We now reach a greater level still of complexity by introducing the interest rate swap, one of the most widely traded derivatives in the market. An interest rate swap is defined as where one party pays a fixed coupon C with a certain frequency and the other party pays, with a potentially different frequency, a floating rate L fixing at j; Di are the discount factors and dt are the day count fractions, dti = Ti –Ti-1. Up to now we have used the term floating rate: from now on we will use it, unless too inappropriate, interchangeably with the term TIBOR. A LIBOR (London Interbank Offer Rate) is an interest rate representing the rate at which investment banks agree to lend to each other: it is set in London every day. Most liquid currencies have a LIBOR of at least one maturity (e.g., six months); the most liquid currencies such as USD and EUR

have several LIBORs. Market practitioners tend to perpetrate (when speaking, not when trading) an imprecision by defining LIBOR as any rate that is the most important rate of choice for banks in a certain currency to lend to each other. In reality it will have a different, if similar, name: for example EUR has two rates, the EUR-LIBOR set in London and the EURIBOR set in Frankfurt; Australian Dollar (AUD) has an AUD-LIBOR set in London and a more liquid one set in Sydney. When speaking, a practitioner will often call all these LIBOR and say that a swap pays a floating LIBOR. We shall follow this practice. In Figure 2.5 we see quotes for interest rate swaps in Norwegian Krone (NOK): what is quoted is the fixed coupon C. Swaps are quoted at par, meaning that the value of C quoted is the one that will make, at the moment of entering the contract, both sides of Equation 2.4 equal.

With interest rate swaps, we now start to see a level of complexity absent from the previous instruments. This complexity easily turns into a considerable computational challenge. Let us illustrate this with an example. Let us imagine that we want to build a NOK curve for maturities up to two years and we have at our disposal a six-month cash deposit R6, a six-month in six-months FRA L6.6, and a two-year interest rate swap where, as one can read from Figure 2.5, the fixed rate C is paid annually and the floating rate semiannually.

Let us first write the interest rate swap, with times given in months, in full.

(2.5)

At first it seems that apart from C in the above equation we have eight unknowns; however, things simplify a little. First of all, L0,6 is the six-month rate starting today so it must be known and we can say (in our simple world) that it is equal to R6. Second, D6 can be found also from R6, as shown in Equation 2.2. Third, we also know the value of L6,6, given from the FRA. Finally, D12 can also be obtained easily using a second iteration^{[1]} on Equation 2.2, mainly

We are now left with four unknowns: the forward rates and discount factors beyond one year. In order to find them we need to use a solver, that is, a process that will try to fit two functions, ft(L) and Dt, so that they will generate, respectively, the forward rates L and discount factors DT that will make the swap price at par. More and more often we shall try to call ft(L) the index curve and Dt the discount curve. (While the use of a solver would certainly be appropriate, in some simple situations, as we shall describe in the numerical example concluding this chapter, one can find simpler solutions based on assumptions and interpolations.)

We are starting to see what the two principles outlined in Section 2.1 might mean. All the information given above (deposit, FRAs, and interest rate swaps) needs to be used simultaneously and only that information will be repriced correctly. This second point is very important, and to stress it, let us imagine that although is traded in the market as an FRA, we chose to ignore it. After using the information given by the cash deposit, the solver needs to find ft(L) and Dt fitting six unknowns instead of four. One of these will be a rate found after solving. Should we expect this rate to be equal to the rate L6,6, quoted by the FRA? The answer should be a resounding no, and to expect otherwise, a rather common mistake is to forget that in any calibration process if one wants a definite output this has to also be an input.

With the second principle of curve construction being so important, let us elaborate further by again using our simple example. We are all familiar with the concept and requirement of an arbitrage-free framework: when modeling finance in a risk-neutral framework we assume that it is impossible to make a riskless profit. This means that the same instrument cannot have two different prices. This leads to a safe assumption that two different instruments implying the same financial quantity cannot have two different prices, which in turn results in the requirement that our discount and index curves should be unique. Even in our brief review of financial instruments we have seen that many of these overlap in time, from which we can ask is it possible that two different instruments being used to calibrate the same portion of the discount and/or index curve could lead to two different results? The answer is sadly yes. One would think that all instruments fit in properly, but it is not the case.^{[2]} Computationally this means that there could be situations where there is an information glut and the calculation would fail. Let us imagine that in our simple world a further piece of information is available, the one-year NOK/USD FX forward. We know that from it we can derive the 12-month discount factor, however, we already have the 12-month discount factor. In practice we are asking

and the answer is, not necessarily. In this case, not only because we cannot always expect two different instruments to imply the same financial information but also because, in the specific case of FX forwards, we have seen that the outcome relies on the assumption of the USD curve construction. In the presence of overspecification, a curve construction process will fail, so what should one do? One should choose one or the other instrument and be ready to accept that the outcome is a framework potentially unable to price the excluded one. The choice could be based on liquidity: the more liquid instrument should be given preference. The choice could be based on portfolio affinity: if we trade a considerable amount of, say, FX forwards it would be more sensible to use those as input in our curve construction process. The choice could also be based on operational issues. While this example is simple, there could be cases in which we need to exclude instruments simply because we are not able to process the information they give us.

A final point is one about consistency: within an institution, curves should be constructed in the same way to avoid internal arbitrage. Let us imagine that within an institution a group trades mainly FX forwards and another trades mainly cash deposits. By applying the suggestions above and ignoring consistency, each will choose a different input in their calibration process, meaning that to each, a cash flow in six months will have a different present value.

[1] This is based on the idea that if we invest a unit amount for a period T2 ata rate R2, this has to be equivalent to investing it for a period T1 at a rate R1 and then reinvesting it from T1 to T2 at a forward rate L1,2, that is, (2.6)

[2] However, before jumping to the conclusion that the world is full of arbitrage, I would like to quote an old hand at Morgan Stanley who used to say, if you think there is arbitrage, show me the money. Most of the situations where one thinks there is arbitrage to exploit, when one goes through the trouble of exploiting it, there is no profit at all since it will have been eaten away by transaction costs, bid-offers, and so on.

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