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2.4.2.1 Collateral in a Foreign Currency

The problem of multiple currencies arises in two ways: first, we need to see what happens to the discount factors in the other currency if we are pricing a leg in a different currency,[1] and second, what happens when the collateral posted is in a foreign currency.

Imagine that, as a U.S.-based institution, we need to discount a stream of cash flows in EUR. Up to the introduction of collateralized curve construction we would have discounted it using information obtained from the currency basis swap market. We have seen that this reflected the cost of funding an institution in a foreign market. The information is expressed as a spread over TIBOR, that is, over a risky rate. With the introduction of collateral we turn toward different types of instruments. The first question is, in what currency is the collateral posted? We should not assume that the collateral is in EUR; it could be in either USD or EUR.[2] Over time the currencies and the ways in which collateral is posted become more and more numerous so that the different options below almost follow an evolution in time.

In the case that the collateral is posted in USD, one could convert each cash flow with the appropriate FX forward and then discount using the OIS-driven discount function. We could write this (showing only the EUR leg) as

(2.17)

where FWDi is the FX forward rate at time 7) and the discount factors are the same as in Equation 2.15 (we have stressed the currency for absolute clarity). Someone might wonder whether the FX forward would not introduce an interest rate element that would make things murkier. To this very interesting point one would reply that, first of all, the interest rate component of an FX forward is there only when we try to extract it. This cryptic, Schrodinger-like argument needs clarifying. First, when an FX trader is trading forwards, he is not thinking about interest rates. The fact that we can imply them is just a handy financial argument that works most of the time. The fact that we need to make assumptions about one of the two curves shows that the information never really comes out completely. Second, didn't we say that the sense of greater safety derived from the trading of FX forwards originated from the fact that they were fully collateralized, that is, the principals were exchanged at the beginning of the trade? This means that FX forwards were never meant to be proxies of LIBOR levels, instead they are particularly suited to the argument that we need to take into account the collateral posted. The interest rate intrinsic to the FX forward should be very close to an overnight rate and therefore it does not introduce murkiness into Equation 2.17.

With collateral being posted in different currencies, the above does not seem to be the ideal solution, which instead would be to use some form of swap indexing the overnight rate for EUR discounting. One cannot tell with certainty whether the appearance in the market of an OIS-type swap for currency X motivates institutions to post collateral in currency X or is the posting in that currency that gives birth to the trading in the overnight index swap as the two tend to appear together (see Andersen and Piterbarg [4] for a few interesting considerations on the proliferation of instruments used for collateralized discounting, and Fujii et al. [42] for a rigorous treatment of collateralized curve construction). With this tool at hand we can obtain discount factors of the form and discount the EUR leg of a swap as

(2.18)

Collateral is being posted in several (major) currencies nowadays, and operationally, since a certain collateral is linked to a certain counterparty, the tendency is to assume that that counterparty will always post, to collateralize a certain currency, the same currency (e.g., bank XYZ will always post EUR to collateralize Hungarian Florins trades). We shall now ask ourselves, does the change in methodology really matter?

  • [1] We keep it general so that the leg can either belong to an interest rate swap in a foreign currency or it can be the foreign leg in a cross currency swap.
  • [2] Technically, collateral can be posted in any currency and there are traders who actively search and profit from seeking the most attractive form of collateral (cheapest to deliver). Let us ignore, however, these more perverse situations.
 
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