Term Structure Models: IEOR E4710 Spring 2010 …

1 Term Structure Models: IEOR E4710 Spring 2010c 2010 by Martin HaughMarket ModelsOne of the principal disadvantages of short rate models, and HJM models more generally, is that they focus onunobservable instantaneous interest rates. The so-called market models that were developed1in the late90 sovercome this problem by directly modeling observable market rates such as LIBOR2and swap rates. Thesemodels are straightforward to calibrate and have quickly gained widespread acceptance from practitioners. Thefirst market models were actually developed in the HJM framework where the dynamics of instantaneous forwardrates are used via It o s Lemma to determine the dynamics of zero-coupon bonds. The dynamics of zero couponbond prices were then used, again via It o s Lemma, to determine the dynamics of LIBOR.

2 Market models aretherefore not inconsistent with HJM models. In these lecture notes, however, we will prefer to specify themarket models directly rather than derive them in the HJM framework. In the process, we will derive Black sformulae for caplets and swaptions thereby demonstrating the consistency of these formulae with martingalepricing these notes, we will ignore the possibility of default or counter-party risk and treat LIBOR interestrates as the fundamental rates in the market. Zero-coupon bond prices are then computed using LIBOR ratherthan the default-free rates implied by the prices of government securities. This does result in a minorinconsistency in that we price derivative securities assuming no possibility of default yet the interest ratesthemselves that play the role of underlying security , LIBOR and swap rates, implicitly incorporate thepossibility of default.

3 This inconsistency actually occurs in practice when banks trade caps, swaps and otherinstruments with each other, and ignore the possibility of default when quoting prices. Instead, the associatedcredit risks are kept to a minimum through the use ofnettingagreements and by counter-parties limiting thetotal size of trades they conduct with one another. This approach can also be justified when counter-partieshave a similar credit rating and similar exposures to one another. Finally, we should mention that it is indeedpossible3, and sometimes necessary, to explicitly model credit risk even when we are pricing standard securitiessuch as caps and swaps. It goes without saying of course, that default risk needs to be modeled explicitly whenpricing credit derivatives and related LIBOR, Swap Rates and Black s Formulae for Caps andSwaptionsWe now describe two particularly important market interest rates, namely LIBOR and swap rates.

4 We firstdefine LIBOR and forward LIBOR, and then describe Black s formula for caplets. After defining LIBOR we thenproceed to discuss swap rates and forward swap rates as well as describing Black s formula for swaptions. Inpractice, the underlying security for caps and swaptions are LIBOR and LIBOR-based swap rates. Thereforeby modeling the dynamics of these rates directly we succeed in obtaining more realistic models than thosedeveloped in the short-rate or HJM Miltersen, Sandmann and Sondermann (1997), Brace, Gatarek and Musiela (1997), Jamshidian (1997) and Musiela andRutkowski (1997).2 These models apply equally well to Euribor chapter 11 of Cairns for a model where swaps are priced taking the possibility of default explicitly into Models2 LIBORThe forward rate at timetbased on simple interest for lending in the interval[T1, T2]is given by4F(t, T1, T2) =1T2 T1(ZT1t ZT2tZT2t)(1)where, as before,ZTtis the timetprice of a zero-coupon bond maturing at timeT.

5 Note also that if wemeasure time in years, then (1) is consistent withF(t, T1, T2)being quoted as an annual rates are quoted assimply-compoundedinterest rates, and are quoted on an annual basis. Theaccrual period ortenor,T2 T1, is usually fixed at = 1/4or = 1/2corresponding to3months and6months,respectively. With a fixed value of in mind we can define the -year forward rate at timetwith maturityTasL(t, T) :=F(t, T, T+ ) =1 (ZTt ZT+ tZT+ t).(2)Note that the -year spot LIBOR rate at timetis then given byL(t, t).Remark1 LIBOR or theLondon Inter-Bank Offered Rate, is determined on a daily basis when theBritishBankers Association(BBA) polls a pre-defined list of banks with strong credit ratings for their interest highest and lowest responses are dropped and then the average of the remainder is taken to be the LIBOR rate.

6 Because there is some credit risk associated with these banks, LIBOR will be higher than thecorresponding rates on government treasuries. However, because the banks that are polled have strong creditratings the spread between LIBOR and treasury rates is generally not very large and is often less than100basispoints. Moreover, the pre-defined list of banks is regularly updated so that banks whose credit ratings havedeteriorated are replaced on the list with banks with superior credit ratings. This has the practical impact ofensuring that forward LIBOR rates will still only have a very modest degree of credit risk associated with s Formula for CapletsConsider now a caplet with payoff (L(T, T) K)+at timeT+ . The timetprice,Ct, is given byCt=BtEQt[ (L(T, T) K)+BT+ ]= ZT+ tEPT+ t[(L(T, T) K)+].

7 Where(Bt, Q)is an arbitrary numeraire-EMM pair and(ZT+ t, PT+ )is the forward measure-numeraire market convention is to quote caplet prices using Black s formula which equatesCtto a Black-Scholes likeformula so thatCt= ZT+ t[L(t, T) (log(L(t, T)/K) + 2(T t)/2 T t) K (log(L(t, T)/K) 2(T t)/2 T t)](3)where ( )is the CDF of a standard normal random variable. Note that (3) is what you would get forCtif youassumed thatdL(t, T) = L(t, T)dWT+ (t)whereWT+ (t)is aPT+ -Brownian motion and is an implied volatility that is used to quote s formula for caps is to equate the cap price with the sum of caplet prices given by (3) but where acommon is assumed. Similar formulae exist for floorlets and follows from a simple arbitrage argument. Prove it!Market Models3 Swap RatesConsider a payerforward startswap where the swap begins at some fixed timeTnin the future and expires attimeTM Tn.

8 We assume the accrual period is of length . Since payments are made in arrears, the firstpayment occurs atTn+1=Tn+ and the final payment atTM+1=TM+ . Then martingale pricing impliesthat the timet < Tnvalue,SWt, of this forward start swap isSWt= EQt M j=nBtBTj+1(L(Tj, Tj) R) whereRis the fixed (annualized) rate specified in the contract. A standard argument using the properties offloating-rate bond prices implies thatSWTn= 1 ZTM+1Tn R M+1 j=n+1 ZTjTn.(4)Exercise1 Prove (4).Equation (4) in turn easily implies (why?) that fort < Tnwe haveSWt=ZTnt ZTM+1t R M+1 j=n+ swap rateat timetis the valueR=R(t, Tn, TM)for whichSWt= 0. In particular,we obtainR=R(t, Tn, TM) =ZTnt ZTM+1t M+1j=n+1 ZTjt.(5)Thespotswap rate is then obtained by takingt=Tnin (5).Now consider the timetprice5of apayer-swaptionthat expires at timeTn> tand with payments of theunderlying swap taking place at timesTn+1.

9 , TM+1. Assuming a fixed rate of R(annualized) and a notionalprinciple of$1, the value of the option at expiration is given by the positive part of (4). It satisfiesCTn= 1 ZTM+1Tn R M+1 j=n+1 ZTjTn +.(6)Using (5) att=Tnwe can substitute for1 ZTM+1 Tnin (6) and find thatCTn= [R(Tn, Tn, TM) R]M+1 j=n+1 ZTjTn += M+1 j=n+1 ZTjTn [R(Tn, Tn, TM) R]+.(7)5 Note that in (6) we have implicitly assumed that the strike isk= Models4 Therefore we see that the swaption is like a call option on the swap rate. The timetvalue of the swaption,Ct,is then given by theQ-expectation of the right-hand-side of (7), suitably deflated by the s Formula for SwaptionsMarket convention, however, is to quote swaption prices via Black s formula which equatesCtto aBlack-Scholes-like formula so thatCt= M+1 j=n+1 ZTjt [R(t, Tn, TM) (log(R(t, Tn, TM)/ R) + 2(Tn t)/2 Tn t) R (log(R(t, Tn, TM)/ R) 2(Tn t)/2 Tn t)](8)where again is an implied volatility that is used to quote prices.

10 Note that the expression in (8) is what wewould obtain for the expectation of M+1 j=n+1 ZTjt [R(Tn, Tn, TM) R]+ifdR(t, Tn, TM) = R(t, Tn, TM) should be stated that Black s formulae for caps and swaptions did not originally correspond to prices that arisefrom the application of martingale pricing theory to some particular model . As originally conceived, they merelyprovided a framework for quoting market prices. The market models of these lecture notes will provide a belatedjustification for these formulae. We shall see that the justifications are mutually inconsistent, however, in that itis impossible for both formulae to hold simultaneously within the one The Term Structure of VolatilityThe term Structure of volatility6is a graph of volatility plotted against time to maturity.