Swaps basis

The credit default swap basis and trading issues 6 Bibliography 11... 

THE CREDIT DEFAULT SWAP BASIS AND TRADING ISSUES... 

Choudhry, M., 2006. The Credit Default Swap Basis. Bloomberg Press, New York. 

Asset Swap Basis The asset swap basis is the spread on the CDS minus the spread on a par asset swap of a bond with a similar average life. A positive basis exists when this difference is positive. 

Several borrowers issue bonds in multiple currencies, which tend to trade at different spreads, even on a fully asset swapped basis. The reasons for this divergence include name recognition, and technical factors such as liquidity and the ability to enter into a repo. 

Such a method has seldom been used with systems containing an aqueous fluid, probably because the complexity of the solution s free energy surface and the wide range in aqueous solubilities of the elements complicate the numerics of the calculation (e.g., Harvie el al., 1987). Instead, most models employ a procedure of elimination. If the calculation described fails to predict a system at equilibrium, the mineral assemblage is changed to swap undersaturated minerals out of the basis or supersaturated minerals into it, following the steps in the previous chapter the calculation is then repeated. 

Ratios greater than one identify supersaturated minerals that need to be swapped into the basis and allowed to precipitate. 

Once there are no undersaturated minerals, the procedure checks for supersaturated minerals. If any exist, the most supersaturated mineral, identified by the largest Qi/K[, is swapped into the basis and the governing equations are solved. Precipitating a new mineral, however, may dissolve another away, so now the process begins anew by checking for undersaturated minerals. Once a solution has been found that includes neither undersaturated nor supersaturated minerals, the true equilibrium state has been located. 

Sometimes when a mineral becomes supersaturated, there is no logical aqueous species in the basis with which to swap the mineral. Such a situation occurs when no species appear in the reaction to form the mineral. Wolery (1979) and Reed (1982) refer to such a situation as an apparent violation of the phase rule, because adding the mineral to the basis would produce more phases in the system than there are components. 

Since we could not possibly store each possible variation on the basis, it is important for us to be able at any point in the calculation to adapt the basis to match the current system. It may be necessary to change the basis (make a basis swap, in modeling vernacular) for several reasons. This chapter describes how basis swaps can be accomplished in a computer model, and Chapter 11 shows how this technique can be applied to automatically balance chemical reactions and calculate equilibrium constants. 

The modeler first encounters basis swapping in setting up a model, when it may be necessary to swap the basis to constrain the calculation. The thermodynamic dataset contains reactions written in terms of a preset basis that includes water and certain aqueous species (Na+, Ca++, K+, Cl-, HCOJ, SO4-, H+, and so on) normally encountered in a chemical analysis. Some of the members of the original basis are likely to be appropriate for a calculation. When a mineral appears at equilibrium or a gas at known fugacity appears as a constraint, however, the modeler needs to swap the mineral or gas in question into the basis in place of one of these species. 

Each entry A -, A k, or A m is a species or mineral that can be formed according to a swap reaction as a combination of the entries in the original basis. If Aj, a secondary species under the original basis, is to be swapped into basis position A (, the corresponding swap reaction is,... 

The third step in changing the basis is to set the equilibrium constants for the revised reactions. The new equilibrium constant K j for a species reaction can be found from its value Kj before the basis swap according to... 

Here, Kpv, K , and K are the equilibrium constants for the reactions by which we swap species, minerals, and gases into the basis. 

Basis entries that do not change over the swap have no effect in these equations, since they are represented by null swap reactions (e.g., H2O = H2O) with equilibrium constants of unity. 

What is the equilibrium constant for CO3 in the example from the previous section The value at 25 °C for the reaction written in terms of the original basis is 1010 34, and the equilibrium constants of the swap reactions for calcite and C02(g) are 101 71 and 10-7 82. The new value according to the equation above is,... 

To calculate the model with react, we swap C02(g) and 02(g) into the basis in place of H+ and 02(aq), and constrain each basis member. The procedure is... 

Next (step 3), the program swaps antigorite, the most supersaturated mineral in the initial fluid, into the basis in place of species MgC03. With antigorite in the system, six minerals are supersaturated. In the next step (step 4), the program chooses to swap dolomite into the basis in place of Si02(aq). This swap seems strange until we write the reaction for dolomite in terms of the current basis ... 

The calculation procedure in react is to swap sphalerite and barite into the basis in place of 02(aq) and Ba++... 

Conveniently, perhaps even miraculously, the equations developed in Chapter 5 to accomplish basis swaps can be used to balance chemical reactions automatically. Once the equations have been coded into a computer program, there is no need to balance reactions, compute equilibrium constants, or even determine equilibrium equations by hand. Instead, these procedures can be performed quickly and reliably on a small computer. 

To balance a reaction, we write swap reactions relating B to B. Returning to the previous example, we wish to compute the reaction by which Ca-clinoptilolite transforms to muscovite and quartz. Reserving the clinoptilolite for the reaction s left side, we write the swap reactions for the basis transformation in matrix form. 

To further illustrate how the basis-swapping algorithm can be used to balance reactions, we consider several ways to represent the dissolution reaction of pyrite, FeS2. Using the program RXN, we retrieve the reaction for pyrite as written in the llnl database... 

Pyrite can dissolve into reducing as well as oxidizing solutions. To find the reaction by which the mineral dissolves to form H2S, we swap this species into the basis in place of the sulfate ion... 

Is there a reaction by which pyrite can dissolve without changing the overall oxidation state of its sulfur To see, we return the sulfate ion to the basis and swap H2S for dissolved oxygen ... 

We repeat the calculation, this time swapping Al(OH) into the basis to represent dissolved aluminum. 

Swapping H2S(aq) into the basis in the place of SO4, on the other hand... 

The model then swaps aqueous species Aj into the basis locations Ak held by the mineral components. The technique for swapping the basis is explained in Chapter 5, and a method for selecting an appropriate species Aj to include in the basis is described in Chapter 4. When the procedure is complete, the equilibrium system contains only the original fluid. 

It is next attempted to change the basis of description from x,y to a new distinct set of variables u, y (swapping u and x), so that differential quantities are expressed in terms of the differentials du and dy. 

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