12. Market LIBOR Models

Transcription

1 12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also, i is he floaing rae commonly used in ineres rae swap agreemens in inernaional financial markes in domesic financial markes as he reference ineres rae for a floaing rae loans i is cusomary o ake a prime rae or a base rae. LIBOR is deermined by rading beween banks and changes coninuously as economic condiions change. For more informaion on marke convenions relaed o he LIBOR and Eurodollar fuures, we refer o Sec In his chaper, we presen an overview of recenly developed mehodologies relaed o he arbirage-free modelling of marke raes, such as LIBORs. In conras o more radiional aproaches, erm srucure models developed recenly by, among ohers, Milersen e al. 1997, Brace e al. 1997, Musiela and Rukowski 1997, Jamshidian 1997a, Hun e al. 1996, 2, Hun and Kennedy 1996, 1997, 1998a, and Andersen and Andreasen 2b, are ailored o handle he mos acively raded ineres-rae opions, such as caps and swapions. For his reason, hey ypically enjoy a higher degree of racabiliy han he classical erm srucure models based on he diffusionype behavior of insananeous spo or forward raes. Recall ha he Heah-Jarrow-Moron mehodology of erm srucure modelling is based on he arbirage-free dynamics of insananeous, coninuously compounded forward raes. The assumpion ha insananeous raes exis is no always convenien, since i requires a cerain degree of smoohness wih respec o he enor i.e., mauriy of bond prices and heir volailiies. An alernaive consrucion of an arbirage-free family of bond prices, making no reference o he insananeous, coninuously compounded raes, is in some circumsances more suiable. The firs sep in his direcion was aken by Sandmann and Sondermann 1994, who focused on he effecive annual ineres rae cf. Sec This idea was furher developed by Goldys e al. 1994, Musiela 1994, Sandmann e al. 1995, Milersen e al and Brace e al I is worh poining ou ha in all hese papers, he HJM framework is adoped a leas implicily. For insance, Goldys e al inroduce a HJM-ype model based on he rae j, T, which is relaed o he insan-

2 Models of LIBOR aneous forward rae hrough he formula 1 + j, T = e f,t. The model pu forward in his paper assumes a deerminisic volailiy funcion for he process j, T. A slighly more general case of nominal annual raes q, T, which saisfy δ represening he duraion of each compounding period 1 + δq, T 1/δ = e f,t, was sudied by Musiela 1994, who assumes he deerminisic volailiy γ, T of each nominal annual rae q, T. This implies he following form of he coefficien σ in he dynamics of he insananeous forward rae σ, T = δ 1 1 e δf,t γ, T, so ha he model is indeed well-defined ha is, insananeous forward raes, and hus also he nominal annual raes, do no explode. Unforunaely, hese models do no give closed-form soluions for zero-coupon bond opions, and hus a numerical approach o opion pricing is required. Milersen e al focus on he acuarial or effecive forward raes a, T, U saisfying 1 + a, T, T + δ δ = exp T +δ T f, u du. They show ha a closed-form soluion for he bond opion price is available when δ = 1. More specifically, an ineres rae cap is priced according o he marke sandard see Sec for deails. However, he model is no explicily idenified and is arbirage-free feaures are no examined, hus leaving open he quesion of pricing oher ineres rae derivaives. These problems were addressed in par in a paper by Sandmann e al. 1995, where a lognormal-ype model based on an add-on forward rae add-on yield f s, T, T + δ, where T +δ 1 + δf s, T, T + δ = exp T f, u du, was analyzed. Finally, using a differen approach, Brace e al explicily idenify he dynamics of all raes f s, T, T +δ under he maringale measure P and analyze he properies of he model. Le us summarize he conen of his chaper. We sar by describing in Sec forward and fuures LIBORs. The properies of he LIBOR in he Gaussian HJM model are also deal wih in his secion. Subsequenly, in Sec we presen various approaches o LIBOR marke models. Furher properies of hese models are examined in Sec In Sec we describe ineres rae cap and floor agreemens. Nex, we provide in Sec he valuaion resuls for hese conracs wihin he framework of he Gaussian HJM model. In Sec. 12.6, we deal wih he valuaion of coningen claims wihin he framework of he lognormal LIBOR marke model. In he las secion, we presen briefly some exensions of his model.

3 12.1 Forward and Fuures LIBORs 12.1 Forward and Fuures LIBORs 433 We shall frequenly assume ha we are given a prespecified collecion of rese/selemen daes T < T 1 < < T n, referred o as he enor srucure. Also, we shall wrie δ j = T j T j 1 for j = 1,..., n. As usual, B, T sands for he price a ime of a T -mauriy zero-coupon bond, P is he spo maringale measure, while P Tj respecively, P Tj+1 is he forward maringale measure associaed wih he dae T j respecively, T j+1. The corresponding Brownian moions are denoed by W and W T j respecively, W T j+1. Also, we wrie F B, T, U = B, T /B, U. Finally, π X is he value ha is, he arbirage price a ime of a European claim X. Our firs ask is o examine hose properies of ineres rae forward and fuures conracs ha are universal, in he sense ha do no rely on specific assumpions imposed on a paricular model of he erm srucure of ineres raes. To his end, we fix an index j, and we consider various ineres raes relaed o he period [T j, T j+1 ] One-period Swap Seled in Arrears Le us firs consider a one-period swap agreemen seled in arrears; i.e., wih he rese dae T j and he selemen dae T j+1 more realisic muli-period swap agreemens are examined in Chap. 13. By he conracual feaures, he long pary pays δ j+1 κ and receives B 1 T j, T j+1 1 a ime T j+1. Equivalenly, he pays an amoun Y 1 = 1 + δ j+1 κ and receives Y 2 = B 1 T j, T j+1 a his dae. The values of hese payoffs a ime T j are π Y 1 = B, T j δ j+1 κ, π Y 2 = B, T j. The second equaliy above is rivial, since he payoff Y 2 is equivalen o he uni payoff a ime T j. Consequenly, for any fixed T j, he value of he forward swap rae ha makes he conrac worhless a ime can be found by solving for κ = κ he following equaion π Y 1 = B, T j δ j+1 κ = B, Tj = π Y 2. I is hus apparen ha κ = B, T j B, T j+1, [, T j ]. δ j+1 B, T j+1 Noe ha κ coincides wih he forward LIBOR L, T j which, by convenion, is se o saisfy 1 + δ j+1 L, T j def = B, T j B, T j+1. I is also useful o observe ha 1 + δ j+1 L, T j = F B, T j, T j+1 = E PTj+1 B 1 T j, T j+1 F, 12.1

4 Models of LIBOR where he las equaliy is a consequence of he definiion of he forward measure P Tj+1. We conclude ha in order o deermine he forward LIBOR L, T j, i is enough o find he forward price of he claim B 1 T j, T j+1 for he selemen dae T j+1. Furhermore, i is eviden ha he process L, T j necessarily follows a maringale under he forward probabiliy measure P Tj+1. Recall ha in he HJM framework, we have df B, T j, T j+1 = F B, T j, T j+1 b, T j b, T j+1 dw T j under P Tj+1, where b, T is he price volailiy of he T -mauriy zero-coupon bond. On he oher hand, L, T j can be shown o admi he following represenaion dl, T j = L, T j λ, T j dw T j+1 for a cerain adaped process λ, T j. Combining he las wo formulas wih 12.1, we arrive a he following fundamenal relaionship δ j+1 L, T j 1 + δ j+1 L, T j λ, T j = b, T j b, T j+1, [, T j ] I is worh sressing ha equaliy 12.3 will play an essenial role in he consrucion of he so-called lognormal LIBOR marke model. For insance, in he consrucion based on he backward inducion, relaionship 12.3 will allow us o specify uniquely he forward measure for he dae T j, provided ha P Tj+1, W T j+1 and he volailiy λ, T j are known we may posulae, for insance, ha λ, T j is a given deerminisic funcion. Recall ha in he HJM framework he Radon-Nikodým densiy of P Tj wih respec o P Tj+1 is known o saisfy dp Tj = E Tj dp Tj+1 In view of 12.3, we hus have b, Tj b, T j+1 dw T j dp Tj δ j+1 L, T j = E Tj dp Tj δ j+1 L, T j λ, T j dw T j+1. For our furher purposes, i is also useful o observe ha his densiy admis he following represenaion dp Tj dp Tj+1 = cf B T j, T j, T j+1 = c 1 + δ j+1 LT j, T j, P Tj+1 -a.s., 12.5 where c > is a normalizing consan, and hus we have ha dp Tj dp Tj+1 F = cf B, T j, T j+1 = c 1 + δ j+1 L, T j, P Tj+1 -a.s., for any dae T j.

5 P Tj 12.1 Forward and Fuures LIBORs 435 Finally, he dynamics of he process L, T j under he probabiliy measure are given by a somewha involved sochasic differenial equaion δj+1 L, T j λ, T j 2 dl, T j = L, T j d + λ, T j dw Tj. 1 + δ j+1 L, T j As we shall see in wha follows, i is neverheless no hard o deermine he probabiliy law of L, T j under he forward measure P Tj a leas in he case of he deerminisic volailiy funcion λ, T j One-period Swap Seled in Advance Consider now a similar swap ha is, however, seled in advance ha is, a ime T j. Our firs goal is o deermine he forward swap rae implied by such a conrac. Noe ha under he presen assumpions, he long pary formally pays an amoun Y 1 = 1 + δ j+1 κ and receives Y 2 = B 1 T j, T j+1 a he selemen dae T j which coincides here wih he rese dae. The values of hese payoffs a ime T j admi he following represenaions and π Y 1 = B, T j 1 + δ j+1 κ, π Y 2 = B, T j E PTj B 1 T j, T j+1 F. The value κ = κ of he modified forward swap rae ha makes he swap agreemen seled in advance worhless a ime can be found from he equaliy π Y 1 = π Y 2, where π Y 1 = B, T j 1 + δ j+1 κ and I is clear ha π Y 2 = B, T j E PTj B 1 T j, T j+1 F. κ = δ 1 j+1 E PTj B 1 T j, T j+1 F 1. We are in a posiion o inroduce he modified forward LIBOR L, T j by seing L, T j def = δ 1 j+1 E PTj B 1 T j, T j+1 F 1, [, T j ]. Le us make wo remarks. Firs, i is clear ha finding he modified LIBOR L, T j is essenially equivalen o pricing he claim B 1 T j, T j+1 a T j more precisely, we need o know he forward price of his claim for he dae T j. Second, i is useful o observe ha 1 BTj, T j+1 L, T j = E PTj F = E PTj LT j, T j F. δ j+1 BT j, T j+1

6 Models of LIBOR In paricular, i is eviden ha a he rese dae T j he wo forward LIBORs inroduced above coincide, since manifesly LT j, T j = 1 BT j, T j+1 δ j+1 BT j, T j+1 = LT j, T j. To summarize, he sandard forward LIBOR L, T j saisfies wih he iniial condiion L, T j = E PTj+1 LT j, T j F, [, T j ], L, T j = B, T j B, T j+1, δ j+1 B, T j+1 while for he modified LIBOR L, T j we have wih he iniial condiion L, T j = E PTj LT j, T j F, [, T j ], L, T j = δ 1 j+1 E PTj B 1 T j, T j+1 1. Noe ha he las condiion depends no only on he iniial erm srucure, bu also on he volailiies of bond prices see, e.g., formula below Eurodollar Fuures As was menioned in Sec. 9.3, Eurodollar fuures conrac is a fuures conrac in which he LIBOR plays he role of an underlying asse. By convenion, a he conrac s mauriy dae T j, he quoed Eurodollar fuures price ET j, T j is se o saisfy ET j, T j def = 1 δ j+1 LT j. Equivalenly, in erms of he price of a zero-coupon bond we have ET j, T j = 2 B 1 T j, T j+1. From he general properies of fuures conracs, i follows ha he Eurodollar fuures price a ime T j equals and hus E, T j def = E P ET j, T j = 1 δ j+1 E P LTj, T j F E, T j = 2 E P B 1 T j, T j+1 F Recall ha he probabiliy measure P represens he spo maringale measure in a given model of he erm srucure. I seems naural o inroduce he concep of he fuures LIBOR, associaed wih he Eurodollar fuures conrac, hrough he following definiion.

7 12.1 Forward and Fuures LIBORs 437 Definiion Le E, T j be he Eurodollar fuures price a ime for he selemen dae T j. The implied fuures LIBOR L f, T j saisfies E, T j = 1 δ j+1 L f, T j, [, T j ] I follows immediaely from ha he following equaliy is valid 1 + δ j+1 L f, T j = E P B 1 T j, T j+1 F. Equivalenly, we have L f, T j = δ 1 j+1 E P B 1 T j, T j+1 F 1 = E P LT j, T j F. I is hus clear ha he fuures LIBOR follows a maringale under he spo maringale measure P LIBOR in he Gaussian HJM Model In his secion, we make a sanding assumpion ha he bond price volailiies b, T j are deerminisic funcions, ha is, we place ourselves wihin he Gaussian HJM framework. In his case, i is no hard o express forward and fuures LIBORs in erms of bond prices and bond price volailiies. Furhermore, as soon as he dynamics of various raes under forward probabiliy measures are known explicily, i is sraighforward o value ineres-rae sensiive derivaives. Recall ha in he HJM framework we have df B, T j, T j+1 = F B, T j, T j+1 bs, T j bs, T j+1 dw T j+1 wih he erminal condiion F B T j, T j, T j+1 = B 1 T j, T j+1. Also, he spo and forward Brownian moions are known o saisfy and dw Tj = dw Tj+1 bs, T j bs, T j+1 d, 12.8 dw = dw T j + bs, T j d In view of he relaionships above, i is quie sandard o esablish he following proposiion see Flesaker 1993b for relaed resuls. I is worh poining ou ha in he presen framework here are no ambiguiies in he definiion of he spo probabiliy measure his should be conrased wih he case of he discree-enor lognormal model of forward LIBORs, in which he spo measure is no uniquely defined. For conciseness, we shall frequenly wrie F B = F B, T j, T j+1. Also, we wrie, as usual, γ, T j, T j+1 = b, T j b, T j+1 o denoe he volailiy of he process F B, T j, T j+1.

8 Models of LIBOR Proposiion Assume he Gaussian HJM model of he erm srucure of ineres raes. Then he following relaionships are valid 1 + δ j+1 L, T j = F B, T j, T j+1, 12.1 Tj γu,t 1 + δ j+1 L, Tj = F B, T j, T j+1 e j,t j+1 2 du, δ j+1 L f, T j = F B, T j, T j+1 e Tj bu,t j+1 γu,t j,t j+1 du Proof. For breviy, we shall wrie F B = F B, T j, T j+1. The firs formula is in fac universal see For he second, noe ha cf dw Tj df B = F B γ, T j, T j+1 + γ, T j, T j+1 d. Consequenly, T j dw Tj F B T j = F B exp γ u u + γ u du 1 2 Tj γ u 2 du, where we wrie γ u = γu, T j, T j+1. Since B 1 T j, T j+1 = F B T j, upon aking condiional expecaion wih respec o he σ-field F, we obain Furhermore, we have df B = F B γ dw b, T j+1 d and hus T j F B T j = F B exp γ u dw u bu, T j+1 du 1 2 This leads o equaliy Tj γ u 2 du. Dynamics of he forward LIBORs are also easy o find, as he following corollary shows. Corollary We have dl, T j = δ 1 j δj+1 L, T j γ, T j, T j+1 dw Tj+1, d L, T j = δ 1 j δj+1 L, Tj γ, T j, T j+1 dw Tj, dl f, T j = δ 1 j δj+1 L f, T j γ, T j, T j+1 dw Proof. Formula is an immediae consequence of 12.1 combined wih Expressions and can be derived by applying Iô s rule o equaliies and respecively. From Corollary , i is raher clear ha closed-form expressions for values of opions wrien on forward or fuures LIBORs are no available in he Gaussian HJM framework.

9 12.2 Ineres Rae Caps and Floors 12.2 Ineres Rae Caps and Floors 439 An ineres rae cap known also as a ceiling rae agreemen, or briefly CRA is a conracual arrangemen where he granor seller has an obligaion o pay cash o he holder buyer if a paricular ineres rae exceeds a muually agreed level a some fuure dae or daes. Similarly, in an ineres rae floor, he granor has an obligaion o pay cash o he holder if he ineres rae is below a preassigned level. When cash is paid o he holder, he holder s ne posiion is equivalen o borrowing or deposiing a a rae fixed a ha agreed level. This assumes ha he holder of a cap or floor agreemen also holds an underlying asse such as a deposi or an underlying liabiliy such as a loan. Finally, he holder is no affeced by he agreemen if he ineres rae is ulimaely more favorable o him han he agreed level. This feaure of a cap or floor agreemen makes i similar o an opion. Specifically, a forward sar cap or a forward sar floor is a srip of caples floorles, each of which is a call pu opion on a forward rae respecively. Le us denoe by κ and by δ j he cap srike rae and he lengh of a caple respecively. We shall check ha an ineres rae caple i.e., one leg of a cap may also be seen as a pu opion wih srike price 1 per dollar of principal ha expires a he caple sar day on a discoun bond wih face value 1 + κδ j mauring a he caple end dae. This propery makes he valuaion of a cap relaively simple; essenially, i can be reduced o he problem of opion pricing on zero-coupon bonds. Similarly o he swap agreemens examined in he nex chaper, ineres rae caps and floors may be seled eiher in arrears or in advance. In a forward cap or floor wih he noional principal N seled in arrears a daes T j, j = 1,..., n, where T j T j 1 = δ j he cash flows a imes T j are and N LT j 1, T j 1 κ + δj N κ LT j 1, T j 1 + δj respecively, where he spo LIBOR LT j 1, T j 1 is deermined a he rese dae T j 1, and i formally saisfies BT j 1, T j 1 = 1 + δ j LT j 1, T j The arbirage price a ime T of a forward cap, denoed by FC, is FC = n j=1 E P B B Tj LTj 1, T j 1 κ + δj F We have assumed here, wihou loss of generaliy, ha he noional principal N = 1. This convenion will be in force hroughou he res of his chaper. Le us consider a caple i.e., one leg of a cap wih rese dae T j 1 and selemen dae T j = T j 1 + δ j.

10 Models of LIBOR The value a ime of a caple equals for simpliciy, we wrie δ j = 1+κδ j { B Cpl = E P δ 1 j BTj 1, T j 1 1 +δj } κ F B Tj { B 1 = E P B Tj BT j 1, T j δ + } j F { B 1 = E P B Tj 1 BT j 1, T j δ + BTj 1 } j E P FTj 1 F B Tj { B = E P 1 B δ + } j BT j 1, T j F Tj 1 = B, T j 1 E PTj 1 { 1 δj BT j 1, T j + F }, where he las equaliy was deduced from Lemma I is apparen ha a caple is essenially equivalen o a pu opion on a zero-coupon bond; i may also be seen as an opion on a one-period forward swap. Since he cash flow of he j h caple a ime T j is a F Tj 1 -measurable random variable, we may also use Corollary o express he value of he cap in erms of expecaions under forward measures. Indeed, from 9.38 we have n FC = B, T j 1 E PTj 1 BT j 1, T j LT j 1, T j 1 κ + δj F. j=1 Consequenly, using we ge once again he equaliy n 1 FC = B, T j 1 E PTj 1 δj BT j 1, T j + F, j=1 which is valid for every [, T j 1 ]. Finally, he equivalence of a cap and a pu opion on a zero-coupon bond can be explained in an inuiive way. For his purpose, i is enough o examine wo basic feaures of boh conracs: he exercise se and he payoff value. Le us consider he j h caple. A caple is exercised a ime T j 1 if and only if LT j 1 κ >, or equivalenly, if BT j 1, T j 1 = 1 + LT j 1, T j 1 T j T j 1 > 1 + κδ j = δ j. 1 The las inequaliy holds whenever BT j 1, T j < δ j. This shows ha boh of he considered opions are exercised in he same circumsances. If exercised, he caple pays δ j LT j 1, T j 1 κ a ime T j, or equivalenly, δ j BT j 1, T j LT j 1, T j 1 κ = δ δ 1 j j BT j 1, T j a ime T j 1. This shows once again ha he j h caple, wih srike level κ and nominal value 1, is essenially equivalen o a pu opion wih srike price 1 + κδ j 1 and nominal value 1 + κδ j wrien on he corresponding zero-coupon bond wih mauriy T j.

11 12.3 Valuaion in he Gaussian HJM Model 441 The price of a forward floor a ime [, T ] equals FF = n j=1 E P B B Tj κ LTj 1, T j 1 + δj F Using a rivial equaliy κ LTj 1, T j 1 + δj = LT j 1, T j 1 κ + δj LT j 1, T j 1 κ δ j, we find ha he following cap-floor pariy relaionship is saisfied a any ime [, T ] he hree conracs are assumed o have he same paymen daes Forward Cap Forward Floor = Forward Swap, For a descripion of a muli-period forward swap, we refer o he nex chaper. This relaionship can also be verified by a sraighforward comparison of he corresponding cash flows of boh porfolios. Le us finally menion ha by a cap respecively, floor, we mean a forward cap respecively, forward floor wih = T Valuaion in he Gaussian HJM Model We assume ha he bond price volailiy is a deerminisic funcion, ha is, we place ourselves wihin he Gaussian HJM framework. Recall ha for any wo mauriy daes U, T we wrie F B, T, U = B, T /B, U, so ha he funcion γ, T, U = b, T b, U represens he volailiy of F B, T, U Plain-vanilla Caps and Floors The following lemma is an immediae consequence of Proposiion and he equivalence of a caple and a specific pu opion on a zero-coupon bond. Lemma Consider a caple wih selemen dae T, accrual period δ, and srike level κ, ha pays a ime T + δ he amoun LT, T κ + δ. Is arbirage price a ime [, T ] in he Gaussian HJM se-up equals Cpl = B, T N e 1, T δf B, T + δ, T N e 2, T, where δ = 1 + κδ and e 1,2, T = ln F B, T, T + δ ln δ ± 1 2 v2, T v, T wih v 2, T = T γu, T, T + δ 2 du.

12 Models of LIBOR The nex resul, which is an almos immediae consequence of Lemma , provides a generic pricing formula for a forward cap in he Gaussian HJM se-up. Proposiion Assume he Gaussian HJM framework, so ha he volailiies γ, T j 1, T j are deerminisic. Then he arbirage price a ime T of an ineres rae cap wih srike level κ, seled in arrears a imes T j, j = 1,..., n, equals FC = n j=1 where δ j = 1 + κδ j and wih B, T j 1 N e 1, T j 1 δ j F B, T j, T j 1 N e j 2, T j 1 e 1,2, T j 1 = ln F B, T j 1, T j ln δ j ± 1 2 v2, T j 1 v, T j 1 v 2, T j 1 = Tj 1 γu, T j 1, T j 2 du. Proof. We represen he price of a forward cap in he following way FC = = = n j=1 n j=1 n j=1 E P E P E P { B B Tj LTj 1, T j 1 κ + δj F } { B B Tj BTj 1, T j 1 1 δ 1 j { B 1 B δ + } j BT, T j F = T +δj } κ F n Cpl j, where Cpl j sands for he price a ime of he j h caple. The asserion now follows from Lemma To derive he valuaion formula for a floor, i is enough o make use of he cap-floor pariy, ha is, he universal relaionship FC FF = FS. By combining he valuaion formulas for caps and swaps, we find easily ha, under he assumpions of Proposiion , he arbirage price of a floor is given by he expression FF = j=1 n δj B, T j N e j 2 B, T j 1 N e j 1. j=1 In he derivaion of he las formula we have used, in paricular, he universal i.e., model independen valuaion formula 13.2 for swaps, which will be esablished in Sec below.

13 12.3 Valuaion in he Gaussian HJM Model Exoic Caps A large variey of exoic caps is offered o insiuional cliens of financial insiuions. In his secion, we develop pricing formulas for some of hem wihin he Gaussian HJM se-up. Dual-srike caps. The dual-srike cap known also as a N-cap is an ineres rae cap ha has a lower srike κ 1, an upper srike κ 2 wih κ 1 κ 2, and a rigger, say l. So long as he floaing rae L is below he level l, he N- cap owner enjoys proecion a he lower srike κ 1. For periods when L is a or above he level l, he N-cap owner has proecion a he upper srike level κ 2. Le us consider an N-cap on principal 1 seled in arrears a imes T j, j = 1,..., n, where T j T j 1 = δ j and T = T. I is clear ha he cash flow of he N-cap a ime T j equals c j = LT j 1 κ 1 + δj 1 {LTj 1 < l } + LT j 1 κ 2 + δj 1 {LTj 1 l }. where LT j 1 = LT j 1, T j 1. I is no hard o check ha he N-cap price a ime [, T ] is NC = where n 1 B, T j N h j 2, l N h j 2, κ 1 l + N h j 2, κ 2 l j= 1 + κ 1 δ j+1 B, T j+1 N h j 1, l N h j 1, κ 1 l n 1 j= 1 + κ 2 δ j+1 B, T j+1 N h j 1, κ 2 l, n 1 j= h j 1,2, κ = ln1 + κδ j+1 ln F B, T j, T j+1 ± 1 2 v2, T j v, T j and v 2, T j is given in Proposiion Bounded caps. A bounded cap or a B-cap consiss of a sequence of caples in which he difference beween he fixed and floaing levels is paid only if he oal paymens o dae are less han some prescribed level b le us sress ha oher kinds of B-caps exis. Le us firs consider a paricular B-caple mauring a a rese dae T j 1. The corresponding cash flow will be paid in arrears a ime T j only if he accumulaed cash flows a ime T j 1, due o reses a imes T k and cash flows of he B-cap paid a imes T k+1, k =,..., j 2, are sill less han b. More formally, he cash flow of a B-caple mauring a T j 1 equals c j κ, b = LT j 1 κ + δ j 1 D,

14 Models of LIBOR where D sands for he following se { j D = LTk 1 κ { + δk b} j = BTk 1, T k 1 δ } + k b, k=1 where, as usual, δ k = 1 + κδ k. The amoun c j κ, b is paid a ime T j. The arbirage price of a B-caple a ime T herefore equals { B + } BCpl = E P LT j 1 κ δj 1 D F, B Tj or equivalenly, BCpl = E P k=1 { B 1 B δ + } j BT j 1, T j 1D F. Tj 1 Using he sandard forward measure mehod, he las equaliy can be given he following form BCpl = B, T j 1 E PTj 1 {1 δ + } j BT j 1, T j 1D F, where E PTj 1 sands for he expecaion under he forward measure P Tj 1. Furhermore, BT k 1, T k = B, T Tk 1 k B, T k 1 exp γ k u dwu Tj 1 1 Tk 1 Tk 1 γ k u 2 du γ k u γu, T k 1, T j 1 du, 2 where γ k u = γu, T k 1, T k. The random variable ξ 1,..., ξ j, where ξ k = Tk 1 γ k u dw T j 1 u for k = 1,..., j, is independen of he σ-field F under he forward measure P Tj 1. Furhermore, is probabiliy law under P Tj 1 is Gaussian N, Γ, where he enries of he marix Γ are noice ha γ kk = vk 2 γ kl = Tk 1 T l 1 γ k u γ l u du. I is apparen ha BCpl = E PTj 1 { B, T j 1 δ j B, T j e ξ j 1 v 2 j 1 /2 + 1Dj 1 F }, where D j 1 sands for he se { j D j 1 = k=1 B, Tk 1 B, T k e ξ k+α kj +v 2 k /2 δ j + b },

15 and Denoing α kj = A j 1 = 12.3 Valuaion in he Gaussian HJM Model 445 Tk 1 we arrive a he following expression γ k u γu, T k 1, T j 1 ds. { ξ j 1 ln δ j B, T j B, T j } 2 v2 j 1, BCpl = B, T j 1 P Tj 1 {A j 1 D j 1 } δ j B, T j E PTj 1 exp ξ j 1 v 2 j 1/2 1 Aj 1 D j Capions Since a caple is essenially a pu opion on a zero-coupon bond, a European call opion on a caple is an example of a compound opion. More exacly, i is a call opion on a pu opion wih a zero-coupon bond as he underlying asse of he pu opion. Hence, he valuaion of a call opion on a caple can be done similarly as in Chap. 6 provided, of course, ha he model of a zero-coupon bond price has sufficienly nice properies. A call opion on a cap, or a capion, is hus a call on a porfolio of pu opions. To price a capion observe ha is payoff a expiry is n +, CC T = Cpl j T K j=1 where as usual Cpl j T sands for he price a ime T of he jh caple of he cap, T is he call opion s expiry dae and K is is srike price. Suppose ha we place ourselves wihin he framework of he spo rae models of Chap. 9.5 for insance, he Hull and Whie model. Typically, he caple price is an increasing funcion of he curren value of he spo rae r. Le r be he criical level of ineres rae, which is implicily deermined by he equaliy n j=1 Cpl j T r = K. I is clear ha he opion is exercised when he rae r T is greaer han r. Le us inroduce numbers K j by seing K j = Cpl j T r for j = 1,..., n. I is easily seen ha he capion s payoff is equal o he sum of he payoffs of n call opions on paricular caples, wih K j being he corresponding srike prices. Consequenly, he capion s price CC a ime T 1 is given by he formula CC = n j=1 C Cpl j, T, K j, where C Cpl j, T, K j is he price a ime of a call opion wih expiry dae T and srike level K j wrien on he j h caple see Hull and Whie An opion on a cap or floor can also be sudied wihin he Gaussian HJM framework see Brace and Musiela However, resuls concerning capion valuaion wihin his framework are less explici han in he case of he Hull and Whie model.

16 Models of LIBOR 12.4 LIBOR Marke Models The goal of his secion is o presen various approaches o he direc modelling of forward LIBORs. We focus here on he model s consrucion, is basic properies, and he valuaion of he mos ypical derivaives. For furher deails, he ineresed reader is referred o he papers by Musiela and Sondermann 1993, Sandmann and Sondermann 1993, Goldys e al. 1994, Sandmann e al. 1995, Brace e al. 1997, Jamshidian 1997a, Milersen e al. 1997, Musiela and Rukowski 1997, Rady 1997, Sandmann and Sondermann 1997, Rukowski 1998b, 1999a, Yasuoka 21, Galluccio and Huner 23, 24, and Glasserman and Kou 23. The issues relaed o he model s implemenaion, including model calibraion and he valuaion of exoic LIBOR and swap rae derivaives, are reaed in Brace 1996, Brace e al. 1998, 21a, Hull and Whie 1999, Schlögl 1999, Uraani and Usunomiya 1999, Loz and Schlögl 1999, Schoenmakers and Coffey 1999, Andersen 2, Andersen and Andreasen 2b, Brace and Womersley 2, Dun e al. 2, Hull and Whie 2, Glasserman and Zhao 2, Sidenius 2, Andersen and Broheron-Racliffe 21, De Yong e al. 21a, 21b, Pelsser e al. 22, Wu 22, Wu and Zhang 22, d Aspremon 23, Glasserman and Merener 23, Galluccio e al. 23a, Jäckel and Rebonao 23, Kawai 23, Pelsser and Pieersz 23, and Pierbarg 23a, 23c. The main moivaion for he inroducion of he lognormal LIBOR model was he marke pracice of pricing caps and swapions by means of Black- Scholes-like formulas. For his reason, we shall firs describe how marke praciioners value caps. The formulas commonly used by praciioners assume ha he underlying insrumen follows a geomeric Brownian moion under some probabiliy measure, Q say. Since he formal definiion of his probabiliy measure is no available, we shall informally refer o Q as he marke probabiliy Black s Formula for Caps Le us consider an ineres rae cap wih expiry dae T and fixed srike level κ. Marke pracice is o price he opion assuming ha he underlying forward ineres rae process is lognormally disribued wih zero drif. Le us firs consider a caple ha is, one leg of a cap. Assume ha he forward LIBOR L, T, [, T ], for he accrual period of lengh δ follows a geomeric Brownian moion under he marke probabiliy, Q say. More specifically, dl, T = L, T σ dw, where W follows a one-dimensional sandard Brownian moion under Q, and σ is a sricly posiive consan. The unique soluion of is L, T = L, T exp σw 1 2 σ2 2, [, T ].

17 12.4 LIBOR Marke Models 447 The marke price a ime of a caple wih expiry dae T and srike level κ is now found from he formula Cpl = δb, T + δ E Q LT, T κ + F. More explicily, for any [, T ] we have Cpl = δb, T + δ L, T N ê 1, T κn ê 1, T, where ê 1,2, T = lnl, T /κ ± 1 2 ˆv2, T ˆv, T and ˆv, 2 T = σ 2 T. This means ha marke praciioners price caples using Black s formula, wih discoun from he selemen dae T + δ. A cap seled in arrears a imes T j, j = 1,..., n, where T j T j 1 = δ j, T = T, is priced by he formula n FC = δ j B, T j L, T j 1 N ê 1, T j 1 κn ê 2, T j 1, j=1 where for every j =,..., n 1 ê 1,2, T j 1 = lnl, T j 1/κ ± 1 2 ˆv2, T j 1 ˆv, T j 1 and ˆv 2, T j 1 = σj 2T j 1 for some consans σ j, j = 1,..., n. The consan σ j is referred o as he implied volailiy of he j h caple. Thus, for a fixed srike κ we obain in his way he erm srucure of caple volailiies. Since he implied caple volailiies usually depend on he srike level, we also observe he volailiy smile in he caples marke. In pracice, caps are quoed in erms of implied volailiies, assuming a fla erm srucure for underlying caples. The erm srucure of caples volailiies can be sripped from marke prices of caps. The marke convenion described above implicily assumes a leas in he case of fla caple volailiies ha for any mauriy T j he corresponding forward LIBOR has a lognormal probabiliy law under he marke probabiliy. As we shall see in wha follows, he valuaion formula obained for caps and floors in he lognormal LIBOR marke model agrees wih marke pracice. Recall ha in he general framework of sochasic ineres raes, he price of a forward cap equals see formula where FC = n j=1 E P for every j = 1,..., n. B B Tj LTj 1 κ + δj F = n Cpl j, 12.2 j=1 Cpl j = B, T j E PTj LTj 1 κ + δj F 12.21

18 Models of LIBOR Milersen, Sandmann and Sondermann Approach The firs aemp o provide a rigorous consrucion a lognormal model of forward LIBORs was done by Milersen, Sandmann and Sondermann in heir paper published in 1997 see also Musiela and Sondermann 1993, Goldys e al. 1994, and Sandmann e al for relaed sudies. As a saring poin of heir analysis, Milersen e al posulae ha he forward LIBOR process L, T saisfies dl, T = µ, T d + L, T λ, T dw, wih a deerminisic volailiy funcion λ, T +δ. I is no difficul o deduce from he las formula ha he forward price of a zero-coupon bond saisfies df, T + δ, T = F, T + δ, T 1 F, T + δ, T λ, T dw T. Subsequenly, hey focus on he parial differenial equaion saisfied by he funcion v = v, x ha expresses he forward price of he bond opion in erms of he forward bond price. The PDE for he opion s price is v λ, T 2 x 2 1 x 2 2 v x 2 = wih he erminal condiion, vt, x = K x +. I is ineresing o noe ha he PDE was previously solved by Rady and Sandmann 1994 who worked wihin a differen framework, however. In fac, hey were concerned wih he valuaion of a bond opion for he Bühler and Käsler 1989 model. By solving he PDE 12.22, Milersen e al derived no only he closed-form soluion for he price of a bond opion his goal was already achieved in Rady and Sandmann 1994, bu also he marke formula for he caple s price. I should be sressed, however, ha he exisence of a lognormal family of LIBORs L, T wih differen mauriies T was no formally esablished in a definiive manner by Milersen e al alhough some parial resuls were provided. The posiive answer o he problem of exisence of such a model was given by Brace e al. 1997, who also sar from he coninuous-ime HJM framework Brace, G aarek and Musiela Approach To inroduce formally he noion of a forward LIBOR, we assume ha we are given a family B, T of bond prices, and hus also he collecion F B, T, U of forward processes. Le us fix a horizon dae T. In conras o he previous secion, we shall now assume ha a sricly posiive real number δ < T represening he lengh of he accrual period, is fixed hroughou. By definiion, he forward δ-libor rae L, T for he fuure dae T T δ prevailing a ime is given by he convenional marke formula

19 12.4 LIBOR Marke Models δl, T = F B, T, T + δ, [, T ]. Comparing his formula wih 9.4, we find ha L, T = f s, T, T + δ, so ha he forward LIBOR L, T represens in fac he add-on rae prevailing a ime over he fuure ime period [T, T +δ]. We can also re-express L, T direcly in erms of bond prices, as for any T [, T δ] we have 1 + δl, T = B, T, [, T ] B, T + δ In paricular, he iniial erm srucure of forward LIBORs saisfies L, T = B, T B, T + δ δb, T + δ Assume ha we are given a family F B, T, T of forward processes saisfying df B, T, T = F B, T, T γ, T, T dw T. on Ω, F [,T ], P T, where W T is a sandard Brownian moion under P T. Then i is no hard o derive he dynamics of he associaed family of forward LIBORs. For insance, one finds ha under he forward measure P T +δ we have dl, T = δ 1 F B, T, T + δ γ, T, T + δ dw T +δ, where W T +δ and P T +δ are defined by see 9.42 W T +δ = W T γu, T + δ, T du. The process W T +δ is a sandard Brownian moion wih respec he probabiliy measure P T +δ P T defined on Ω, F T by means of he Radon-Nikodým densiy see 9.43 dp T +δ dp T = E T +δ This means ha L, T solves he equaion γu, T + δ, T dw T u P-a.s. dl, T = δ δl, T γ, T, T + δ dw T +δ subjec o he iniial condiion Suppose ha forward LIBORs L, T are sricly posiive. Then formula can be rewrien as follows where for every [, T ] we have dl, T = L, T λ, T dw T +δ, λ, T = 1 + δl, T δl, T γ, T, T + δ We hus see ha he collecion of forward processes uniquely specifies he family of forward LIBORs.

20 Models of LIBOR The consrucion of a model of forward LIBORs relies on he following se of assumpions. LR.1 For any mauriy T T δ, we are given a R d -valued, bounded F-adaped process λ, T represening he volailiy of he forward LIBOR process L, T. LR.2 We assume a sricly decreasing and sricly posiive iniial erm srucure B, T, T [, T ], and hus an iniial erm srucure L, T of forward LIBORs L, T = B, T B, T + δ, T [, T δ]. δb, T + δ Noe ha he volailiy λ is a sochasic process, in general. In he special case when λ, T is a bounded deerminisic funcion, a model we are going o consruc is ermed lognormal LIBOR marke model for a fixed accrual period. Remarks. Needless o say ha he boundedness of λ can be weakened subsanially. In fac, we shall frequenly posulae, o simplify he exposiion, ha a volailiy process or funcion is bounded in order bu i is clear ha a suiable inegrabiliy condiions are sufficien. To consruc a model saisfying LR.1-LR.2, Brace e al place hemselves in he HJM se-up and hey assume ha for every T [, T ], he volailiy b, T vanishes for every [T δ, T ]. The consrucion presened in Brace e al relies on forward inducion, as opposed o he backward inducion, which will be used in wha follows. They sar by posulaing ha he dynamics of L, T under he maringale measure P are governed by he following SDE dl, T = µ, T d + L, T λ, T dw, where λ is known, bu he drif coefficien µ is unspecified. Recall ha he arbirage-free dynamics of he insananeous forward rae f, T are df, T = σ, T σ, T d + σ, T dw. In addiion, we have he following relaionship cf T +δ 1 + δl, T = exp T f, u du Applying Iô s formula o boh sides of 12.28, and comparing he diffusion erms, we find ha σ, T + δ σ, T = T +δ T σ, u du = δl, T λ, T. 1 + δl, T

21 12.4 LIBOR Marke Models 451 To solve he las equaion for σ in erms of L, i is necessary o impose some kind of iniial condiion on he process σ, or, equivalenly, on he coefficien σ in he dynamics of f, T. For insance, by seing σ, T = for T + δ his choice was posulaed in Brace e al. 1997, we obain he following relaionship [δ 1 T ] b, T = σ δl, T kδ, T = λ, T kδ δl, T kδ k=1 The exisence and uniqueness of soluions o he SDEs ha govern he insananeous forward rae f, T and he forward LIBOR L, T for σ given by can be shown raher easily, using he forward inducion. Taking his resul for graned, we conclude ha he process L, T saisfies, under he spo maringale measure P, or equivalenly, dl, T = L, T σ, T λ, T d + L, T λ, T dw, dl, T = L, T λ, T dw T +δ under he forward measure P T +δ. In his way, Brace e al were able o specify compleely heir generic model of forward LIBORs. In paricular, in he case of deerminisic volailiies λ, T, we obain he lognormal model of forward LIBORs, ha is, he model in which he process L, T is lognormal under P T +δ for any mauriy T > and for a fixed δ. Le us noe ha his model is someimes referred o as he LLM model ha is, Lognormal LIBOR Marke model model or as he BGM model ha is, Brace-G aarek-musiela model Musiela and Rukowski Approach As an alernaive o forward inducion, we describe he backward inducion approach o he modelling of forward LIBORs. We shall now focus on he modelling of a finie family of forward LIBORs ha are associaed wih a prespecified collecion T < < T n of rese/selemen daes. The consrucion presened below is based on he one given by Musiela and Rukowski Le us sar by recalling he noaion. We assume ha we are given a predeermined collecion of rese/selemen daes T < T 1 < < T n referred o as he enor srucure. Le us wrie δ j = T j T j 1 for j = 1,..., n, so ha T j = T + j i=1 δ i for every j =,..., n. Since δ j is no necessarily consan, he assumpion of a fixed accrual period δ is now relaxed, and hus a model will be more suiable for pracical purposes. Indeed, in mos LIBOR derivaives he accrual period day-coun fracion varies over ime, and hus i is essenial o have a model of LIBORs ha is capable of mimicking his imporan real-life feaure.

22 Models of LIBOR We find i convenien o se T = T n and n Tm = T δ j = T n m, m =,..., n. j=n m+1 For any j =,..., n 1, we define he forward LIBOR L, T j by seing L, T j = B, T j B, T j+1, [, T j ]. δ j+1 B, T j+1 Le us inroduce he noion of a maringale probabiliy associaed wih he forward LIBOR L, T j 1. Definiion Le us fix j = 1,..., n. A probabiliy measure P Tj on Ω, F Tj, equivalen o P, is said o be he forward LIBOR measure for he dae T j if, for every k = 1,..., n, he relaive bond price U n j+1, T k def = B, T k B, T j, [, T k T j ], follows a local maringale under P Tj. I is clear ha he noion of forward LIBOR measure is formally idenical wih ha of a forward maringale measure for a given dae. The sligh modificaion of our previous erminology emphasizes our inenion o make a clear disincion beween various kinds of forward probabiliies, which we are going o sudy in he sequel. Also, i is rivial o observe ha he forward LI- BOR L, T j necessarily follows a local maringale under he forward LIBOR measure for he dae T j+1. If, in addiion, i is a sricly posiive process and he underlying filraion is generaed by a Brownian moion, he exisence of he associaed volailiy process can be jusified easily. In our furher developmen, we shall go he oher way around; ha is, we will assume ha for any dae T j, he volailiy λ, T j of he forward LIBOR L, T j is exogenously given. Basically, i can be a deerminisic R d -valued funcion of ime, an R d -valued funcion of he underlying forward LIBORs, or a d-dimensional sochasic process adaped o a Brownian filraion. For simpliciy, we assume ha he volailiies of forward LIBORs are bounded of course, his assumpion can be relaxed. Our aim is o consruc a family L, T j, j =,..., n 1 of forward LIBORs, a collecion of muually equivalen probabiliy measures P Tj, j = 1,..., n, and a family W Tj, j =,..., n 1 of processes in such a way ha: i for any j = 1,..., n he process W T j is a d-dimensional sandard Brownian moion under he probabiliy measure P Tj, ii for any j = 1,..., n 1, he forward LIBOR L, T j saisfies he SDE wih he iniial condiion dl, T j = L, T j λ, T j dw Tj+1, [, T j ],

23 L, T j = B, T j B, T j+1. δ j+1 B, T j LIBOR Marke Models 453 As was menioned already, he consrucion of he model is based on backward inducion. We sar by defining he forward LIBOR wih he longes mauriy, T n 1. We posulae ha L, T n 1 = L, T 1 is governed under he underlying probabiliy measure P by he following SDE noe ha, for simpliciy, we have chosen he underlying probabiliy measure P o play he role of he forward LIBOR measure for he dae T wih he iniial condiion Pu anoher way, we have dl, T 1 = L, T 1 λ, T 1 dw, L, T1 = B, T 1 B, T δ n B, T. L, T 1 = B, T 1 B, T δ n B, T E λu, T1 dw u for [, T 1 ]. Since B, T 1 > B, T, i is clear ha he L, T 1 follows a sricly posiive maringale under P T = P. The nex sep is o define he forward LIBOR for he dae T 2. For his purpose, we need o inroduce firs he forward maringale measure for he dae T 1. By definiion, i is a probabiliy measure Q, equivalen o P, and such ha processes U 2, T k = B, T k B, T 1 are Q-local maringales. I is imporan o observe ha he process U 2, T k admis he following represenaion U 2, T k = U 1, T k δ n L, T The following auxiliary resul is a sraighforward consequence of Iô s rule. Lemma Le G and H be real-valued adaped processes, such ha dg = α dw, dh = β dw. Assume, in addiion, ha H > 1 for every and wrie Y = 1 + H 1. Then dy G = Y α Y G β dw Y β d. I follows immediaely from Lemma ha du 2, Tk = η k dw δ nl, T1 1 + δ n L, T1 λ, T 1 d for a cerain process η k.

24 Models of LIBOR I is herefore enough o find a probabiliy measure under which he process W T 1 def δ n Lu, T 1 = W 1 + δ n Lu, T1 λu, T 1 du = W γu, T1 du, where [, T1 ], follows a sandard Brownian moion he definiion of γ, T1 is clear from he conex. This can easily be achieved using Girsanov s heorem, as we may pu dp T 1 dp = E T 1 γu, T1 dw u, P-a.s. We are in a posiion o specify he dynamics of he forward LIBOR for he dae T2 under P T 1, namely we posulae ha wih he iniial condiion dl, T 2 = L, T 2 λ, T 2 dw T 1, L, T 2 = B, T 2 B, T 1 δ n 1 B, T 1. Le us now assume ha we have found processes L, T1,..., L, Tm. In paricular, he forward LIBOR measure P T m 1 and he associaed Brownian moion W T m 1 are already specified. Our aim is o deermine he forward LIBOR measure P T m. I is easy o check ha U m+1, T k = U m, T k δ n m L, T m + 1. Using Lemma , we obain he following relaionship W T m = W T m 1 δ n m Lu, T m 1 + δ n m Lu, T m λu, T m du for [, Tm]. The forward LIBOR measure P T m can hus be found easily using Girsanov s heorem. Finally, we define he process L, Tm+1 as he soluion o he SDE wih he iniial condiion dl, T m+1 = L, T m+1 λ, T m+1 dw T m, L, Tm+1 = B, T m+1 B, Tm δ n m B, Tm. Remarks. If he volailiy coefficien λ, T m : [, T m ] R d is deerminisic, hen, for each dae [, T m ], he random variable L, T m has a lognormal probabiliy law under he forward maringale measure P Tm+1. In his case, he model is referred o as he lognormal LIBOR model.

25 12.4 LIBOR Marke Models Jamshidian s Approach The backward inducion approach o modelling of forward LIBORs presened in he preceding secion was re-examined and modified by Jamshidian 1997a. In his secion, we presen briefly his alernaive approach o he modelling of forward LIBORs. As was made apparen in he previous secion, in he direc modelling of LIBORs, no explici reference is made o he bond price processes, which are used o define formally a forward LIBOR hrough equaliy Neverheless, o explain he idea ha underpins Jamshidian s approach, we shall emporarily assume ha we are given a family of bond prices B, T j for he fuure daes T j, j =,..., n. By definiion, he spo LIBOR measure is ha probabiliy measure equivalen o P, under which all relaive bond prices are local maringales, when he price process obained by rolling over one-period bonds, is aken as a numeraire. The exisence of such a measure can be eiher posulaed, or derived from oher condiions. Le us define for every [, T ], where we se m G = B, T m B 1 T j 1, T j, 12.3 m = inf {k N T + j=1 k δ i } = inf {k N T k }. i=1 I is easily seen ha G represens he wealh a ime of a porfolio ha sars a ime wih one uni of cash invesed in a zero-coupon bond of mauriy T, and whose wealh is hen reinvesed a each dae T j, j =,..., n 1, in zero-coupon bonds mauring a he nex dae; ha is, a ime T j+1. Definiion A spo LIBOR measure P L is any probabiliy measure on Ω, F T equivalen o a reference probabiliy P, and such ha he relaive prices B, T j /G, j = 1,..., n, are local maringales under P L. Noe ha B, T k+1 G = m j=1 1 + δj LT j 1, T j 1 1 k j=m δj L, T j 1 1, so ha all relaive bond prices B, T j /G, j = 1,..., n are uniquely deermined by a collecion of forward LIBORs. In his sense, he rolling bond G is he correc choice of he numeraire asse in he presen se-up. We shall now concenrae on he derivaion of he dynamics under P L of forward LIBOR processes L, T j, j = 1,..., n. Our aim is o show ha he join dynamics of forward LIBORs involve only he volailiies of hese processes as opposed o volailiies of bond prices or some oher processes.

26 Models of LIBOR Pu differenly, we shall show ha i is possible o define he whole family of forward LIBORs simulaneously under a single probabiliy measure of course, his feaure can also be deduced from he previously examined consrucion. To faciliae he derivaion of he dynamics of L, T j, we posulae emporarily ha bond prices B, T j follow Iô processes under he underlying probabiliy measure P, more explicily db, T j = B, T j a, T j d + b, T j dw for every j = 1,..., n, where, as before, W is a d-dimensional sandard Brownian moion under an underlying probabiliy measure P i should be sressed, however, ha we do no assume here ha P is a forward or spo maringale measure. Combining 12.3 wih 12.31, we obain dg = G a, Tm d + b, T m dw. Furhermore, by applying Iô s rule o equaliy we find ha where and µ, T j = 1 + δ j+1 L, T j = B, T j B, T j+1, dl, T j = µ, T j d + ζ, T j dw, B, T j a, Tj a, T j+1 ζ, T j b, T j+1 δ j+1 B, T j+1 ζ, T j = B, T j b, Tj b, T j δ j+1 B, T j+1 Using and 12.33, we arrive a he following relaionship b, T m b, T j+1 = j k=m δ k+1 ζ, T k 1 + δ k+1 L, T k By he definiion of a spo LIBOR measure P L, each relaive price process B, T j /G follows a local maringale under P L. Since, in addiion, P L is assumed o be equivalen o P, i is clear because of Girsanov s heorem ha i is given by he Doléans exponenial, ha is, dp L dp = E T h u dw u, P-a.s. for some adaped process h. I i no hard o check, using Iô s rule, ha h needs o saisfy, for every [, T j ] and every j = 1,..., n, a, T j a, T m = b, T m h b, Tj b, T m.

27 Combining wih he formula above, we obain 12.4 LIBOR Marke Models 457 B, T j a, Tj a, T j+1 = ζ, T j b, T m h, δ j+1 B, T j+1 and his in urn yields dl, T j = ζ, T j b, Tm b, T j+1 h d + dw. Using he las formula, and Girsanov s heorem, we arrive a he following resul, due o Jamshidian 1997a. Proposiion For any j =,..., n 1, he process L, T j saisfies dl, T j = j k=m δ k+1 ζ, T k ζ, T j 1 + δ k+1 L, T k d + ζ, T j dw L, where he process W L = W h u du is a d-dimensional sandard Brownian moion under he spo LIBOR measure P L. To furher specify he model, we posulae ha he processes ζ, T j, j = 1,..., n are exogenously given. Specifically, le ζ, T j = λ j, L, Tj, L, T j+1,..., L, T n, [, T j ], where λ j : [, T j ] R n j+1 R d are given funcions. This leads o he following sysem of SDEs j δ k+1 λ k, L k λ j, L j dl, T j = d + λ j, L j dw L, 1 + δ k+1 L, T k k=m where, for breviy, we wrie L j = L, T j, L, T j+1,..., L, T n. Under sandard regulariy assumpions imposed on he se of coefficiens λ j, his sysem of SDEs can be solved recursively, saring from he SDE for he process L, T n 1. In his way, one can produce a large variey of alernaive versions of a forward LIBOR model, including he CEV LIBOR model and a simple version of displaced-diffusion model hese sochasic volailiy versions of a LIBOR model are presened briefly in Sec Le us finally observe ha he lognormal LIBOR marke model corresponds o he choice of ζ, T j = λ, T j L, T j, where λ, T j : [, T j ] R d is a deerminisic funcion for every j =,... n 1. In his case we deal wih he following sysem of SDEs dl, T j L, T j = j k=m δ k+1 L, T k λ, T k λ, T j 1 + δ k+1 L, T k d + λ, T j dw L. If we decide o use he probabiliy measure P L o value a given coningen claim X, is arbirage price will be expressed in unis of he rolling bond G.

28 Models of LIBOR 12.5 Properies of he Lognormal LIBOR Model We make he sanding assumpions he volailiies of forward LIBORs L, T j for j =,..., n 1 are deerminisic. In oher words, we place ourselves wihin he framework of he lognormal LIBOR model. I is ineresing o noe ha in all approaches, here is a uniquely deermined correspondence beween forward measures and forward Brownian moions associaed wih differen daes i is based on relaionships 12.3 and On he oher hand, however, here is a considerable degree of ambiguiy in he way in which he spo maringale measure is specified in some insances, i is no inroduced a all. Consequenly, he fuures LIBOR L f, T j, which equals cf. Sec L f, T j = E P LT j, T j F = E P LT j, T j F, is no necessarily specified in he same way in various approaches o he LIBOR marke model. For a given funcion g : R R and a fixed dae u T j, we are ineresed in he payoff of he form X = g Lu, T j ha seles a ime T j. Paricular cases of such payoffs are X 1 = g B 1 T j, T j+1, X 2 = g BT j, T j+1, X 3 = g F B u, T j+1, T j. Recall ha B 1 T j, T j+1 = 1 + δ j+1 LT j = 1 + δ j+1 LTj = 1 + δ j+1 L f T j. The choice of he pricing measure is hus largely maer of convenience. Similarly, we have BT j, T j+1 = δ j+1 LT j, T j = F BT j, T j+1, T j More generally, he forward price of a T j+1 -mauriy bond for he selemen dae T j equals F B u, T j+1, T j = Bu, T j+1 Bu, T j = δ j+1 Lu, T j To value a European conigen claim X = glu, T j = gf B u, T j+1, T j seling a ime T j we may use he risk-neural valuaion formula π X = B, T j E PTj X F, [, T j ]. I is hus clear ha o value a claim in he case u T j, i is enough o know he dynamics of eiher L, T j or F B, T j+1, T j under he forward maringale measure P Tj. When u = T j, we may equally well use he dynamics under P Tj of eiher he process L, T j, or he L f, T j.

29 12.5 Properies of he Lognormal LIBOR Model 459 For insance, π X 1 = B, T j E PTj B 1 T j, T j+1 F, or equivalenly, π X 1 = B, T j E PTj F 1 B T j, T j+1, T j F, bu also π X 1 = B, T j 1 + δ j+1 E PTj ZT j F, where ZT j = LT j = LT j = L f T j Transiion Densiy of he LIBOR We shall now derive he ransiion probabiliy densiy funcion of he process L, T j under he forward maringale measure P Tj. Le us firs prove he following relaed resul ha is of independen ineres i was firs esablished by Jamshidian Proposiion Le u T j. Then we have E PTj Lu, Tj F = L, Tj + δ j+1var PTj+1 Lu, Tj F 1 + δ j+1 L, T j In he case of he lognormal model of forward LIBORs, we have E PTj Lu, Tj F = L, Tj 1 + δ j+1l, T j e v 2 j,u 1, 1 + δ j+1 L, T j where v 2 j, u = Var PTj+1 u λs, T j dw T j+1 s = u λs, T j 2 ds. Proof. Combining 12.5 wih he maringale propery of L, T j under P Tj+1, we obain E PTj δj+1 Lu, T j Lu, T j F E PTj Lu, Tj F =, 1 + δ j+1 L, T j so ha E PTj Lu, Tj F = L, Tj + δ j+1 E PTj+1 Lu, Tj L, T j 2 F In he case of he lognormal model, we have 1 + δ j+1 L, T j Lu, T j = L, T j e η,u 1 2 v2 j,u,..

30 Models of LIBOR where Consequenly, η, u = u λs, T j dw T j+1 s E PTj+1 Lu, Tj L, T j 2 F = L 2, T j e v2 j,u 1. This gives he desired equaliy. To derive he ransiion probabiliy densiy funcion of he process L, T j, noe ha for any u T j and any bounded Borel measurable funcion g : R R we have glu, T j 1 + δ j+1 Lu, T j F E PTj glu, Tj F = E PTj δ j+1 L, T j The following simple lemma appears o be useful in wha follows. Lemma Le ζ be a nonnegaive random variable on a probabiliy space Ω, F, P wih he probabiliy densiy funcion f P. Le Q be a probabiliy measure equivalen o P. Suppose ha for any bounded Borel measurable funcion g : R R we have E P gζ = E Q 1 + ζgζ. Then he probabiliy densiy funcion f Q of ζ under Q saisfies f P y = 1 + yf Q y. Proof. The asserion is in fac rivial since, by assumpion, gyf P y dy = gy1 + yf Q y dy for any bounded Borel measurable funcion g : R R. Assume he lognormal LIBOR model, and fix x R. Recall ha for any u we have Lu, T j = L, T j e η,u 1 2 Var P Tj+1 η,u, where η, u is given by so ha i is independen of he σ-field F. The Markovian propery of L, T j under he forward measure P Tj+1 is hus apparen. Denoe by p L, x, u, y he ransiion probabiliy densiy funcion under P Tj+1 of he process L, T j. Elemenary calculaions involving Gaussian densiies yield 1 lny/x + 1 p L, x, u, y = { 2πvj, uy exp 2 v2 j, u 2 } 2vj 2, u.

31 for any x, y > and arbirary < u, where 12.5 Properies of he Lognormal LIBOR Model 461 p L, x, u, y = P Tj+1 {Lu, T j = y L, T j = x}. Taking ino accoun Lemma , we conclude ha he ransiion probabiliy densiy funcion of he process 1 L, T j, under he forward maringale measure P Tj, saisfies p L, x, u, y = P Tj {Lu, T j = y L, T j = x} = 1 + δ j+1y 1 + δ j+1 x p L, x, u, y. We are now in a posiion o sae he following resul. Corollary The ransiion probabiliy densiy funcion under P Tj of he forward LIBOR L, T j equals 1 + δ j+1 y lny/x + 1 p L, x, u, y = { 2πvj, u y1 + δ j+1 x exp 2 v2 j, u 2 } 2vj 2, u for any < u and arbirary x, y > Transiion Densiy of he Forward Bond Price Observe ha he forward bond price F B, T j+1, T j saisfies F B, T j+1, T j = B, T j+1 B, T j = δ j+1 L, T j Firs, his implies ha in he lognormal LIBOR model, he dynamics of he forward bond price F B, T j+1, T j are governed by he following sochasic differenial equaion, under P Tj, df B = F B 1 F B λ, T j dw T j, 12.4 where we wrie F B = F B, T j+1, T j. If he iniial condiion in 12.4 saisfies < F B < 1, hen his equaion can be shown o admi a unique srong soluion i saisfies < F B < 1 for every >. This makes i clear ha he forward bond price F B, T j+1, T j, and hus also he LIBOR L, T j, are Markovian under P Tj. Using he formula esablished in Corollary and relaionship 12.39, one can find he ransiion probabiliy densiy funcion of he Markov process F B, T j+1, T j under P Tj ; ha is, p B, x, u, y = P Tj {F B u, T j+1, T j = x F B, T j+1, T j = y}. We have he following resul see Rady and Sandmann 1994, Milersen e al and Jamshidian 1997a. 1 The Markov propery of L, T j under P Tj follows from he properies of he forward price, which will be esablished in Sec

32 Models of LIBOR Corollary The ransiion probabiliy densiy funcion under P Tj of he forward bond price F B, T j+1, T j equals 2 x ln x1 y p B, x, u, y = 2πvj, uy 2 1 y exp y1 x v2 j, u 2vj 2, u for any < u and arbirary < x, y < 1. Proof. Le us fix x, 1. Using 12.39, i is easy o show ha p B, x, u, y = y 2 p L, 1 x δx, u, 1 y, δy where δ = δ j+1. The formula now follows from Corollary Goldys 1997 esablished he following resul we find i convenien o defer he proof of equaliy o Sec Proposiion Le X be a soluion of he following sochasic differenial equaion dx = X 1 X λ dw, X = x, where W follows a sandard Brownian moion under P, and λ : R + R is a locally square inegrable funcion. Then for any nonnegaive Borel measurable funcion g : R R and any u > we have E P gx u = 1 xe Q gh1 ζ + xe Q gh2 ζ, where ζ has, under Q, a Gaussian law wih zero mean value and variance Var Q ζ = v 2, u = u λs 2 ds. Furhermore, 1 h 1,2 y = 1 + exp y + ln 1 x ± 1 x 2 v2, u, y R. Before we end his chaper, we shall check ha formula 12.39, which gives he ransiion probabiliy densiy funcion of he forward bond price, can be re-derived using formula According o 12.42, we have E PTj gx u = I 1 + I 2, where and I 1 = I 2 = 1 x 2πv, u gh 1 z e z2 /2v 2,u dz x gh 2 z e z2 /2v 2,u dz. 2πv, u

33 12.5 Properies of he Lognormal LIBOR Model 463 Firs, le us se y = h 1 z in I 1, so ha dy = y1 ydz. Then we obain where we se I 1 = 1 x 1 gy kx, y dy 2πv, u y1 y ln x1 y kx, y = exp y1 x 1 2 v2, u 2v 2, u 2. Equivalenly, we have where I 1 = 1 x 1 xgy 2πv, u 1 xy kx, 2 y dy. ln x1 y kx, y = exp y1 x v2, u 2v 2, u 2. Similarly, he change of variable y = h 2 z in I 2 so ha once again dy = y1 y dz leads o he following equaliy I 2 = x 1 gy 2πv, u y1 y kx, y dy. Simple algebra now yields recall ha E PTj gx u = I 1 + I 2 I 1 + I 2 = 1 1 2πv, u xgy y 2 1 y exp ln x1 y y1 x v2, u 2v 2, u 2 dy. I is ineresing o noe ha he formula above generalizes easily o he case < u. Indeed, i is enough o consider he sochasic differenial equaion wih he iniial condiion X = x a ime. As a resul, we obain he following formula for he condiional expecaion E PTj gx u X = x = 1 1 2πv, u xgy y 2 1 y exp ln x1 y y1 x v2, u 2v 2, u 2 dy. I is worh observing ha an applicaion of he las formula o he process X = F B, T j+1, T j leads o an alernaive derivaion of he formula esablished in Corollary

34 Models of LIBOR 12.6 Valuaion in he Lognormal LIBOR Model We sar by considering he valuaion of plain-vanilla caps and floors. Subsequenly, we shall sudy he case of a generic pah-independen claim Pricing of Caps and Floors We shall now examine he valuaion of caps wihin he lognormal LIBOR model of Sec To his end, we formally assume ha k n. Dynamics of he forward LIBOR process L, T j 1 under he forward maringale measure P Tj are known o be dl, T j 1 = L, T j 1 λ, T j 1 dw T j, where W T j is a d-dimensional Brownian moion under he forward measure P Tj, and λ, T j 1 : [, T j 1 ] R d is a deerminisic funcion. Consequenly, for every [, T j 1 ] we have L, T j 1 = L, T j 1 E λu, T j 1 dw T j u. To he bes of our knowledge, he cap valuaion formula was firs esablished in a rigorous way by Milersen e al. 1997, who focused on he dynamics of he forward LIBOR for a given dae. Equaliy was subsequenly re-derived independenly in Goldys 1997 and Rady 1997, who used probabilisic mehods hough hey deal wih a European bond opion, heir resuls are essenially equivalen o equaliy Finally, he same was esablished by means of he forward measure approach in Brace e al. 1997, where an arbirage-free coninuous-ime model of all forward LIBORs was presened. I is insrucive o compare he valuaion formula wih he formula of Proposiion , which holds for a Gaussian HJM case. The following proposiion is an immediae consequence of formulas , combined wih dynamics Since he proof of he nex resul is raher sandard, i is provided for he sake of compleeness. Proposiion Consider an ineres rae cap wih srike level κ, seled in arrears a imes T j, j = 1,..., k. Assuming he lognormal LIBOR model, he price of a cap a ime [, T ] equals where and FC = k j=1 δ j B, T j L, T j 1 N ẽ j 1 κ N ẽ j 2, ẽ j 1,2 = lnl, T j 1/κ ± 1 2 ṽ2 j ṽ j ṽ 2 j = Tj 1 λu, T j 1 2 du.

35 12.6 Valuaion in he Lognormal LIBOR Model 465 Proof. We fix j and we consider he j h caple, wih he payoff a ime T j Cpl j T j = δ j LT j 1 κ + = δ j LT j 1 1 D δ j κ 1 D, where D = {LT j 1 > κ} is he exercise se. Since he caple seles a ime T j, i is convenien o use he forward measure P Tj o find is arbirage price. We have Cpl j = B, T j E PTj Cpl j T j F, [, T j ]. Obviously, i is enough o find he value of a caple for [, T j 1 ]. In view of 12.45, i suffices o compue he following condiional expecaions Cpl j = δ j B, T j E PTj LTj 1 1 D F κδj B, T j P Tj D F = δ j B, T j I 1 I 2, where he meaning of I 1 and I 2 is clear from he conex. Recall ha he spo LIBOR LT j 1 = LT j 1, T j 1 is given by he formula T j 1 LT j 1 = L, T j 1 exp λ j 1 u dw T j u 1 2 Tj 1 λ j 1 u 2 du, where we se λ j 1 u = λu, T j 1. Since λ j 1 is a deerminisic funcion, he probabiliy law under P Tj of he Iô inegral ζ, T j 1 = Tj 1 λ j 1 u is Gaussian, wih zero mean and he variance Var PTj ζ, T j 1 = Tj 1 dw Tj u λu j 1 2 du. I is hus sraighforward o check ha lnl, Tj 1 ln κ 1 2 I 2 = κ N v2 j. v j To evaluae I 1, we inroduce an auxiliary probabiliy measure ˆP Tj, equivalen o P Tj on Ω, F Tj 1, by seing dˆp Tj = E Tj 1 dp Tj λ j 1 u Then he process Ŵ T j, given by he formula Ŵ Tj = W Tj dwu Tj. λ j 1 u du, [, T j 1 ], is he d-dimensional sandard Brownian moion under ˆP Tj.

36 Models of LIBOR Furhermore, he forward price LT j 1 admis he following represenaion under ˆP Tj, for [, T j 1 ], T j 1 LT j 1 = L, T j 1 exp λ j 1 u dŵ u Tj + 1 Tj 1 2 Since T j 1 I 1 = L, T j 1 E PTj 1 D exp λ j 1 u dw Tj u 1 2 Tj 1 λ j 1 u 2 du. λ j 1 u 2 F du from he absrac Bayes rule we ge I 1 = L, T j 1 ˆP Tj D F. Arguing in much he same way as for I 2, we hus obain ln L, Tj 1 ln κ I 1 = L, T j 1 N v2 j. v j This complees he proof of he proposiion. As before, o derive he valuaion formula for a floor, i is enough o make use of he cap-floor pariy Hedging of Caps and Floors I is clear he replicaing sraegy for a cap is a simple sum of replicaing sraegies for caples. I is herefore enough o focus on a paricular caple. Le us denoe by F C, T j he forward price of he j h caple for he selemen dae T j. From 12.44, i is clear ha F C, T j = δ j L, T j 1 N ẽ j 1 κ N ẽ j 2, so ha an applicaion of Iô s formula yields he calculaions here are essenially he same as in he classical Black-Scholes model df C, T j = δ j N ẽ j 1 dl, T j 1. Le us consider he following self-financing rading sraegy in he T j -forward marke, ha is, wih all values expressed in unis of T j -mauriy zero-coupon bonds. We sar our rade a ime wih F C, T j unis of zero-coupon bonds; we need hus o inves Cpl j = F C, T j B, T j of cash a ime. A any ime T j 1, we ake ψ j = N ẽ j 1 posiions in one-period forward swaps over he period [T j 1, T j ]. The associaed gains process G, in he T j forward marke, saisfies G = and 2 d G = δ j ψ j dl, T j 1 = δ j N ẽ j 1 dl, T j 1 = df C, T. 2 To ge a more inuiive insigh ino his formula, i is convenien o consider firs a discreized version of ψ.

37 12.6 Valuaion in he Lognormal LIBOR Model 467 Consequenly, F C T j 1, T j = F C, T j + Tj 1 δ j ψ j dl, T j 1 = F C, T j + G Tj 1. I should be sressed ha dynamic rading is resriced o he inerval [, T j 1 ] only. The gains/losses involving he iniial invesmen are incurred a ime T j, however. All quaniies in he las formula are expressed in unis of T j -mauriy zero-coupon bonds. Also, he caple s payoff is known already a ime T j 1, so ha i is compleely specified by is forward price F C T j 1, T j = Cpl j T j 1 /BT j 1, T j. I is hus clear ha he sraegy ψ inroduced above replicaes he j h caple. Formally, he replicaing sraegy also has he second componen, η j say, represening he number of forward conracs, wih he selemen dae T j, on T j -mauriy bond. Noe ha F B, T j, T j = 1 for every T j, and hus df B, T j, T j =. Hence, for he T j -forward value of our sraegy, we ge and Ṽ ψ j, η j = η j = F C, T j dṽψ j, η j = ψ j δ j dl, T j 1 + η j df B, T j, T j = δ j N ẽ j 1 dl, T j 1. I should be sressed ha, wih he excepion for he iniial invesmen a ime in T j -mauriy bonds, no rading in bonds is required for replicaion of a caple. In pracical erms, he hedging of a cap wihin he framework of he lognormal LIBOR model in done exclusively hrough dynamic rading in he underlying one-period forward swaps. In his inerpreaion, he componen η j represens simply he fuure i.e., as of ime T j 1 effecs of coninuous rading in forward conracs. The same remarks and similar calculaions apply also o floors. Alernaively, replicaion of a caple can be done in he spo ha is, cash marke, using wo simple porfolios of bonds. Indeed, i is easily seen ha for he process we have and V ψ j, η j = B, T j 1 Ṽψ j, η j = Cpl j V ψ j, η j = ψ j B, Tj 1 B, T j + η j df B, T j, T j dv ψ j, η j = ψ j d B, T j 1 B, T j + η j db, T j = N ẽ j 1 d B, T j 1 B, T j + η j db, T j. In his inerpreaion, he componens ψ j and η j represen he number of unis of porfolios B, T j 1 B, T j and B, T j ha are held a ime.

38 Models of LIBOR Valuaion of European Claims We follow here he approach due o Goldys Le X be a soluion o he sochasic differenial equaion { dx = X 1 X λ dw, X = x, where we assume ha he funcion λ : [, T ] R d is bounded and measurable and, as usual, W is a sandard Brownian moion defined on a filered probabiliy space Ω, F, P. For any x, 1, he exisence of a unique global soluion o can be deduced easily from he general heory of sochasic differenial equaions. However, in Lemma below we provide a direc proof by means of a simple ransformaion which is also crucial for he furher calculaions. Consider he following sochasic differenial equaion dz = 1 1 e Z e Z λ 2 d λ dw wih he iniial condiion Z = z. Since he drif erm in his equaion is represened by a bounded and globally Lipschiz funcion, equaion is known o have a unique srong non-exploding soluion for any iniial condiion z R see, e.g., Theorem in Karazas and Shreve 1998a. Lemma For any x, 1, he process where z saisfies X = 1 + e Z 1, [, T ], Z = z = ln x 1 x is he unique srong and non-exploding soluion o equaion Moreover, < X < 1 for every [, T ]. Proof. I is easy o see ha equaion can be rewrien in he form dz = X 1 2 λ 2 d λ dw. Hence, applying he Iô formula o he process X given by formula 12.48, we find ha dx = X 1 X λ dw. Hence, he process X is indeed a soluion o Conversely, if X is any local weak soluion o hen i is in fac a srong soluion because he diffusion coefficien in is locally Lipschiz. Moreover, using he Iô formula again, one can check ha he process Z = ln X 1 X, [, T ],

39 12.6 Valuaion in he Lognormal LIBOR Model 469 is a srong soluion of Thus, i can be coninued o a global one which is unique. The las par of he lemma follows from he definiion of he process X, and he uniqueness of soluions o The nex resul provides a represenaion of he expeced value E P gx in erms of he inegral wih respec o he Gaussian probabiliy law. Proposiion Le g : [, 1] R be a nonnegaive Borel funcion such ha he random variable gx is P-inegrable for some iniial condiion x, 1 and some >. Then he expeced value E P gx is given by he formula E P gx = x1 xe v2, 8 E Q { η 1 + η g 1 + η 2 1}, where η = exp z + ζ x, z = ln 2 1 x, and he random variable ζ has under Q a Gaussian law wih zero mean value and he variance v 2, = λu 2 du. Proof. The proof of he proposiion, due o Goldys 1997, is based on a simple idea ha for any P-inegrable random variable, U say, we have E P U = E P X U + E P Y U, where Y = 1 X. Nex, i is essenial o observe ha X = x E Y u λu dw u and 12.5 Y = 1 x E X u λu dw u Furhermore, from i follows ha dz = λ dw X 1 2 λ d, or equivalenly, dz = λ dw Y λ d. Le us inroduce he auxiliary probabiliy measures P and ˆP on Ω, F by seing d P dp = E Xu 2 1 def λu dwu = η and

40 Models of LIBOR Noe ha where he processes dˆp dp = E Z = z W = W Ŵ = W Yu λu dwu def = ˆη. λu d W u = z Xu 1 2 λu du Yu λu du λu dŵu, are known o be sandard Brownian moions under P and ˆP respecively. Simple manipulaions show ha η = E X u λu dw u and hus, using 12.51, we obain Similarly, we have exp 1 2 Y = 1 x e v2, 1 8 η exp 2 ˆη = E Y u λu dw u Hence, in view of 12.5, we ge The las equaliy yields 1 exp 2 X = x e v2, 8 ˆη exp 1 2 E P X gx = x e v2, 8 E ˆP and gives E P Y gx = 1 x e v2, 8 E P λu d W u λu 2 du λu d W u λu dŵu { gx exp 1 2 Using and 12.49, we conclude ha E P gx = e v2, 8 E Q {g λu dŵu { 1 gx exp e z+ζ. λu 2 du. λu dŵu }, λu d W u }. xe 1 2 ζ + 1 xe 1 2 ζ},

41 12.6 Valuaion in he Lognormal LIBOR Model 471 where ζ is, under Q, a Gaussian random variable wih zero mean value and variance v 2,. The las formula is equivalen o he formula in he saemen of he proposiion. To esablish direcly formula 12.42, a slighly differen change of he underlying probabiliy measure P is convenien. Namely, we pu Now, under P and ˆP we have respecively, where W = W d P dp = E dˆp dp = E X u λu dw u Y u λu dw u. dz = 1 2 λ d λ d W dz = 1 2λ d λ dŵ X u λu du, Ŵ = W Y u λu du, [, T ], are sandard Brownian moions under he corresponding probabiliies. Formula can hus be derived easily from represenaion combined wih equaliies Corollary Le g : R R be a nonnegaive Borel funcion. Then for every x, 1 and any < < T he condiional expecaion E P gx T F equals E P gx T F = kx, where he funcion k :, 1 R is given by he formula kx = x1 xe v2,t 8 E Q {g e z+ζ e z+ζ/2 + e z+ζ/2} wih z = ln x/1 x, and he random variable ζ has under Q a Gaussian law wih zero mean value and variance v 2, T = T λu 2 du Bond Opions Our nex goal is o esablish he bond opion valuaion formula wihin he framework of he lognormal LIBOR model. I is ineresing o noice ha an idenical formula was previously esablished by Rady and Sandmann 1994, who adoped he PDE approach, and who worked wih a differen model, however. In his secion, in which we follow Goldys 1997, we presen a probabilisic approach o he bond opion valuaion. His derivaion of he bond opion price is based on Corollary Noe ha his resul can be used o he valuaion of an arbirary European claim.

42 Models of LIBOR Proposiion The price C a ime T j 1 of a European call opion, wih expiraion dae T j 1 and srike price < K < 1, wrien on a zero-coupon bond mauring a T j = T j 1 + δ j, equals where and C = 1 KB, T j N l j 1 KB, T j 1 B, T j N l j 2, l j 1,2 = ln 1 KB, T j ln K B, T j 1 B, T j ± 1 2ṽj ṽ j ṽ 2 j = Tj 1 λu, T j 1 2 du. Proof. Le us wrie T = T j 1 and T + δ = T j. In view of Corollary , i is clear ha C = B, T E PT F T, T + δ, T K + F = B, T kx, where x = F, T + δ, T and gy = y K +. Using he noaion ṽ = ṽ j and l = ṽ 1 x1 K ln, K1 x we obain kx = x1 x e 1 8 ṽ2 l e ỹ K e 1 2 ỹ + e 1 ỹ 2 ny dy, where ỹ = z + ṽy, and n sands for he sandard normal densiy. Le us se Then kx = x1 x e 1 8 ṽ2 Equivalenly, where and I 1 = l Consequenly, we find ha hy = e 1 2 z+ṽy + e 1 2 z+ṽy. l hyny dy K 1 + e z+ṽy kx = x1 x e 1 8 ṽ2 I 1 KI 2, e 1 2 z+ṽy ny dy = e 1 8 ṽ z 1 N l 1 2ṽ I 2 = I 1 + e 1 8 ṽ2 1 2 z 1 N l + 1 2ṽ. kx = x1 x1 Ke 1 2 z 1 N l 1 2ṽ K x1 xe 1 2 z 1 N l + 1 2ṽ, l hyny dy.

43 12.7 Exensions of he LLM Model 473 or, afer simplificaion, Since 1 1 kx = x1 KNl + 2ṽ K1 xnl 2ṽ. x = F, T + δ, T = B, T + δ/b, T, he proof of he proposiion is compleed. Using he pu-call pariy relaionship for opions wrien on a zero-coupon bond, C P = B, T j KB, T j 1, i is easy o check ha he price of he corresponding pu opion equals P = K 1B, T j N l 1 K B, T j B, T j 1 N l 2. Recall ha he j h caple is equivalen o he pu opion wrien on a zerocoupon bond, wih expiry dae T j 1 and srike price K = δ j = 1 + κd j 1. 1 More precisely, he opion s payoff should be muliplied by he nominal value δ j = 1 + κd j. Hence, using he las formula, we obain Cpl j = B, T j 1 B, T j N l j 2 κδ j B, T j N l j 1, since clearly K 1 = κδ δ 1 j j. To show ha he las formula coincides wih , i is enough o check ha if K = d j, hen he erms l j 2 and l j 1 coincide wih he erms ẽj 1 and ẽj 2 of Proposiion In his way, we obain an alernaive probabilisic derivaion of he cap valuaion formula wihin he lognormal LIBOR marke model. Remarks. Proposiion shows ha he replicaion of he bond opion using he underlying bonds of mauriy T j 1 and T j is no sraighforward. This should be conrased wih he case of he Gaussian HJM se-up in which hedging of bond opions wih he use of he underlying bonds is done in a sandard way. This illusraes our general observaion ha each paricular model of he erm srucure should be ailored o a specific class of derivaives and hedging insrumens Exensions of he LLM Model Le us emphasize ha he consrucions of he LIBOR marke model presened in Sec do no require ha he volailiies of LIBORs be deerminisic funcions. We have seen ha hey may be adaped sochasic processes, or some deerminisic or random funcions of he underlying forward LIBORs. The mos popular choice of deerminisic volailiies for forward LIBORs leads ineviably o he so-called lognormal LIBOR marke model also known as he LLM model or he BGM model.

44 Models of LIBOR Empirical sudies have shown ha he implied volailiies of marke prices of caples and swapions end o be decreasing funcions of he srike level. Hence, i is of pracical ineres o develop sochasic volailiy versions of he LIBOR marke model capable of maching he observed volailiy smiles of caples. CEV LIBOR model. A sraighforward generalizaion of he lognormal LIBOR marke model was examined by Andersen and Andreasen 2b. In heir approach, he assumpion ha he volailiies are deerminisic funcions was replaced by a suiable funcional form of he volailiy coefficien. The main emphasis in Andersen and Andreasen 2b is pu on he use of he CEV process 3 as a model of a forward LIBOR. To be more specific, hey posulae ha, for every [, T j ], dl, T j = L β, T j λ, T j dw Tj+1, where β > is a sricly posiive consan. Under his specificaion of he dynamics of forward LIBORs wih he exponen β 1, hey derive closed-form soluions for caple prices in erms of he cumulaive disribuion funcion of a non-cenral χ 2 probabiliy disribuion. They show also ha, depending on he choice of he parameer β, he implied Black volailiies for caples, when considered as a funcion of he srike level κ >, exhibi downward- or upward-sloping skew. Sochasic volailiy LIBOR model. In a recen paper by Joshi and Rebonao 23, he auhors examine a sochasic volailiy displaced-diffusion exension of a LIBOR marke model. Recall ha he sochasic volailiy displaced-diffusion approach o he modelling of sochasic volailiy was presened in Sec Basically, Joshi and Rebonao 23 posulae ha df, T i + α = f, T i + α µ α, T i d + σ α, T i dw, where he drif erm can be found explicily, and where by assumpion he volailiy σ α, T i is given by he expression: σ α, T i = a + b T i e c T i + d for some mean-revering diffusion processes a, b, ln c and ln d. They argue ha such a model is sufficienly flexible o be capable of describing in a realisic way no only he oday s implied volailiy surface, bu also he observed changes in he marke erm srucure of volailiies. For oher examples of sochasic volailiy exensions of a LIBOR marke model, he ineresed reader is referred o Rebonao and Joshi 21, G aarek 23, and Pierbarg 23a. 3 In he conex of equiy opions, he CEV consan elasiciy of variance process was inroduced by Cox and Ross For more deails, see Sec. 7.2.

45 hp://