Under this approach we consider now an application of the binomial model to value a convertible security. Following the usual conditions of an option pricing model such as Black-Scholes (1973) or Cox-Ross-Rubinstein (1979), we assume no dividend payments, no transaction costs, a risk-free interest rate, and no bid-offer spreads. 

Application of the binomial model requires a binomial tree detailing the price outcomes from the start period, which is shown at FIGURE 13.3. In the case of a convertible bond this will refer to the prices for the underlying asset, which is the ordinary share of the issuing company. 

The binomial model reviewed previously will calculate the fair value for a convertible where certain parameters have been specified. It is apparent that altering any of the inputs to the model will have an impact on the price calculation. We consider now the effect of changing one of these parameters. 

A change in the volatility of the underlying share price will affect the price of the convertible, given that under the binomial model its value has been... 

Concerning the results of the Greenwood Yule (1920) study, in nearly all of the cases the pure chance model did not fit the data, whereas the best fit was obtained in conjunction with the negative binomial model calculated on the basis of the assumed unequal liability. This result is representative of a large number of follow-up studies (e.g. Adelstein 1952, Mintz Blum 1949, Burkardt 1962, 1970), in which the negative binomial distribution fit the data better than all other models did. 

Complexity of systems arise when a specific combination of components do not function entirely in series or parallel, or a specified number of components needs to operate for mission success (r of N components). For component configurations where r of N components needs to operate the component reliability can be quantified using the Binomial model. An example of needing Y of N components is seen in the structure of the HRS for a single Brayton system where one of two HRLs associated with a given radiator needs to operate for mission success. The binomial model is as follows ... 

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