# USING MULTIPLE INSTRUMENTS TO BUILD A CURVE

We introduced in the previous section the importance of basis swaps. We explained the introduction of the tenor basis swap as a way of literally putting a price, in an uncertain environment, on the inherent riskiness of lending for a longer period. It is a principle fairly easy to accept. The introduction of the currency basis swap is also in nature linked to credit, but it is considerably more complex.

First of all let us state the obvious: a cross currency basis swap is a trade with two currencies. This means that there is an FX element involved and when this happens we lose the idea of an absolute frame of reference– everything can be seen in two ways. What is there to be seen? We will show in great detail when explaining the role of the treasury desk within a financial institution, that the activity of an investment bank can be summed as borrowing short term and lending long term. Instead of lending, there can be other forms of trading, but the principle is that the money an investment bank uses needs to be borrowed somehow. Being global institutions, investment banks can choose to borrow (or fund themselves) in their native currencies (USD for American banks, EUR for European, etc.) or in other currencies abroad. The level at which one borrows and the discounting one applies to a future cash flow are strictly linked: the way we see now a payment in the future has to be the same way we see the money we have to repay ourselves at the same moment in the future.

Borrowing rates have everything to do with the perception the lender has of the borrower. This means that shifting perceptions have a direct influence on the way we discount. As we have previously mentioned, before 2005 (the date is very approximate; for a good description of the changing environment, see Piterbarg [70] and Johannes and Sundaresan [55]) a, say, U.S.-based institution would borrow funds in EUR paying EUR LIBORs (or very close to it) and hence would believe that a floating leg in EUR would price at par. The only instrument needed (assuming the short end of the curve is taken careof with deposits and/or interest rate futures) to fit its index function and discount function would be an interest rate swap of the form

(2.10)

The market then changed and perceptions shifted. The market wondered whether it is true that all institutions can borrow EUR at the same level, or any other currency for the matter. A currency basis gives a numerical value to this: the market believes that not everyone should be able to borrow currency *X* at the same level. The basis gives a level of this belief and institutions are then charged different rates. This is one of those situations where it is difficult to distinguish the cause from the effect. Was it the difficulty of financial institutions to borrow abroad that caused the market to doubt their ability, or was it the market's doubt which caused their inability? In, say, 2001 a U.S.-based institution would have borrowed, for 10 years, Euro paying EUR LIBOR basically flat or Japanese Yen paying JPY LIBOR basically flat. Now, looking at Figure 2.7, it would pay, respectively, EUR LIBOR minus 37.25 basis points (mid level) and JPY LIBOR minus 64.75 basis points. Now the same institution cannot construct its EUR curve simply using Equation 2.10, the currency basis information needs to be applied. Equation 2.10 needs to be combined with

(2.11)

Equations 2.10 and 2.11 need to be solved simultaneously, which is not an easy task if we consider how many terms each summation sign could be expanded to (let us imagine, for example, a 40-year swap paying every three months). Not only a numerical solver is needed but a further consideration can simplify things considerably. From Figure 2.7 we noticed that all currency basis are quoted against a USD LIBOR leg flat. This is

FIGURE 2.7 A few examples of quotes for common cross currency basis swaps quoted as USD three-month flat versus foreign currency three-month rate plus basis. *Source:* Thomson Reuters Eikon.

probably because U.S.-based institutions are the largest market participants,^{[1]} but in any case it means that the view is, in general, USD-centric. From this we can assume that at least in USD (and for a U.S. institution, as we shall discuss soon), a flat floating leg does indeed price at par. In an instant, this wipes out any USD-denominated variable from Equation 2.11 and we are left with having to solve simultaneously the following set of equations

(2.12)

What does the basis signify and why are they different from currency to currency? Theories abound on this point and few people agree, however, one could see the basis as an indication of the average ease with which a U.S.- based institution can borrow in the currency of country X and/or the perception within country X of the creditworthiness of the average U.S.-based institution. A further question is, should an institution based in a country with currency *X* include the currency basis when constructing the discount curve for X? The question does not have a clear answer and there are many arguments about it. We believe (a very elegant point is made by Fujii et al. [41] and by Benhamou [11]) that they should not include the basis. The move toward collateralized discounting, however, which we shall introduce later, has rendered the argument almost irrelevant. It would be interesting to insert here an argument often raised by local currency traders^{[2]} at the time of the move toward a curve construction including currency basis: I agree one should include the basis if one trades cross currency swaps (i.e., a trade in two frames of reference, so to speak), but if all my trading activity takes place in local currency, why should I include the basis? First of all, let us say that a trader will instinctively resist any move toward a different price resulting in a financial loss, particularly if driven by a reason obscure to him. The answer, and the goal of this chapter, is that one should try to take into consideration the wider activity of a financial institution (see for example Tuckman and Home [80]). While it is true the pseudo-scientific argument that in the frame of reference of the local currency (which we have seen is the point of view of an institution in that country) there is no basis, the money the trader is using to fund the trade has come from a frame of reference where there is basis.

Another argument used against two different discounting (one with and one without currency basis according to the country of origin) is the following: the swap rate quoted, that is, actually traded, is one, how can there be two discounting? We should not forget that the swap rate is the starting point, not the arrival point. A swap rate of C means that one party agrees to pay C and the other to pay a series of floating rates. This is the only contractual agreement and the only thing that matters. Each party will then construct its discount function accordingly and both will price the swap correctly. Of course, as we shall see in the next sections, what is true at inception, when the swap is at par, is not true when the swap seasons. Eater in the life of the trade the two parties–the one using the currency basis and the one not using it–will disagree on the price of the swap, and this is another reason for the recent move away from LIBOR-driven discounting.

It is probably important to mention here the current argument (see^{[3]} Carver [27]) that the new advances in discounting are dangerously flirting with the idea of abandoning the concept of one price. We have mentioned earlier that a pillar, if not *the* pillar, of financial modeling in the risk-neutral framework is the fact that the same instrument cannot have two different prices, this of course is because it would lead to arbitrage. When trading a swap, is the existence of one price given by the uniqueness of the swap rate value (for example, the fixed rate C) or by the fact that both parties see the value of the swap as initially at par? In the past the two have almost always, and certainly at inception, coincided, but we should not forget that the former, the swap rate, is the traded quantity, while the latter is the fruit of an institution-dependent calculation, the act of implying discount factors from market quantities. Of course, the fact that two market participants see the present value of the same stream of cash flows in different ways is rather unsettling, and it is certainly cause for discussion. One should not, however, fear for risk neutrality as this is preserved by the fact that both agree that one should pay the other a fixed amount at specific intervals.

We have not offered quite the full picture yet. In Equation 2.12 we have been vague with the frequency of reset, but in reality, the EUR LIBOR in the interest rate swap resets every six months, whereas the EUR LIBOR in the cross currency basis swaps resets every three months. We know from the mere existence of a tenor basis swap in EUR that these two rates are not the same. The tenor basis information needs to be included in our simultaneous curve construction, so what we actually need to solve is

(2.13)

Our solver will need to fit three functions now: the discount function and two index functions, and .

From Equation 2.13 we can easily see how, whenever a new tenor basis swap appears featuring a combination between two rates of different length,

it needs to be included in our curve construction and we will need to solve for an extra index curve. As an aside, solvers needed to fit multiple index and discount functions, which requires considerable computational sophistication that is not available to all institutions. The mathematical problem is the one of finding the (global) minima of a function with an extremely high number of dimensions: this problem, together with the one of finding the factors of a number, is known to be among the most challenging numerically.

We have said that Equation 2.13 can be expanded beyond three instruments to "-instruments. The final concept that needs to be addressed is the one of collateralized curve construction.

- [1] Currency basis for some Eastern European currencies are quoted against EUR LIBOR flat and, in a bout of quirkiness, for Mexican and Chilean Pesos the swap is against USD but the basis is on the USD leg.
- [2] Local currency, used particularly when it comes to very small and/or emerging market currencies, is another admittedly confusing way to say foreign currency.
- [3] This article is part of a wider debate that can be found in Laughton and Vaisbrot [60] and in Hull and White ([48] and [49])