Research Paper Series. No. 64. Yield Spread Options under the DLG Model. July, 2009

Transcription

1 Research Paper Series No. 64 Yield Spread Opions under he LG Model Masaaki Kijima, Keiichi Tanaka and Tony Wong July, 2009 Graduae School of Social Sciences, Tokyo Meropolian Universiy Graduae School of Social Sciences, Tokyo Meropolian Universiy Mizuho Securiies Co., Ld.

2 Yield Spread Opions under he LG Model Masaaki Kijima, Keiichi Tanaka and Tony Wong ecember 31, 2008 Absrac In his aricle, we consider opions wrienonyieldspreadssuchasswapspreads and basis swap spreads under he LG model developed by Kijima e al. (2009). For his purpose, we exend he LG model, by shifing he shor raes wih deerminisic funcions of ime, so ha he iniial yield curves implied by he LG model are consisen wih he observed curves in he marke. This is imporan no only for risk managemen purposes, bu also because he drif erm of shor raes affecs he price of spread opions. Some numerical examples are given o discuss he impac of model parameers, in paricular of correlaions, on opion prices for he quadraic Gaussian model and he Hull-Whie model. 1 Inroducion Ineres-rae producs such as bonds, swaps and basis swaps are frequenly raded in he financial marke. However, because of fricions exising in he marke, yield spreads such as swap spreads and basis swap spreads are observed, and marke paricipans need o wach hem carefully in order o manage heir exposures o liquidiy and credi. A swap spread is a swap rae minus a bond yield (or a par yield) wih he same mauriy, whereas a basis swap spread is a fixed spread o be added on eiher leg of a basis swap o exchange LIBORs (London Inerbank Offered Raes) of wo currencies wih principal exchange. From he pricing poin of view, we can regard such a spread conrac as a swap conrac beween wo paries o exchange cash flows ha depend on a yield spread. These spreads are flucuaing o reflec several fricions and/or condiions of demand and supply in he marke. For example, a swap spread is likely o ge wider when fligh o qualiy akes place in he bond marke. A US/JPY basis swap ofen moves when a company issues a JPY denominaed bond and swaps he raised fund ino US. 1 Such a yield curve spread implies a differen qualiy of he wo yield curves. The LG model 2 developed by Kijima e al. (2009) allows one o formulae many yield curves wih differen qualiy under he no-arbirage framework. For risk managemen purposes, opion conracs wrien on such spreads are beneficial o he marke paricipans who pay aenion o he movemen of yield curve spreads. Hence, he pricing of spread opions becomes imporan for hem, in paricular, due Graduae School of Social Sciences, Tokyo Meropolian Universiy, kijima@mu.ac.jp Graduae School of Social Sciences, Tokyo Meropolian Universiy, anaka-keiichi@mu.ac.jp Mizuho Securiies Co. Ld., skw813@gmail.com 1 The lieraure and he background of basis swaps are found in Kijima e al. (2009) and references herein. 2 sands for iscoun, L for LIBOR, and G for Governmen, respecively. 1

3 o he growing concern on credi risk and liquidiy risk. See Carmona and urrleman (2003) for a survey of he pricing of spread opions when he dynamics of underlying asses follows a geomeric Brownian moion. A spread we focus in his aricle follows a more complicaed dynamics han he geomeric Brownian moion, because swaps and basis swaps involve many cash flows during heir lives. However, he spread conrac can be seen as a porfolio of zero-coupon bonds, and an opion on a spread as a swapion wih a floaing srike rae. Employing his idea, his aricle considers an opion on a spread conrac in one currency under he LG model, and derives he opion value ha is naurally quoed in one currency. Our resul is disinc from Brigo and Mercurio (2007) who discussed several quano derivaives on wo currency curves. One of crucial issues on ineres-rae models for praciioners is he fiing of he iniial yield curves. I is useful for risk managemen purposes if he iniial yield curves implied by a model are consisen wih he yield curves observed in he marke. Hull and Whie (1994) firs showed ha i is possible by adding a ime dependen funcion in he drif erm of he shor rae in he Vasicek model (1977). 3 This is equivalen o shifing he shor rae by a deerminisic funcion. Brigo and Mercurio (2001) applied his idea o several shor rae models. Noe ha he drif erm of he shor-rae processes does affec he price of spread opions, while i does no for bond opions in general. This is so, because bonds are raded asses while spreads are no. Under he risk-neural measure, any raded asse has he risk-free shor rae as he drif erm, and his erm will disappear afer he change of measure o he forward measure. See, e.g., Brigo and Mercurio (2007) for deails. In his aricle, we exend he LG model so ha he iniial yield curves implied by he model are consisen wih he observed curves in he marke. The idea is based on he deerminisic shif of shor raes as in Brigo and Mercurio (2001). To his end, we carry ou boosrapping of he discoun facors and he forward raes from observed yield curves. Oher model parameers can be calibraed from he opion prices. This aricle is organized as follows. In he nex secion, we briefly describe he LG model and sudy opions wrien on a swap spread and a basis swap spread. The issue of iniial curve fiing is discussed in Secion 3 for wo shor-rae models, he quadraic Gaussian model and he Hull-Whie model. In Secion 4, we show some numerical examples of yield curves and opion prices under he wo shor rae models. Secion 5 concludes his aricle. 2 The LG Model and Spread Opions In his secion, we provide a brief summary of he LG model developed by Kijima e al. (2009), and sudy opions wrien on a swap spread and a basis swap spread. The LG model was consruced in order o rea muli-qualiy of yield curves under muliple currencies wihin he no-arbirage framework. 2.1 The LG model Consider a marke in which ineres-rae swaps, basis swaps and governmen bonds are raded among marke paricipans. I is assumed ha here exis he hree yield curves, -curve, L-curve and G-curve, in each currency. The -curve is used o discoun cash 3 According o Inui and Kijima (1998), he Hull-Whie model is a special case of he Heah, Jarrow and Moron (HJM) model (1992) wih Markovian sae variables. 2

4 flows. The L-curve deermines he LIBORs so ha i is relaed o swap raes and basis swap raes. The G-curve deermines he governmen bond ( Gov ) raes. Each curve in each currency is associaed wih he shor rae r k () and he zero-coupon bond prices P k (, T ), k =, L, G. Roughly speaking, he spread beween he -curve and he L-curve affecs he basis swap spreads, while he spread beween he L-curve and he G-curve deermines he erm srucure of he swap spread ha is equal o a swap rae minus a bond yield. Alhough we concenrae on he hree curves in his aricle, i is sraighforward o exend he model so as o include oher curves such as corporae bonds. Among currencies, he currency US is supposed o suppor enough liquidiy of he fund o all marke paricipans, so ha US has no fricion beween he -curve and he L-curve. This assumpion is equivalen o saying ha any floaing rae noe wih coupons of US LIBOR is worh a par. 4 Under his seing, i is sufficien o focus on he hree yield curves of he oher currency, say JPY. The uncerainy is represened by a probabiliy space (Ω, F,Q ) on which he shor raes r k (), k =, L, G, are defined by using a hree-dimensional sandard Brownian moion W () =(W (),W L (),W G ()). Here, sands for ransposiion of vecors and marices. The filraion generaed by he Brownian moion is denoed by {F }. The probabiliy measure Q is he risk-neural measure, since we are ineresed in he pricing of financial producs. Hence, he shor rae r () isregardedasherisk-free ineres rae. The expecaion of random variable X condiional on F wih respec o probabiliy measure P is denoed by E P [X]. The forward raes on he L-curve and he G-curve are imporan ingrediens for he pricing of swaps and governmen bonds. The zero-coupon bond price of he k-curve, k =, L, G, is defined by P k (, T )=E Q [ { T exp (r k (s) 12 ) T }] λ k(s) 2 ds λ k (s) dw (s), (1) where λ k () =(λ k (),λl k (),λg k ()) represens he marke prices of risk of he k-curve relaive o he -curve. Hence, we assume λ () =0forhe-curve. For he period [T 1,T 2 ], he ime- forward LIBOR L(, T 1,T 2 ) and forward Gov rae G(, T 1,T 2 ) are calculaed, respecively, by making use of he forward measure Q T 2 ( and 1 L(, T 1,T 2 )= T 2 T 1 ( 1 G(, T 1,T 2 )= T 2 T 1 E QT 2 E QT 2 ) [ PL (T 1,T 2 ) 1] 1 as (2) ) [ PG (T 1,T 2 ) 1] 1. (3) See Kijima e al. (2009) for deails. AswapraeS(, T 0,T N ) is a fixed rae a ime o be exchanged wih he floaing rae L(T i 1,T i 1,T i )aimet i, i =1, 2,,N,forheperiod[T 0,T N ]. On he oher hand, for a basis swap conrac enered a ime, he US LIBOR is exchanged by he JPY LIBOR L(T i 1,T i 1,T i ) plus a basis swap spread bs(, T 0,T N )aimet i, i =1, 2,,N, for he period [T 0,T N ], wih a principal exchange a boh he saring dae and he mauriy. A governmen bond wih mauriy T N is supposed o pay a coupon C(T N )for he period [T 0,T N ], and he ime- bond price is denoed by V (, T N ). 4 One can remove he assumpion if he price of he floaing rae noe wih any mauriy is known and he dynamics of he foreign exchange rae is formulaed in he model. 3

5 In wha follows, we assume ha he same day-coun convenion is applied o all he producs, and he relevan daes T 0 <T 1 < <T = T N are se a regularly spaced ime inervals wih δ = T i T i 1 for all i. The curren ime is any ime on or prior o T 0. Kijima e al. (2009) obained he following hree fundamenal equaions: N i=1 S(, T 0,T N ) = L(, T i 1,T i )P (, T i ) N i=1 P, (4) (, T i ) bs(, T 0,T N ) = P (, T 0 ) P (, T N ) δ N i=1 P S(, T 0,T N ), (5) (, T i ) and N V (, T N )=P (, T 0 )+δ (C(T N ) G(, T i 1,T i )) P (, T i ). (6) i=1 The swap rae (4) is an average of he forward LIBOR s wih he weighs of zerocoupon bond prices implied by he -curve. The basis swap spread (5) can be viewed as a difference beween he swap rae wihou fricions implied by he -curve, i.e. S (, T 0,T N )= P (, T 0 ) P (, T N ) δ N i=1 P, (7) (, T i ) and he swap rae S(, T 0,T N ) wih fricions implied by boh he -curve and he L- curve. If he L-curve coincides wih he -curve compleely, he difference beween he wo swap raes diminishes. Hence, he basis swap spread appears due o such fricions ha are observed as a spread beween he -curve and he L-curve in he marke. The bond price (6) is derived by regarding he bond ransacion as a swap conrac o exchange he fixed coupon C(T N ) wih he floaing Gov raes G(T i 1,T i 1,T i ). I follows ha, in he LG model, he par yield of a governmen bond is given by N i=1 Y (, T 0,T N )= G(, T i 1,T i )P (, T i ) N i=1 P, (8) (, T i ) in he form of a swap rae as if G(T i 1,T i 1,T i ) were he floaing raes. See Kijima e al. (2009) for deails. 2.2 Boosrapping Boosrapping is a mehod o obain he unobservable variables P,L,Grecursively from observed raes and prices in (4), (5) and (6). enoing he observed or implied prices in he marke by superscrip M, one can carry ou boosrapping o obain P M (0,T i ) = P M(0,T 0) δ ( S M (0,T 0,T i )+bs M (0,T 0,T i ) ) i 1 j=1 P M(0,T j) 1+δ (S M (0,T 0,T i )+bs M, (0,T 0,T i )) (9) L M (0,T i 1,T i ) = SM (0,T 0,T i ) i j=1 P M(0,T j) S M (0,T 0,T i 1 ) i 1 j=1 P M(0,T j) P M(0,T, i) G M (0,T i 1,T i ) = C(T i ) 1 [ δp M (0,T V M (0,T i ) P M (0,T 0 ) i) j=1 (10) i 1 ( δ C(Ti ) G M (0,T j 1,T j ) ) P j)] M (0,T, (11) 4

6 where S M (0,T 0,T i ),bs M (0,T 0,T i ),V M (0,T i ), i = 1, 2,,N are he observed yield curves. See Kijima e al. (2009) for deails. Le us compare he above resuls wih he classical boosrapping of swap raes in which basis swaps are no aken ino consideraion. By he classical boosrapping, one obains he ordinary discoun facors P (T i )andforwardlibor s L(T i 1,T i )as P (T i )= P (T 0 ) δs M (0,T 0,T i ) i 1 j=1 P (T j ) 1+δS M (0,T 0,T i ) (12) and L(T i 1,T i )= SM (0,T 0,T i ) i P j=1 (T j ) S M (0,T 0,T i 1 ) i 1 P j=1 (T j ), (13) P (0,T i ) respecively. If one regards a swap as an exchange of a fixed coupon bond and a floaing coupon bond, he values of he fixed leg and he floaing leg of he swap are boh zero, i.e. 0 = P (T 0 )+δ i L(T j 1,T j ) P (T j )+ P (T i ) j=1 = P (T 0 )+δs M (0,T 0,T i ) i P (T j )+ P (T i ). (14) Also, under he classical boosrapping, he classical forward LIBOR is calculaed as ( ) P (Ti 1 ) L(T i 1,T i )=δ 1. (15) P (T i ) Hence, he classical model (14) canno express a non-zero basis swap spread, because 0 P (T 0 )+δ i j=1 j=1 ) ( L(Tj 1,T j )+bs M (0,T 0,T i ) P (Tj )+ P (T i ) in general. On he oher hand, boosrapping of he LG model (9) (10) implies ha he legs are no worh zero bu he minus of he basis swap muliplied by he annuiy, i.e. I follows ha 0= P M (0,T 0 )+δ bs M (0,T 0,T i ) = P M (0,T 0)+δ i P M (0,T j) j=1 i L M (0,T j 1,T j )P M (0,T j)+p M (0,T i) j=1 = P M (0,T 0)+δS M (0,T 0,T i ) i P M (0,T j)+p M (0,T i). (16) j=1 i ( L M (0,T j 1,T j )+bs M (0,T 0,T i ) ) P M (0,T j )+P M (0,T i ). j=1 Hence, he LG model is consisen wih a non-zero basis swap spread. 5

7 Equaion (16) shows ha negaive basis swap spreads lead o higher discoun facors han he classical ones, i.e. P M (0,T) > P (T ). Neverheless, he forward LIBOR s are similar, i.e. L M (0,T i 1,T i ) L(T i 1,T i ), as we shall see laer in numerical examples. These findings have imporan implicaions in he LG model. As Kijima e al. (2009) saed, on-he-marke swap values are always zero, independen of he boosrapping mehod, while off-he-marke swap values may be disinc over he wo boosrapping mehods due o he exisence of basis swap spreads. 2.3 Opions on spread Le us consider opions on a swap spread and a basis swap spread by making use of he muli-qualiy curves. Needless o say, an opion on a spread is an insurance agains undesired movemens of he spread. In his aricle, a call opion on a swap spread is defined as a conrac o give he opion buyer he righ o ener ino a swap o receive coupons of a swap spread S(T 0,T 0,T N ) Y (T 0,T 0,T N ) prevailing a he opion expiry T 0 by paying coupons of a fixed spread k, called he srike spread, for he same period [T 0,T N ]. The opion buyer will see a benefi when he swap spread ges wider han he srike spread a he opion expiry. Since he period for he observaion o deermine he spread a he expiry is mached wih he period of he underlying swap, he call opion on a swap spread is clearly equivalen o a payer swapion o pay a fixed rae of Y (T 0,T 0,T N )+k versus o receive LIBOR. This fac is also confirmed by he following opion price formula a ime : ( N ) P (, T 0 )E QT 0 δ (S(T 0,T 0,T N ) Y (T 0,T 0,T N ) k) P (T 0,T i ) = P (, T 0 )E QT 0 i=1 ( N ) δ (L(T 0,T i 1,T i ) (Y (T 0,T 0,T N )+k)) P (T 0,T i ) i=1 where (x) + =max{x, 0}. Here, equaliy follows from (4). Recall from (8) ha we can calculae he bond par yield as a swap rae consruced by he -curve and G-curve. Thus, in he LG model, he call opion can be calculaed as a payer swapion wih he floaing srike rae of Gov swap rae plus he srike spread agains LIBOR. A pu opion on a swap spread is parallel o he call opion and equivalen o a receiver swapion wih a floaing srike rae. 5 A call opion on a basis swap spread gives he buyer he righ o ener ino a swap o receive coupons of a basis swap spread bs(t 0,T 0,T N ) prevailing a he opion expiry T 0 by paying coupons of a fixed spread k for he same period [T 0,T N ]. Since he basis swap spread is a spread beween wo swap raes by (5), i.e. S (T 0,T 0,T N ) S(T 0,T 0,T N ), he opion price is wrien as ( N ) P (, T 0 )E QT 0 δ (bs(t 0,T 0,T N ) k) P (T 0,T i ) = P (, T 0 )E QT 0 i=1 ( N ) δ ((S (T 0,T 0,T N ) k) L(T 0,T i 1,T i )) P (T 0,T i ) i= ,. 5 An opion o ener ino an asse swap can be formulaed similarly. 6

8 This implies ha he call opion is equivalen o a receiver swapion o receive a fixed rae of S (T 0,T 0,T N ) k versus o pay LIBOR. The same argumens apply o a pu opion on a basis swap. Remark 2.1. The above discussions can be generalized o opions wih mauriy T m of he swap period o exchange cash flows ha is differen from he mauriy T N of swaps and/or bonds. Tha is, we hen have ( m ) P (, T 0 )E QT 0 δ (S(T 0,T 0,T N ) Y (T 0,T 0,T N ) k) P (T 0,T i ). i=1 Typical examples are caps and floors on hese spreads. The analyical expressions of he above opion prices reveal ha i reduces o a calculaion of an opion on a rae wih convexiy adjusmen, which has been well sudied for CMS (consan mauriy swaps). A Mone Carlo simulaion is a simple means o evaluae such opions, alhough i is ofen very ime-consuming. However, as demonsraed in Tanaka e al. (2007), he usage of bond momens is efficien when pricing opions for a cerain class of underlying ineres rae models, including affine erm srucure models and quadraic Gaussian models. In Secion 4, we presen some numerical examples of sandard spread opions wih m = N. 3 Yield Curve Fiing Following he spiri of Hull and Whie (1994) and Brigo and Mercurio (2001), his secion exends he resuls of Kijima e al. (2009) so ha he iniial curves implied by he LG model are consisen wih observed raes in he marke by using deerminisic shifing funcions. Simpler versions of he shor rae models sudied in his secion have been discussed in Kijima e al. (2009) wihou he iniial curve fiing. 3.1 Quadraic Gaussian model The firs example is a quadraic Gaussian model where he -curve is consruced by a quadraic Gaussian model of Pelsser (1997) while each of he L-curve and he G-curve has a Gaussian spread over he -curve. Namely, we assume ha r () = (x ()+α + β) 2 + ϕ (), r L () = r ()+h L (), h L () =x L ()+ϕ L (), r G () = r ()+h G (), h G () =x G ()+ϕ G (), where x k () are he Ornsein-Uhlenbeck processes give by dx k () = a k x k ()d + σ k dw k (), x k (0) = 0, k =, L, G, and where ϕ k () are deerminisic funcions of ime ha are o be joinly deermined from observed iniial curves. 6 The Brownian moions W (), W L (), W G () are independen of each oher under Q. The marke prices of risk are assumed o be given by λ () λl () λg () λ L () λl L () λg L () = 0 λ L 0 (17) λ G () λl G () λg G () 0 0 λ G 6 The shifing funcion ϕ () can be negaive, so ha he shor rae r () may become negaive, alhough r () ϕ () mus be nonnegaive. + 7

9 wih some consans λ L,λ G. For he implemenaion purpose, i is enough o consider he inegral of ϕ k, k =, L, G, givenby Φ k (T ) T 0 ϕ k ()d. (18) According o Pelsser (1997), he zero-coupon bond price of he -curve is an exponenial of he quadraic funcion of x.moreprecisely,wehave { T } P (, T )=exp ϕ (s)ds + A (, T ) B (, T )x () C (, T )x () 2, (19) where γ = a 2 +2σ2, ( ) 1 F (, T ) = 2γe γ(t ) (γ + a )e 2γ(T ) + γ a, ( )( ) 1 C (, T ) = e 2γ(T ) 1 (γ + a )e 2γ(T ) + γ a, T α + βs B (, T ) = 2F (, T ) F (s, T ) ds and T ( ) 1 A (, T )= 2 σ2 B (s, T ) 2 σ 2 C (s, T ) (α + βs) 2 ds. Explici formulas for B and A are obained in Kijima e al. (2008). For he reader s convenience, hey are shown in Appendix. Noe ha, by seing = 0 in (19), Φ mus be given by Φ (T ) = ln P M (0,T)+A (0,T). (20) As o he oher curves, we need o calculae he forward raes (2) and (3). To his end, define for j, k = L, G B k (, T )= 1 ( ) 1 e a k(t ), B jk (, T )= 1 e (a j +a k )(T ). a k a j + a k From he dynamics of he Ornsein-Uhlenbeck processes x k (), k = L, G, wehave T Hence, he inegral T E Q Var Q T x k (s)ds = B k (, T )x k () σ k B k (s, T )dw k (s). x k (s)ds is normally disribued wih [ T ] x k (s)ds = B k (, T )x k (), [ T ] x k (s)ds = σ2 k a 2 (2B k (, T ) B kk (, T )+T ) k and [ T Cov Q ] x k (s)ds, W k (T ) W k () = σ k (B k (, T )+T ). a k 8

10 By making use of he independence of Brownian moions, we hen obain from (1) wih k = L ha where P L (, T )=P (, T )exp{b L (, T )x L ()+A L (, T )}, A L (, T ) = Φ L (T )+Φ L ()+ σ2 L 2a 2 (2B L (, T ) B LL (, T )+T ) L σ L λ L (B L (, T )+T ). (21) a L The forward LIBOR (2) is given by L(, T i 1,T i )= 1 ( ) P (, T i 1 ) δ P (, T i ) K L(, T i 1,T i ) 1, (22) where { K L (, T i 1,T i ) = exp A L (T i 1,T i ) σ2 L 2 B L(T i 1,T i ) 2 B LL (, T i 1 ) } e a L(T i 1 ) B L (T i 1,T i )x L (). (23) By comparing (22) wih (15), we find ha he funcion K L represens he effec of basis swap spreads. On he oher hand, by he resuls of boosrapping (9) (10), we know he iniial value of K L implied by he observed raes in he marke as KL M (0,T i 1,T i )= P M(0,T i) ( 1+δL M P M(0,T (0,T i 1,T i ) ). (24) i 1) Therefore, by plugging (21) ino (23) wih =0,Φ L mus saisfy Φ L (T i ) = Φ L (T i 1 )+lnk M L (0,T i 1,T i ) + σ2 L 2a 2 (2B L (T i 1,T i ) B LL (T i 1,T i )+T i T i 1 ) (25) L σ L λ L (B L (T i 1,T i )+T i T i 1 )+ σ2 L a L 2 B L(T i 1,T i ) 2 B LL (0,T i 1 ), whichisaformulaoconsrucφ L (T i )wihφ L (0) = 0. Noe ha some inerpolaion mehod, such as he cubic spline mehod, should be applied o calculae Φ L () for (T i,t i+1 ). However, by consrucion, Φ and Φ L defined in his way can generae iniial curves of swap raes and basis swap spreads which are consisen wih he observed raes in he marke. For he evaluaion of swaps and swapions, he following represenaion may be useful: P (, T ) = P M(0,T) { P M(0,) exp A (, T ) A (0,T)+A (0,) B (, T )x () C (, T )x () 2}, (26) L(, T 1,T 2 ) = 1 ( P M (0,T 1 ) { δ P M(0,T 2) KM L (0,T 1,T 2 )exp (B (, T 1 ) B (, T 2 ))x () (C (, T 1 ) C (, T 2 ))x () 2 e a L(T 1 ) B L (T 1,T 2 )x L () } ) + M L (, T 1,T 2 ) 1, (27) M L (, T 1,T 2 ) = A (, T 1 ) A (0,T 1 ) A (, T 2 )+A (0,T 2 ) σ2 L 2 B L(T 1,T 2 ) 2 (B LL (, T 1 ) B LL (0,T 1 )). 9

11 One may apply Mone Calro simulaion o (26) and (27) under he T 0 -forward measure Q T 0. Since he vecor process (W T 0 (),WT 0 L (),WT 0 G ()) defined by dw T 0 () = dw () σ (B (, T 0 )+2C (, T 0 )x ()) d, dw T 0 L () = dw L(), dw T 0 G () = dw G(), follows he hree-dimensional sandard Brownian moion under T 0 -forward measure Q T 0 by he Girsanov heorem, he dynamics of x k () under Q T 0 are given by dx () = (σ B (, T 0 )+(2σ C (, T 0 ) a ) x ()) d + σ dw T 0 (), dx L () = a L x L ()d + σ L dw T 0 L (), dx G () = a G x G ()d + σ G dw T 0 G (). Hence, x k (T 0 ), k =, L, G, are again normally disribued under he T 0 -forward measure. Finally, using he same argumens, we can derive an explici formula for he G-curve in a compleely parallel form. In fac, i is enough o replace he noaion L wih G. The only difference is wheher he iniial curve is calculaed in raes in (10) or bond prices in (11). If he iniial yield curve of bond yields is given as he par yields raher han bond prices, he formula of Φ G is compleely he same as Φ L due o he form of (8). 3.2 The Hull-Whie model The second example is a correlaed Gaussian model a he sacrifice of non-negaiviy in he shor rae. Tha is, following he idea of Hull and Whie (1994), suppose ha r () = x ()+ϕ (), r L () = r ()+h L (), h L () =x L ()+ϕ L (), r G () = r ()+h G (), h G () =x G ()+ϕ G (), where x k () are he Ornsein-Uhlenbeck processes given by dx k () = a k x k ()d + σ k dw k (), x k (0) = 0, k =, L, G, and where ϕ k () are deerminisic funcions of ime. Again, i is sufficien o specify he inegral Φ k (T ) T 0 ϕ k ()d, k =, L, G. (28) In order o inroduce correlaions among x k () k =, L, G, we assume ha he Brownian moions W k () are correlaed as dw ()dw L () =ρ L d, dw ()dw G () =ρ G d, dw L ()dw G () =ρ LG d. The marke prices of risk are assumed o be given by (17). efine he funcions B k (, T )= 1 ( ) 1 e a k(t ), B jk (, T )= 1 e (a j +a k )(T ) a k a j + a k 10

12 for j, k =, L, G. Then, we obain { P (, T ) = exp Φ (T )+Φ ()+B (, T )x () + σ2 } 2a 2 (2B (, T ) B (, T )+T ). Thus, Φ is given by Φ (T ) = ln P M (0,T)+ σ2 2a 2 (2B (0,T) B (0,T)+T). (29) Nex, we consider he forward LIBOR L(, T i 1,T i ). Wrie Equaion (1) for k = L as P L (, T )=E Q [e X ], where X is a Gaussian random variable given by T X = ( x (s)+ϕ (s)+x L (s)+ϕ L (s)+ 1 2 λ2 L = B (, T )x ()+σ T ) T ds λ L dw L (s) T B (s, T )dw (s)+b L (, T )x L ()+σ L Φ (T )+Φ () Φ L (T )+Φ L () 1 2 λ2 L(T ) λ L (W L (T ) W L ()). Since he random variable X has he mean and he variance E Q [X] = B (, T )x ()+B L (, T )x L () Φ (T )+Φ () Φ L (T )+Φ L () 1 2 λ2 L(T ) B L (s, T )dw L (s) Var Q [X] = σ2 a 2 (2B (, T ) B (, T )+T )+ σ2 L a 2 (2B L (, T ) B LL (, T )+T ) L + λ 2 L(T ) 2λ L ρ L σ a (B (, T )+T ) 2λ L σ L a L (B L (, T )+T ) +2 ρ Lσ σ L a a L (B (, T )+B L (, T ) B L (, T )+T ), we can calculae he expecaion o yield P L (, T ) = P (, T )exp{b L (, T )x L ()+A L (, T )+A L (, T )}, where A L and A L are defined as and A L (, T ) = Φ L (T )+Φ L ()+ σ2 L 2a 2 (2B L (, T ) B LL (, T )+T ) L σ L λ L (B L (, T )+T ) a L A L (, T ) = σ2 ρ L σ 2a 2 (2B (, T ) B (, T )) λ L (B (, T )+T ) a + ρ Lσ σ L (B (, T )+B L (, T ) B L (, T )+T ), a a L 11

13 respecively. Noe ha A L is he erm relaed o he L-curve only while A L is he erm relaed o an ineracion beween he L-curve and he -curve. The forward LIBOR is calculaed as L(, T i 1,T i )= 1 ( ) P (, T i 1 ) δ P (, T i ) K L(, T i 1,T i ) 1, where { K L (, T i 1,T i ) = exp A L (T i 1,T i ) A L (T i 1,T i ) 1 2 B L(T i 1,T i ) 2 σl 2 B LL(, T i 1 ) ( B L (T i 1,T i ) x L ()e a L(T i 1 ) ρ ) Lσ σ } L (B L (, T i 1 ) B L (, T i 1 )). a I follows ha Φ L mus saisfy Φ L (T i ) = Φ L (T i 1 )+lnk M L (0,T i 1,T i ) + σ2 L 2a 2 (2B L (T i 1,T i ) B LL (T i 1,T i )+T i T i 1 ) L σ L λ L (B L (T i 1,T i )+T i T i 1 ) a L + A L (T i 1,T i )+ 1 2 B L(T i 1,T i ) 2 σlb 2 LL (0,T i 1 ) ρ Lσ σ L a B L (T i 1,T i )(B L (0,T i 1 ) B L (0,T i 1 )), (30) where KL M is given by (24). The remaining calculaions are carried ou in he same way as he quadraic Gaussian case. A las, similar o he quadraic Gaussian case, he following expressions will be useful for he evaluaion of swaps and swapions: P (, T ) = P M(0,T) P M(0,) exp {A (, T )+B (, T )x ()}, A (, T ) = σ2 2a 2 ( 2B (, T ) B (, T ) (2B (0,T) B (0,T)) ) +2B (0,) B (0,). ( P M (0,T 1 ) { P M(0,T 2) KM L (0,T 1,T 2 )exp (B (, T 1 ) B (, T 2 ))x () L(, T 1,T 2 ) = 1 δ } e a L(T 1 ) B L (T 1,T 2 )x L ()+M L (, T 1,T 2 ) ) 1 M L (, T 1,T 2 ) = A (, T 1 ) A (, T 2 ) σ2 L 2 B L(T 1,T 2 ) 2 (B LL (, T 1 ) B LL (0,T 1 )) + ρ Lσ σ L B L (T 1,T 2 )(B L (, T 1 ) B L (, T 1 ) (B L (0,T 1 ) B L (0,T 1 ))). a 4 Numerical examples In his secion, we presen some numerical examples of opion evaluaion implied by he LG model whose iniial curve is consisen wih observed marke raes. Suppose ha a ime = 0 we observe swap raes S(T ), basis swap spreads bs(t ) and bond par yields Y (T ) for several mauriies T in he marke as indicaed in Table 1. 12

14 Throughou he numerical examples, swaps and bonds are assumed o have semi-annual coupon paymens. Using he classical boosrapping (12) (13), we calculae he zero rae Z(T )= ln P (T )/T,heforwardLIBOR L(T )= L(T 0.5,T), and he forward Gov rae G(T )= G(T 0.5,T) by assuming ha he bond par yield is a swap rae. For a comparison wih he LG boosrapping (9) (11), we give he adjused zero rae Z(T )+bs(t ) ha reflecs he basis swap spread in he las column of Table 1. The model parameers of he quadraic Gaussian model (QG model) and he Hull- Whie model (HW model) discussed in he previous secion are se as shown in Table 2. σ L is se higher han σ G in each model. The correlaions of he HW model ρ = (ρ L,ρ G,ρ LG ) will be specified laer in each example. Given hese informaion, we can calculae he shifing funcions Φ, Φ L and Φ G in each model. Resuls for he QG model are shown in Table 3. Using he LG boosrapping (9) (11), we obain he iniial zero rae Z(T )= ln P M (0,T)/T,heforward LIBOR L(T )=L M (0,T 0.5,T) and he forward Gov rae G(T )=GM (0,T 0.5,T). I is ineresing o noe ha Z(T ) Z(T )+bs(t ), L(T ) L(T ), G(T ) G(T ). In paricular, he differences in he forward raes are wihin 0.2 basis poins. Roughly speaking, he zero rae Z(T ) is lower han he classical zero rae Z(T ) by he basis swap spread, alhough he forward raes are kep o be he same. Therefore, as explained in Secion 2.2, here exiss a difference in he evaluaion of off-he-marke swaps by he annuiy of he basis swap spreads beween he wo boosrapping mehods. For example, he value of annuiy for 10 years is i=1 P (0,i/2) = in boosrapping of he QG model, while i is 1 20 P 2 i=1 (i/2) = in he classical boosrapping. The shifing funcions in he HW model are shown in Table 4. Noe ha Φ is independen of he correlaions by (29), while Φ L and Φ G depend on he correlaion ρ L and ρ G, respecively, due o (30); bu hey are independen of ρ LG. The boosrapped raes Z(T ),L(T),G(T) in hese models are very close o he resuls obained in he QG model (see Table 3) and are omied. Prices of swapions and bond opions sruck a he ATMF (a-he-money-forward) rae are calculaed by Mone Carlo simulaion wih 100,000 runs. In hese numerical examples of bond opions, we consider opions on ficiious bonds. Namely, an ATMF bond opion means an opion on a bond whose coupon is equal o he ATMF yield wih a srike price of a par, no an opion on a bond wih a srike price of he ATMF price of he bond. I is a benefi of he LG model ha such a bond opion is equivalen o a swapion agains Gov raes consruced by he G-curve. There are wo ypes of he implied volailiy for a price of swapion or bond opion; he yield volailiy and he absolue volailiy. The absolue volailiy is he annual sandard deviaion of movemens of a paricular forward swap rae (or bond yield). The yield volailiy is he Black-Scholes ype volailiy or he relaive volailiy ha equals he absolue volailiy divided by he ATMF rae. Thus, he absolue volailiy assumes a normal disribuion for he underlying raes, while he yield volailiy assumes a lognormal disribuion. Prices and volailiies of ATMF receiver s swapions and ATMF call bond opions are shown in Table 5 for he QG model and Tables 6 8 for he HW model. A he firs glance of he volailiy erm srucure in he QG model, we observe a decreasing volailiy along he underlying mauriies. Opions on shor-daed underlyings have high volailiies compared wih opions on long-daed underlyings wih he same expiry. The HW model exhibis flaer erm srucures of volailiy han he QG model. Le us compare he prices of swapions wih hose of bond opions, where he correlaions ρ L and ρ G play an imporan role for opion prices bu ρ LG does no. In he case of non-negaive correlaions (see Tables 6 and 7), he prices of swapions are slighly 13

15 higher han hose of bond opions wih he same expiry and underlying mauriy, since he volailiy σ L is se higher han σ G. Therefore, in his case, he absolue volailiies of swapions are slighly higher han hose of bond opions, while he yield volailiies of swapions are lower han hose of bond opions due o relaively higher ATMF raes of swaps han ATMF yields of bonds. On he oher hand, negaive correlaions produce lower absolue volailiies of swapions han hose of bond opions (see Table 8). By comparing Table 6 wih Tables 7 and 8 in he HW model, higher correlaion ρ L (or ρ G ) beween he L-curve (or G-curve) and he -curve yields higher volailiies. In Tables 9 13, we show prices of spread opions and he absolue volailiies. 7 Regardless of differen levels of volailiies of he -curves beween he QG model and he HW model, Tables 9 (QG model) and 10 (HW model) show very close prices of spread opions, since he dynamics of x L and x G are he same. By comparing Tables 11 and 12, i is eviden ha he prices of opions on basis swaps are he same because of he same correlaion ρ L, while he prices of swap spread opions are quie differen due o differen correlaion ρ LG. Posiive ρ LG yields low prices of swap spread opions. Posiive (or negaive) ρ L makes high (or low) prices of opions on basis swap as shown in Tables Conclusion The LG model allows us o formulae many yield curves wih differen qualiy under he no-arbirage seing. This aricle demonsraes he usefulness of our model, especially in he pricing of spread opions, and how o consruc a shor rae model whose iniial curves are consisen wih he observed curves in he marke. The consrucion of hese yield curves is carried ou by deerminisic shifing funcions of shor raes and boosrapping of he discoun facors of he cash flows and he forward raes. Oher model parameers can be calibraed from he opion prices. I becomes clear ha correlaions beween curves play a crucial role for he pricing of spread opions. 7 The yield volailiies do no bring meaningful informaion due o he small numbers of he ATMF raes in hese spread opions. 14

16 References [1] Brigo,. and F. Mercurio (2001), A deerminisic-shif exension of analyicallyracable and ime-homogenous shor-rae models, Finance and Sochasics, 5, [2] Brigo,. and F. Mercurio (2007), Ineres Rae Models: Theory and Pracice: Wih Smile, Inflaion and Credi, Second Ediion, Springer-Verlag, New York. [3] Carmona, R. and V. urrleman (2003), Pricing and hedging spread opions, SIAM Review, 45, [4] Heah., R. Jarrow, and A. Moron (1992), Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claims valuaion, Economerica, 60, [5] Hull, J. and A. Whie (1994), Numerical procedures for implemening erm srucure models I: Single-facor models, Journal of erivaives, 2, [6] Inui, K. and M. Kijima (1998), A Markovian framework in muli-facor Heah- Jarrow-Moron models, Journal of Financial and Quaniaive Analysis, 33, [7] Kijima, M., K. Tanaka and T. Wong (2009), A muli-qualiy model of ineres raes, Quaniaive Finance, forhcoming. [8] Pelsser, A. (1997), A racable yield-curve model ha guaranees posiive ineres raes, Review of erivaives Research, 1, [9] Tanaka, K., T. Yamada and T. Waanabe (2007), Applicaions of Gram Charlier expansion and bond momens for pricing of ineres raes and credi risk, Working Paper, Tokyo Meropolian Universiy. [10] Vasicek, O.A. (1977), An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, 5,

17 Table 1: Iniial yield curve (percen) Mauriy Swap Basis swap Bond Swap spread Classical boosrapping T (years) S(T ) bs(t ) Y (T ) S(T ) Y (T ) Z(T ) L(T ) G(T ) Z(T )+bs(t ) Table 2: Model parameers Quadraic Gaussian (QG) model Hull-Whie (HW) model a =0.07, σ =0.0750, α = β =0 a =0.07, σ =0.0090, a L =0.04, σ L =0.0020, λ L =0 a L =0.04, σ L =0.0020, λ L =0 a G =0.04, σ G =0.0010, λ G =0 a G =0.04, σ G =0.0010, λ G =0 Table 3: Boosrapping of he QG model (percen) Mauriy Shifing funcion Boosrapping T Φ (T ) Φ L (T ) Φ G (T ) Z(T ) L(T ) G(T )

18 Table 4: Shifing funcions of he HW model (percen) Mauriy ρ =(0, 0, 0) ρ =(0.4, 0.4, 0.4) T Φ (T ) Φ L (T ) Φ G (T ) Φ (T ) Φ L (T ) Φ G (T ) Table 5: Yields and volailiies in he QG model Swapion Bond opion Opion Expiry Swap Mauriy (years) Bond Mauriy (years) (years) ATMF rae (percen) Opion price (basis poin) Yield volailiy (percen) Absolue volailiy (basis poin)

19 Table 6: Volailiies in he HW model wih ρ =(0, 0, 0) Swapion Bond opion Opion Swap Mauriy Bond Mauriy Expiry Yield volailiy (percen) Absolue volailiy (basis poin) Table 7: Volailiies in he HW model wih ρ =(0.4, 0.4, 0.4) Swapion Bond opion Opion Swap Mauriy Bond Mauriy Expiry Yield volailiy (percen) Absolue volailiy (basis poin) Table 8: Volailiies in he HW model wih ρ =( 0.4, 0.4, 0) Swapion Bond opion Opion Swap Mauriy Bond Mauriy Expiry Yield volailiy (percen) Absolue volailiy (basis poin)

20 Table 9: Spread opions in he QG model Opion on swap spread Opion on basis swap Opion Mauriy Mauriy Expiry ATMF rae (basis poin) ATMF Opion price (basis poin) Absolue volailiy (basis poin) Table 10: Spread opions in he HW model wih ρ =(0, 0, 0) Opion on swap spread Opion on basis swap Opion Mauriy Mauriy Expiry ATMF Opion price (basis poin) Absolue volailiy (basis poin)

21 Table 11: Spread opions in he HW model wih ρ =(0.4, 0.4, 0.4) Opion on swap spread Opion on basis swap Opion Mauriy Mauriy Expiry ATMF Opion price (basis poin) Absolue volailiy (basis poin) Table 12: Spread opions in he HW model wih ρ =(0.4, 0.4, 0.4) Opion on swap spread Opion on basis swap Opion Mauriy Mauriy Expiry ATMF Opion price (basis poin) Absolue volailiy (basis poin) Table 13: Spread opions in he HW model wih ρ =( 0.4, 0.4, 0) Opion on swap spread Opion on basis swap Opion Mauriy Mauriy Expiry ATMF Opion price (basis poin) Absolue volailiy (basis poin)

22 A We have Formulas T B (, T )=2F (, T ) α + βs F (s, T ) ds = 2B 1(, T ) γ 2 A 5 (, T ) and T ( ) 1 A (, T ) = 2 σ2 B (s, ) 2 σ 2 C (s, T ) (α + βs) 2 ds ( ) = σ 2 A4 (, T ) γ 5 A 5 (, T ) + A 6(, T ) α 2 (T ) αβ(t 2 2 ) 1 3 β2 (T 3 3 ), where Γ a = γ a, Γ b = γ + a, A 1a (, T ) = e γ(t ) +4 e γ(t ) (3 + 2γ(T )), A 1b (, T ) = e γ(t ) 4+e γ(t ) (3 2γ(T )), A 2a (, T ) = e γ(t ) (1 γt) 2(1 γ( + T )) + e γ(t ) (1 γ(2 + T )+γ 2 ( 2 T 2 )), A 2b (, T ) = e γ(t ) (1 + γt) 2(1 + γ( + T )) + e γ(t ) (1 + γ(2 + T )+γ 2 ( 2 T 2 )), A 3a (, T ) = 4γ(1 γt) e γ(t ) (1 γt) 2 +e (1+2γ γ(t ) γ 2 (2 2 + T 2 )+ 2 ) 3 γ3 ( 3 T 3 ), A 3b (, T ) = 4γ(1 + γt)+e γ(t ) (1 + γt) 2 +e ( 1+2γ γ(t ) + γ 2 (2 2 + T 2 )+ 2 ) 3 γ3 ( 3 T 3 ), ( A 4 (, T ) = Γ a α 2 γ 2 A 1a (, T )+2αβγA 2a (, T )+β 2 A 3a (, T ) ) ( +Γ b α 2 γ 2 A 1b (, T )+2αβγA 2b (, T )+β 2 A 3b (, T ) ), A 5 (, T ) = Γ a e γ(t ) +Γ b e γ(t ), A 6 (, T ) = 1 2 (T ) ( Γ 1 a Γ 1 ) 1 ( b + Γ 1 a +Γ 1 ) A 5 (, T ) 2γ b ln 2γ and B 1 (, T ) = αγ ( e γt e γ)( Γ a e γ +Γ b e γt) ( ) + β Γ a e γ(t ) (1 γ)+γ b e γ(t ) (1 + γ) Γ a (1 γt) Γ b (1 + γt). 21