Term Srucure Models: IEOR E4710 Spring 2005 c 2005 by Marin Haugh Marke Models One of he principal disadvanages of shor rae models, and HJM models more generally, is ha hey focus on unobservable insananeous ineres raes. The so-called marke models ha were developed 1 in he lae 90 s overcome his problem by direcly modelling observable marke raes such as LIBOR and swap raes. These models are sraighforward o calibrae and have quickly gained widespread accepance from praciioners. The firs marke models were acually developed in he HJM framework where he dynamics of insananeous forward raes are used via Iô s Lemma o deermine he dynamics of zero-coupon bonds. The dynamics of zero coupon bond prices were hen used, again via Iô s Lemma, o deermine he dynamics of LIBOR. Marke models are herefore no inconsisen wih HJM models. In hese lecure noes, however, we will prefer o specify he marke models direcly raher han derive hem in he HJM framework. In he process, we will derive Black s formulae for caples and swapions hereby demonsraing he consisency of hese formulae wih maringale pricing heory. Throughou hese noes, we will ignore he possibiliy of defaul or counerpary risk and rea LIBOR ineres raes as he fundamenal raes in he marke. Zero-coupon bond prices are hen compued using LIBOR raher han he defaul-free raes implied by he prices of governmen securiies. This does resul in a minor inconsisency in ha we price derivaive securiies assuming no possibiliy of defaul ye he ineres raes hemselves ha play he role of underlying securiy, i.e. LIBOR and swap raes, implicily incorporae he possibiliy of defaul. This inconsisency acually occurs in pracice when banks rade caps, swaps and oher insrumens wih each oher, and ignore he possibiliy of defaul when quoing prices. Insead, he associaed credi risks are kep o a minimum hrough he use of neing agreemens and by counerparies limiing he oal size of rades hey conduc wih one anoher. This approach can also be jusified when counerparies have a similar credi raing and similar exposures o one anoher. Finally, we should menion ha i is indeed possible 2, and someimes necessary, o explicily model credi risk even when we are pricing sandard securiies such as caps and swaps. I goes wihou saying of course, ha defaul risk needs o be modelled explicily when pricing credi derivaives and relaed securiies. 1 LIBOR, Swap Raes and Black s Formulae for Caps and Swapions We now describe wo paricularly imporan marke ineres raes, namely LIBOR and swap raes. We firs define LIBOR and forward LIBOR, and hen describe Black s formula for caples. Afer defining LIBOR we hen proceed o discuss swap raes and forward swap raes as well as describing Black s formula for swapions. In pracice, he underlying securiy for caps and swapions are LIBOR and LIBOR-based swap raes. Therefore by modelling he dynamics of hese raes direcly we succeed in obaining more realisic models han hose developed in he shor-rae or HJM framework. LIBOR The forward rae a ime based on simple ineres for lending in he inerval [T 1, T 2 ] is given by 3 F, T 1, T 2 ) = 1 T 2 T 1 ) Z T 1 Z T 2 Z T2 1) 1 See Milersen, Sandmann and Sondermann 1997), Brace, Gaarek and Musiela 1997), Jamshidian 1997) and Musiela and Rukowski 1997). 2 See chaper 11 of Cairns for a model where swaps are priced aking he possibiliy of defaul explicly ino accoun. 3 This follows from a simple arbirage argumen.
Marke Models 2 where, as before, Z T is he ime price of a zero-coupon bond mauring a ime T. Noe also ha if we measure ime in years, hen 1) is consisen wih F, T 1, T 2 ) being quoed as an annual rae. LIBOR raes are quoed as simply-compounded ineres raes, and are quoed on an annual basis. The accrual period or enor, T 2 T 1, is usually fixed a δ = 1/4 or δ = 1/2 corresponding o 3 monhs and 6 monhs, respecively. Wih a fixed value of δ in mind we can define he δ-year forward rae a ime wih mauriy T as L, T ) := F, T, T + δ) = 1 δ Noe ha he δ-year spo LIBOR rae a ime is hen given by L, ). Z T Z T +δ Z T +δ ). 2) Remark 1 LIBOR or he London Iner-Bank Offered Rae, is deermined on a daily basis when he Briish Bankers Associaion BBA) polls a pre-defined lis of banks wih srong credi raings for heir ineres raes. The highes and lowes responses are dropped and hen he average of he remainder is aken o be he LIBOR rae. Because here is some credi risk associaed wih hese banks, LIBOR will be higher han he corresponding raes on governmen reasuries. However, because he banks ha are polled have srong credi raings he spread beween LIBOR and reasury raes is generally no very large and is ofen less han 100 basis poins. Moreover, he pre-defined lis of banks is regularly updaed so ha banks whose credi raings have deerioraed are replaced on he lis wih banks wih superior credi raings. This has he pracical impac of ensuring ha forward LIBOR raes will sill only have a very modes degree of credi risk associaed wih hem. Black s Formula for Caples Consider now a caple wih payoff δlt, T ) K) + a ime T + δ. The ime price, C, is given by C = [ ] B E Q δlt, T ) K) + B T +δ = δz T +δ E P [ T +δ LT, T ) K) + ]. where B, Q) is an arbirary numeraire-emm pair and Z T +δ, P T +δ ) is he forward measure-numeraire pair. The marke convenion is o quoe caple prices using Black s formula which equaes C o a Black-Scholes like formula so ha C = δz T +δ [ L, T )Φ logl, T )/K) + σ 2 ) T )/2 logl, T )/K) σ 2 )] σ T )/2 KΦ T σ T where Φ ) is he CDF of a sandard normal random variable. Noe ha 3) is wha you would ge for C if you assumed ha dl, T ) = σl, T ) dw T +δ ) where W T +δ ) is a P T +δ -Brownian moion and σ is an implied volailiy ha is used o quoe prices. Black s formula for caps is o equae he cap price wih he sum of caple prices given by 3) bu where a common σ is assumed. Similar formulae exis for floorles and floors. Swap Raes Consider a payer forward sar swap where he swap begins a some fixed ime in he fuure and expires a ime T M. We assume he accrual period is of lengh δ. Since paymens are made in arrears, he firs paymen occurs a +1 = + δ and he final paymen a T = T M + δ. Then maringale pricing implies ha he ime < value, SW, of his forward sar swap is M SW = E Q B δ LT j, T j ) R) B Tj+1 j=n 3)
Marke Models 3 where R is he fixed rae annualized) specified in he conrac. A sandard argumen using he properies of floaing-rae bond prices implies ha SW Tn = 1 Z T This in urn easily implies why?) ha for < we have SW = Z Z T Rδ Rδ Definiion 1 The forward swap rae is he value R = R,, T M ) for which SW = 0. In paricular, we obain The swap rae is hen obained by aking = in 4)... R = R,, T M ) = ZTn Z T δ. 4) ZT j Now consider he ime price 4 of a payer-swapion ha expires a ime > and wih paymens of he underlying swap aking place a imes +1,..., T. Assuming a fixed rae of R annualized) and a noional principle of $1, he value of he opion a expiraion is given by C Tn = 1 Z T Rδ Z Tj Subsiuing 4) a = ino 5) we find ha [ C Tn = δ R,, T M ) R ] = δ Z Tj +. 5) + [ R,, T M ) R] +. 6) Therefore we see ha he swapion is like a call opion on he swap rae. The ime value of he swapion, C, is hen given by he Q-expecaion of he righ-hand-side of 6), suiably deflaed by he numeraire. Black s Formula for Swapions Marke convenion, however, is o quoe swapion prices via Black s formula which equaes C o a Black-Scholes-like formula so ha [ C = δ logr,, T M )/ R,, T M )Φ R) ) + σ 2 )/2 σ RΦ logr,, T M )/ R) )] σ 2 )/2 σ 7) where again σ is an implied volailiy ha is used o quoe prices. Noe ha he expression in 7) is wha we would obain for he expecaion of δ [ R,, T M ) R 4 Noe ha in 5) we have implicily assumed ha he srike is k = 0. ] +
Marke Models 4 if dr,, T M ) = σr,, T M ) dw. I should be saed ha Black s formulae for caps and swapions did no originally correspond o prices ha arise from he applicaion of maringale pricing heory o some paricular model. As originally conceived, hey merely provided a framework for quoing marke prices. The marke models of hese lecure noes will provide a belaed jusificaion for hese formulae. We shall see ha he jusificaions are muually inconsisen, however, in ha i is impossible for boh formulae o hold simulaneously wihin he one model. 2 The Term Srucure of Volailiy The erm srucure of volailiy 5 is a graph of volailiy ploed agains ime o mauriy, τ. There are of course many definiions of volailiy and care is needed in specifying which definiion is inended. Some commonly used definiions of he erm srucure of volailiy a ime include: 1. The volailiy of spo raes Y +τ as a funcion of τ. Depending on he model under consideraion, his volailiy may be available in closed form and he model calibraed o hisorical or implied raes. 2. The volaily, σ, + τ), of insananeous forward raes, f, + τ). 3. The implied volailiy, σ, given by Black s formula for caples. This will vary wih ime o mauriy and can be compued a any ime from marke prices for caples. 4. The implied volailiy, σ, given by Black s formula for caps. Again his will vary wih ime o mauriy and can be compued a any ime from marke prices for caps. When calibraing erm srucure models i is common o calibrae using boh marke prices and he erm srucure of volailiy. As a resul we ofen wan o work wih models ha allow for a rich variey of erm srucures of volailiy as well of course, as a rich variey of erm srucures of ineres raes. 3 Numeraires and Zero-Coupon Bond Prices While he cash accoun wih B := exp ) 0 r s ds has been he defaul numeraire o dae, we will no work wih he cash accoun as our numeraire in he conex of marke models. The reason is clear: in marke models we ake LIBOR raes or swap raes) wih a fixed enor, δ, in mind, as our fundamenal ineres raes. I would herefore be very inconvenien as well as defeaing he purpose) if we had o deermine he insananeous shor rae a each poin in ime. As a resul we will generally work wih oher numeraire-emm pairs as described below. Bu firs we will fix he mauriies or enor daes o which our marke models will apply. A ime we could in principal have LIBOR raes, L, T ), available for all T >. This is unnecessary, however, as he prices of mos imporan securiies, e.g. caps, floors, swaps, swapions, Bermudan swapions, ec., are deermined by he raes LIBOR or swap) applying o only a finie se of mauriies. We herefore fix in advance a se of enor daes 6 0 := T 0 < T 1 < T 2 <... < T M < T wih δ i := T i+1 T i, i = 0, 1,..., M. While he δ i s are usually nominally equal, e.g. 1/4 or 1/2, day-coun convenions will resuls in slighly differen values for each δ i. We le Z n denoe he ime price of a zero-coupon bond mauring a ime > for 5 Quans in he fixed-income indusry commonly refer o he erm-srucure of volailiy when discussing fixed-income derivaives and models. In his secion we briefly give some possible definiions of he erm-srucure of volailiy bu we will no need hese definiions elsewhere in he course. 6 The noaion and seup in his secion and he nex will borrow heavily from Secion 3.7 in Mone Carlo Mehods in Financial Engineering by Glasserman.
Marke Models 5 n = 1,..., M. Similarly, we use L n ) o denoe he ime forward rae applying o he period [, +1 ] for n = 0, 1,..., M. In paricular, a simple arbirage argumen hen implies L n ) = Zn Z n+1 δ n Z n+1, for 0, n = 0, 1,..., M. 8) Wih some work we can inver 8) o obain an expression for bond prices in erms of LIBOR raes. We find Z n T i = j=i 1 1 + δ j L j T i ) for n = i + 1,..., M + 1. 9) Equaion 9) only deermines he bonds prices a he fixed mauriy daes. However, for an arbirary dae we can easily check ha Z n = Z φ) j=φ) 1 1 + δ j L j ) where we define φ) o be nex enor dae afer ime. Tha is, for 0. 10) φ) := min {i : < T i}. i=1,..., Remark 2 The presence of Z φ) in 10) suggess ha i may no be sufficien o model only he dynamics of he forward LIBOR raes, L n ), when we specify a marke model since hey are no sufficien o deermine a an arbirary ime. However, as we shall see below, his will no prove o be a problem as he φ) facor vanishes upon deflaing by he numeraire. Z φ) Exercise 1 Prove equaions 9) and 10). Numeraire-EMM Pairs The following numeraire-emm pairs are commonly used in marke models: 1. The spo measure, Q, assumes ha B is he numeraire where B is defined as follows. sar wih $1 a = 0 and hen purchase 1/Z 1 0 of he zero-coupon bonds mauring a ime T 1 a ime T 1 reinves he funds in he zero-coupon bond mauring a ime T 2 by coninuing in his way, we see ha a ime he spo numeraire will be worh B = Z φ) φ) 1 j=0 [1 + δ j L j T j )]. 11) 2. The forward measure, P T, akes he zero-coupon bond mauring a ime T as numeraire. We have seen his numeraire-emm pair already. 3. The swap measure, P X, is useful for pricing swapions analyically. I akes he numeraire o be X = δ M k=1 Zk, which is indeed a posiive securiy price process. Deflaing Zero-Coupon Bond Prices by he Spo Numeraire Equaions 10) and 11) show ha deflaed 7 zero-coupon bond prices, D n, saisfy φ) 1 D n ) = 1 1 for 0. 12) 1 + δ j L j T j ) 1 + δ j L j ) j=0 In paricular, we see ha he facor, Z φ), has vanished. 7 We will ake he spo numeraire o be he defaul numeraire. j=φ)
Marke Models 6 4 Arbirage-Free LIBOR Dynamics Dynamics under he Spo Measure We assume ha he dynamics of he LIBOR raes saisfy dl n ) = µ n )L n ) d + L n )σ n ) T dw ), 0, n = 1,..., M 13) where W ) is a d-dimensional Brownian moion, and µ n ) and σ n ) are adaped processes ha may depend on he curren vecor of ineres raes L) := L 1 ),..., L M )). The assumpion of no arbirage and he posiiviy of deflaed bond prices implies he exisence of an R d -valued process ν n ) such ha dd n ) = D n )ν ) dw ). 14) We could apply Iô s Lemma direcly o our expression for D n ) in 12) bu his would be awkward. Insead we will apply Iô s Lemma o Y n ) := log D n ). We see from 14) ha dy n ) = 1 2 ν n) 2 d + ν ) dw ) 15) We can also find an alernaive expression for dy n ) using 12). In paricular, noing ha he firs facor in 12) is consan beween mauriies, we obain via Iô s Lemma dy n ) = = j=φ) j=φ) d log 1 + δ j L j )) δ j µ j )L j ) 1 + δ j L j ) δ2 j L j) 2 σj T )σ ) j) 2 1 + δ j L j )) 2 d j=φ) δ j L j )σj T ) dw ).16) 1 + δ j L j ) Comparing he volailiy erms in 15) and 16) hen gives us ν n ) = j=φ) δ j L j )σ j ) 1 + δ j L j ). 17) We would now like o find an expression for he µ j s. Towards his end, we could compare he drif erms in 15) and 16), and his is easy o do when n = 2 and φ) = 1. Afer some sraighforward algebra, we easily find 8 More generally, we obain µ 1 ) = σ T 1 )ν 2 ), 0 T 1. µ n ) = σ ) ν n+1 ) = n j=φ) δ j L j )σn T )σ j ). 18) 1 + δ j L j ) We could have obained 18) by again comparing he drif erms in 15) and 16) bu his appears o be very cumbersome. Exercise 2 insead provides a more elegan approach. Exercise 2 Use inducion o esablish ha he drifs, µ n ), mus saisfy 18) under he no-arbirage assumpion. In paricular, firs assume µ 1,..., µ have been chosen consisen wih he Q-maringale assumpion on D 1,..., D n. Show ha D n+1 is a maringale if and only if L n D n+1 and hen apply Iô s Lemma o obain 9 18). 8 Noe ha L 1 ), and herefore µ 1 ), do no have any meaning for > T 1. 9 See Glasserman, page 170.
Marke Models 7 We herefore obain ha he arbirage free Q-dynamics of he forward LIBOR raes are given by n dl n ) = δ j L j )σn T )σ j ) L n ) d + L n )σ n ) T dw ), 0, n = 1,..., M. 19) 1 + δ j L j ) j=φ) Dynamics under he Forward Measure Consider now he case where we use he forward measure, P, and he associaed numeraire, Z. We now find ha deflaed zero-coupon bond prices are given by D n ) = M 1 + δ j L j )). 20) j=n We would like o find he marke-price-of-risk process, η ) R d, ha relaes he Q-Brownian moion W ) o he he P Brownian moion, W ), so ha dw ) = dw ) η) d. 21) There are a number of ways o do his bu perhaps he easies is he approach we followed wih he Vasicek model when we swiched o he forward measure. Equaion 20) implies D M ) = 1 + δ M L M ) so ha dd M ) = δ M dl M ). 22) We now subsiue for dl M ) in 22) using 19) evaluaed a n = M, and hen subsiue for W ) using 21). Since D M ) is a P -maringale we find ha η) = M j=φ) δ j L j )σ j ) 1 + δ j L j ). In paricular, we obain he arbirage-free P -dynamics of he forward LIBOR raes are given by dl n ) = M δ j L j )σn T )σ j ) L n ) d + L n )σ n ) T dw ), 1 + δ j L j ) 0, n = 1,..., M. Black s Formula for Caples We are now in a posiion o derive Black s formula see 3)) for caple prices. If we ake n = M in 23), hen we obain dl M ) = L M )σ M ) T dw ) 24) implying in paricular 10 ha L M ) is a P -maringale. If we assume ha σ M ) is a deerminisic funcion, hen we easily see ha L M ) is lognormally disribued. In paricular, we obain log L M ) N logl M 0)) 1 2 0 σ M s) 2 ds, We can now obain 3) if we le T M = T and reinerpre σ appropriaely. 0 ) σ M s) 2 ds. Noe also ha here is no problem when we ake σ M ) o be deerminisic in 24) which conrass wih he HJM framework. This is because while he numeraors in he drif of 19) are quadraic in L j ), he 1 + δ j L j ) 10 Subjec, as usual, o echnical condiions. 23)
Marke Models 8 erm in he denominaor ensures ha here is no possibiliy of explosion in he SDE. This is a furher advanage of he marke model framework where we model simple LIBOR raes raher han insananeous forward raes. BGM s Approximaion for Swapion Prices In heir original paper, Brace, Gaarek and Musiela BGM) succeeded in deriving Black s formula for caples and hereby demonsraed is consisency wih maringale pricing. Their framework did no enable hem o derive Black s formula for swapions, however. Insead hey provided an analyic approximaion for swapion prices ha we will no describe 11 here. I is worh menioning, however, ha heir approximaion works well in pracice and provides swapion prices ha are very close o hose obained via Mone Carlo simulaion. 5 A Swap Marke Model for Pricing Swapions Consider a payer-swapion ha expires a ime > and wih paymens of he underlying swap aking place a imes +1,..., T. Assuming a fixed rae of R annualized) and a noional principle of $1, we showed in 6) ha he ime price of he swapion is given by C Tn = δ This implies ha he ime price of he swapion, C, saisfies C = X E P x [ R,, T M ) R] +. 25) δ ZT j ) [R,, T M ) R X Tn ] + 26) where X is he ime price of he chosen numeraire securiy and P x is he corresponding EMM. A paricularly convenien choice of numeraire ha we will adop is he porfolio 12 consising of δ unis of each of he zero-coupon bonds mauring a imes +1,..., T. Then X = δ ZT j and we find [ [ C = δ R] ] + E P x R,, T M ) 27) Jamshidian 1997) developed a erm srucure framework where a any ime he curren erm srucure was given in erms of he forward swap raes, R, T i, T M ) for i = φ),..., M. In paricular, he showed ha i was possible o assume ha he P x -dynamics of R,, T M ) saisfy dr,, T M ) = R,, T M )σ) T dw x ) 28) where σ) is a deerminisic vecor of volailiies. This implies ha he forward swap rae is lognormally disribued so we can obain 13 Black s formula for swapion prices 7). Remark 3 When we model swap raes direcly as in 28) we say ha we have a swap marke model. This conrass wih he LIBOR marke models of Secion 4. 11 See Chaper 9 of Cairns for a derivaion. 12 There is no difficuly aking a porfolio of securiies raher han a fixed individual securiy as he numeraire. More generally in fac, we could ake a dynamic self-financed porfolio as he numeraire securiy, assuming of course ha i has sricly posiive value a all imes. 13 Of course we need o reinerpre σ in 7) in erms of he deerminisic funcion σ) in 28).
Marke Models 9 Remark 4 The advanage of Black s swapion formula is ha i is elegan and exac, whereas he BGM formula is cumbersome and only an approximaion. However, he BGM approximaion is consisen wih Black s formulae for caples and caps whereas Black s swapion formula is no. Indeed, i may be shown 14 ha if forward LIBOR raes have deerminisic volailiies hen i i is no possible for swap raes o also have deerminisic volailiies. Therefore Black s formulae for caples and swapions canno boh hold wihin he same model. Tha said, wihin he LIBOR marke framework wih deerminisic volailiies, i can be argued ha forward swap raes are approximaely lognormally disribued. 6 Mone-Carlo Simulaion While i is possible o price many commonly raded derivaive securiies such as caps, floors and swapions in he marke model framework, i is in general necessary o use Mone Carlo mehods o price oher securiies. Indeed, if our marke model has sochasic volailiy funcions hen i will ypically be necessary o also use Mone Carlo mehods o price even caps, floors and swapions. The ypical approach is o use some discreizaion scheme such as he Euler scheme when performing he Mone Carlo simulaion. This does no creae oo much of a compuaional burden as we will only need o simulae he SDE s describing he forward LIBOR dynamics for a finie number of mauriies. This conrass wih he HJM framework where we had infiniely many mauriies which mean i was pracically infeasible o use a very fine discreizaion. This in urn promped he developmen of he discree-ime HJM framework wih he resuling discree-ime arbirage-free resricion on he drif. I is also possible o develop discree-ime arbirage-free marke models in a manner ha is analogous o our discree-ime HJM developmen. As described above, however, he need o do so is no as urgen as i is pracically feasible o simulae he marke model SDE s on a sufficienly fine grid and his is wha is ypically done in pracice. Noneheless, Glasserman s Mone Carlo Mehods for Financial Engineering describes how o build discree-ime arbirage-free marke models. I urns ou o be inconvenien o choose he LIBOR raes as he fundamenal variables ha we choose o discreize. Insead i is more convenien o direcly model deflaed bond prices as discree-ime Q-maringales 15 and o define LIBOR raes in erms of hese bond prices. Oher choices of discreizaion variable are also possible. As usual, we can choose o simulae under any EMM ha we prefer and all of he usual variance reducion echniques may be employed. 14 This is done by applying Iô s Lemma o he forward swap rae given in 4). 15 This ensures he discree-ime model is arbirage-free.