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A financial institution is in possession of a certain amount of cash, some of which has been received as collateral. Due to its liquid nature, cash is most sensitive to the overnight rate, the rate used by institutions to lend to each other from one day to the next. Because of the low risk associated with overnight lending, the overnight rate is usually the lowest available rate. In order to manage overnight rate exposure, an instrument called the overnight index swap (OIS) was introduced. Practice dictates, as it does in the case of LIBOR, that when there is no currency specified after it, OIS is intended to mean a swap linked to the USD overnight rate: we shall use this practice and refer to the similar swaps in other currencies as either X OIS or give them their proper name if it exists (such as EONIA for EUR OIS). In an OIS,[1] one side pays a fixed coupon C (the OIS rate) at certain intervals and the other side pays the daily compounded overnight rate i, that is, where j is a daily counter. OIS are very useful because they allow the user to manage over a long period (OIS can have maturities of up to 40 years) a rate resetting almost continuously. For us they are very useful because, in this form, we can include overnight index swap rates in our curve construction calculations. Before doing so, however, let us try to understand a bit more why collateral is important when pricing swaps.

In Section 2.5.2 we said we believe that a floating leg with principal repayment at the end should price at par value. In Equation 2.9 we have briefly shown how, if one uses the floating rate to obtain the discount factors, this is the case. At that moment we did not mention any credit-related aspect, basically assuming that the process of discounting was inherently as risky as the process of rate generation. Since the floating rate was LIBOR, the discounting was done at the same level. With the introduction of the currency basis, which we have presented as a credit-driven quantity, we saw that already this was no longer true as we were not discounting anymore with the same rate used for the forward rates. Let us repeat why, under a credit point of view, the action of calculating the present value, that is, discounting, is closely related to the rate generation: a certain cash flow will grow from its present discounted value (the way we see things at the moment) to its actual value at the moment of payment at the same rate we are charged to borrow.


A plot highlighting the difference between the overnight rate and the three-month LIBOR over time before, during, and after the peak of the financial crisis.


The rate we are charged to borrow is driven by a perception of credit risk, hence discounting should be driven by the same perception.

If we look at Equation 2.4 and assume that we are in a world without any other type of swaps, we could be tempted to say that rate generation and discounting should be driven by LIBOR. This was indeed the classical theory of swap pricing where (see for example Duffie and Singleton [[32]) swap rates were seen as bond par rates of an issuer at LIBOR level of risk for the life of the trade. This, however, would ignore the role played by collateral. It is true that the cash payments are indexed to the LIBOR, but hasn't the presence of collateral eliminated the credit risk element? This is indeed the case, although it is a fact that has been overlooked for a long time leading to the inconsistency (see for example Tuckman [79]) in which a trade that is risk free and should have been discounted using risk-free rates, was discounted using a risky rate such as LIBOR. Why this is the case is very simple: for a long time it mattered very little. The spread between LIBOR and OIS rate, an indicator of the market's feeling toward credit risk (see Michaud and Upper [63]), had always been around few tens of basis points up to the 2007 to 2009 financial crisis. It is when the same spread went above 300 bps (see Ligure 2.8) that practitioners started to seriously take into consideration a change in the methodology with which discount factors should be calculated.

We shall now see how discounting takes into account collateral in practice and how this consideration has evolved in time.

  • [1] It is common to hear the term OIS swap. Considering that the “S” in OIS already stands for swap, it seems redundant.
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