lesniewski libor and ois lecture1_2012
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INTEREST RATES AND FX MODELS1. LIBOR and OISAndrew LesniewskiCourant Institute of Mathematical SciencesNew York UniversityNew YorkJanuary 26, 20122 Interest Rates & FX ModelsContents1 Introduction 22 LIBOR and LIBOR based instruments 32.1 LIBOR rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Forward rate agreements . . . . . . . . . . . . . . . . . . . . . . 52.3 LIBOR futures . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 OIS and basis swaps 94 Valuation of swaps 124.1 Zero coupon bonds . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Swap rates and valuation of swaps . . . . . . . . . . . . . . . . . 164.4 Valuation of LIBOR / OIS basis swaps . . . . . . . . . . . . . . . 175 Building the OIS and LIBOR curves 185.1 Splines and B-splines . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Curve construction by B-splines tting . . . . . . . . . . . . . . . 211 IntroductionDebt and debt related markets, also known as xed income markets, account forthe lion share of the worlds nancial markets. Instruments trading in these mar-kets fall into the categories of cash instruments, or bonds, and derivative instru-ments, such as interest rate swaps, bond futures and credit default swaps. Bondsare debt instruments issued by various entities (such as sovereigns, corporations,municipalities, US Government Sponsored Enterprizes, etc), with the purposes ofraising funds in the capital markets. Derivatives are synthetic instruments whichextract specic features of commonly traded cash instruments in order to makerisk management and speculation easier. In this course we will focus largely on aclass of derivatives, known as interest rate derivatives.Depending on their purpose, xed income instruments may exhibit very com-plex risk proles. The value of a xed income instrument may depend on thelevel and term structure of interest rates, credit characteristics of the underlyingentity, foreign exchange levels, or prepayment propensity of the collateral poolLIBOR and OIS 3of loans. Understanding, modeling, and managing each of these (and other) risksposes unique challenges. These lectures cover some aspects of the interest raterisk only.Like all other nancial markets, xed income markets uctuate, and the maindriver of this variability is the current perception of future interest rates. Marketparticipants have adopted the convention that a rally refers to falling interest rates,while a sell off refers to rising rates. This somewhat counterintuitive convention isbest rationalized by the fact that interest rate sensitive instruments, such as bonds,typically appreciate in value when rates go down and depreciate in value whenrates go up. By the same token, a xed income portfolio is long the market, if itsvalue goes up with falling rates, and is short the market if the opposite is true.Each xed income instrument is a stream of known or contingent cash ows.Such cash ows are typically periodic, and are computed as a xed or oatingcoupon, applied to a principal (or a notional principal). The market has developedvarious conventions to apply the coupon to the relevant accrual period. Exam-ples of such conventions in the US dollar market are the money market conventionact/360 (actual number of days over a 360 day year), and the bond market con-ventions 30/360 (based on a 30 day month over a 360 day year).2 LIBOR and LIBOR based instruments2.1 LIBOR ratesMuch of the activity in the world capital markets is tied to the LIBOR rates. Theyare widely used as benchmarks for short term(overnight to 1 year) interest rates. ALIBOR (= London Interbank Offered Rate) rate is the interest rate at which banksoffer (at least in principle) unsecured deposits to each other. Daily xings of LI-BOR are published by the British Banking Association on each London businessday at 11 a.m. London time. These xings are calculated fromthe quotes providedby a panel of participating banks. LIBOR rates are not a risk free, but the partici-pating banks have high credit ratings. The details on the composition of the panelsand how the xings are calculated can found on the web site www.bbalibor.comof the British Banking Association.LIBOR is quoted in ten major currencies: US Dollar (USD), British Sterling(GBP), Euro (EUR), Japanese Yen (JPY), Swiss Franc (CHF), Canadian Dollar(CAD), Australian Dollar (AUD), Danish Krone (DKK), Swedish Krona (SEK),and New Zealand Dollar (NZD). For each currency, LIBOR rates of 15 different4 Interest Rates & FX Modelsmaturities are quoted. The LIBOR rates and the market practices surroundingthem vary somewhat from currency to currency. For example, the prevalent ma-turity in the USD is 3 months, and when we refer to the LIBOR rate we mean the3 month rate. On the other hand, the most popular benchmark in the EUR is the6 month rate1. In order to keep things simple, throughout this course, we shallfocus on the the USD market, and designate the term LIBOR to mean the 3 monthLIBOR rate.In the USD market, LIBOR applies to deposits that settle two business daysfrom the current date (this date is called the spot date), and whose maturity ison an anniversary date (say, 3 months) of that settlement date. Determining theanniversary date follows two rules:(a) If the anniversary date is not a business day, move forward to the next busi-ness day, except if this takes you over a calendar month end, in which caseyou move back to the last business day. This rule is known as modiedfollowing business day convention.(b) If the settlement date is the last business day of a calendar month, all an-niversary dates are last business days of their calendar months.In addition to spot transactions, there are a variety of vanilla LIBOR basedinstruments actively trading both on exchanges and over the counter: LIBOR fu-tures, forward rate agreements, and interest rate swaps. The markets for LIBORbased instruments are among the most active derivatives markets. The signicanceof these instruments is that:(a) They allow portfolio managers and other nancial professionals effectivelyhedge their interest rate exposure. We will discuss it in detail later in thiscourse.(b) One can use them to synthetically create desired future cash ows and thuseffectively manage assets versus liabilities.(c) They allow market participants easily express their views on the future leveland shape of the term structure of interest rates, and thus are convenientvehicles of speculation.1The prevalent index rate in the Eurozone is actually the EURIBOR, which is different fromthe euro LIBOR.LIBOR and OIS 52.2 Forward rate agreementsForward rate agreements (FRAs) are over the counter (OTC) transactions. Thismeans that they are arranged between the counterparties without an involvementof an exchange or a clearing house. The trades are typically arranged through asalesman, a voice broker or an electronic platform. In a FRA transaction, one ofthe counterparties (A) agrees to pay the other counterparty (B) LIBOR settling tyears from now applied to a certain notional amount (say, $100 mm). In exchange,counterparty B pays counterparty A a pre-agreed interest rate (say, 3.05%) appliedto the same notional. The contract matures on an anniversary T (say, 3 months)of the settlement date, and interest is computed on an act/360 day count basis.Anniversary dates generally follow the same modied following business dayconvention as the LIBOR. FRAs are quoted in terms of the annualized forwardinterest rate applied to the accrual period of the transaction.Market participants use FRAs for both hedging and speculation. Because oftheir granular structure, they allow one to hedge the exposure or express the viewon rates move on specic days. For example, an FOMC FRA is a FRA whichxes the day after a Federal Open Markets Committee meeting. An FOMC switchinvolves taking opposite positions in a FRA xing before and after an FOMCmeeting. Likewise, a turn switch allow to hedge or express the view on the ratespikes idiosyncratic to the month-, quarter- or year-end.Econometric studies of historical rates data show that forward rates are poorpredictors of future interest rates. Rather, they reect the evolving current con-sensus market sentiment about the future levels of rates. Their true economicsignicance lies in the fact that a variety of instruments whose values derive fromthe levels of forward rates (such as swaps) can be liquidly traded and used tohedge against adverse future levels of rates.2.3 LIBOR futuresLIBOR futures, known also as the Eurodollar futures, are exchange traded futurescontracts on the 3 month LIBOR rate. They trade on the Chicago MercantileExchange, often referred to as the Merc, which also clears and settles the trades.The parties involved in futures trading maintain a cash position in their marginaccounts, which serve as a security deposit against default. The Merc makes dailymark to market adjustments to the margin account in order to reect the changesin values of each of the individual contracts.In many ways, Eurodollar futures are similar to FRAs, except that their terms,6 Interest Rates & FX Modelssuch as contract sizes and settlement dates are standardized.