Discount factors technique

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

This technique of calculating discount factors, known as bootstrapping, is conceptually neat, but may not work so well in practice. Problems arise when you do not have a set of bonds that mature at precise six-month intervals. Liquidity issues connected with individual bonds can also cause complications. This is true because the price of the bond, which is still the sum of the present values of the cash flows, may reflect liquidity considerations (e.g., hard to buy or sell the bond, difficult to find) that do not reflect the market as a whole but peculiarities of that... 

Linear interpolation is simple but not accurate enough to be recommended. Market analysts use multiple regression or spline-based methods instead. One technique is to assume that the discount factors represent a functional form—that is, a higher-order function that takes... 

The products of each system were costed using two different techniques the equality method and the by-product work method (1, 2 ). Every case analyzed accounts for maintenance as well as fuel expenses but excludes other operating costs. Capital investment is amortized at an after-tax discount rate of 8.5%. The effects of income taxes, ad valorem taxes, and depreciation (see (3) for the formulation of the annualization factor accounting for capital, taxes and depreciation) are included, while the effects of inflation were neglected. 

The transforms 0t(z) and S, (z), by itself can be seen as a modified characteristic function. Unfortunately, there exists no closed-form of the transform S,(z), meaning that the standard Fourier inversion techniques can be applied only for the computation of options on discount bonds. On the other hand, the transform S,(n) can he used to compute the n-th moments of the underlying random variable V (7b, 7 ). Then, by plugging the moments (cumulants) in the lEE scheme the price of an option on coupon hearing bond can be computed, even in a multi-factor framework. 

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

If we discount the role of (Dab/TWi) in these three expressions for AB, the separation factors are what would have been achieved in dissociation extraction from an aqueous solution to an organic solvent If, however, species A were to be reextracted back into an aqueous solution for recovery, then the liquid membrane step achieves the same goal using very little solvent and only one device (instead of an extractor and a back extractor). Such a technique is likely to be highly useful in the pharmaceutical industry. 

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