A financial institution receiving collateral is expected to pay interest on the collateral amount to the financial institution posting the collateral. The rate paid is almost always the overnight interest rate. This means that if the cash matching the MTM of the swap grows at the risk-free rate, then the discounting (the inverse of growth) of the same swap should be done using the same rate.^{[1]} OIS is the tool that allows us to imply this new discount function. Let us rewrite Equation 2.14 as where we writeto stress the discount function driven by the overnight rate. If we now wish to value a swap or use a swap to construct a set of LIBORs (let us hrst think of the single currency situation, but let us assume that we deal with rates of more than one length and therefore we need to find and , we use, instead of Equations 2.4 and 2.7,

in conjunction with Equation 2.15, which is needed for the discount factors. The above can be extended, similarly to Equation 2.13, should we need rates of additional lengths. Since the above applies to the single currency situation, what should we do in the presence of multiple currencies?

We see that now we are moving further and further away from the assumption that a floating leg should price at par since now, even more so than when we introduced the currency basis, we are using two completely different rates to generate the forward cash flows and to discount them. We are going to return to this point, but now we need to consider carefully what it

(2.15)

(2.16)

means to price a swap at par (as opposed to a single leg). The real definition of a par swap is one where the net of both legs is zero; however, in the past this was informally extended to the requirement that each leg would be equal to par, that is, zero (or one depending on whether there was an initial payment of principal). Now the change in discounting forces us to go back to the initial and rigorous definition, as we can no longer expect each leg to price at par.

[1] We shall return to this, but this statement is far from being universally accepted (see for example Hull and White [48]).

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