A Joint Framework for Consistently Pricing Interest Rates and Interest Rate Derivatives

Transcription

1 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Vol. 44, No. 3, June 2009, pp COPYRIGHT 2009, MICHAEL G. FOSTER SCHOOL OF BUSINESS, UNIVERSITY OF WASHINGTON, SEATTLE, WA doi: /s A Join Framework for Consisenly Pricing Ineres Raes and Ineres Rae Derivaives Massoud Heidari and Liuren Wu Absrac Dynamic erm srucure models explain he yield curve variaion well bu perform poorly in pricing and hedging ineres rae opions. Mos exising opion pricing pracices ake he yield curve as given, hus having lile o say abou he fair valuaion of he underlying ineres raes. This paper proposes an m + n model srucure ha bridges he gap in he lieraure by successfully pricing boh ineres raes and ineres rae opions. The firs m facors capure he yield curve variaion, whereas he laer n facors capure he ineres rae opions movemens ha canno be effecively idenified from he yield curve. We propose a sequenial esimaion procedure ha idenifies he m yield curve facors from he LIBOR and swap raes in he firs sep and he n opions facors from ineres rae caps in he second sep. The hree yield curve facors explain over 99% of he variaion in he yield curve bu accoun for less han 50% of he implied volailiy variaion for he caps. Incorporaing hree addiional opions facors improves he explained variaion in implied volailiies o over 99%. I. Inroducion In pricing ineres raes and ineres rae opions, he lieraure akes wo disinc approaches. The firs approach, ofen referred o as a dynamic erm srucure model, capures he dynamics of he yield curve wih a finie-dimensional sae vecor. Empirical sudies show ha well-designed dynamic erm srucure models can explain over 90% of he variaion on he yield curve wih as few as hree facors (Dai and Singleon (2000)). However, hese models perform poorly in pricing and hedging ineres rae opions (Li and Zhao (2006)). In pracice, he ask of pricing ineres rae opions has mosly been handled by alernaive approaches ha ake he yield curve as given and focus exclusively Heidari, massoud.heidari@ccm.naixis.com, Caspian Capial Managemen, LLC, 745 5h Ave., 28h floor, New York, NY 10151; Wu, liuren.wu@baruch.cuny.edu, Baruch College, Zicklin School of Business, 1 Bernard Baruch Way, Box B10-225, New York, NY We hank Rene Carmona, Peer Carr, Jefferson Duare (he referee), Damir Filipović, Ayman Hindy, Chi-fun Huang, Gregory Klein, Karl Kolderup, Markus Leippold, Bill Lu, Paul Malaesa (he edior), Kenneh Singleon, and seminar paricipans a Oak Hill Plainum Parners, LLC and Princeon Universiy for commens. We hank Lehman Brohers for providing he daa. 517

2 518 Journal of Financial and Quaniaive Analysis on he specificaion of he volailiy srucure. 1 Ye, by aking he yield curve as given, hese models have lile o say abou he fair valuaion of he underlying ineres raes. Furhermore, accommodaing he whole yield curve ofen necessiaes acceping an infinie-dimensional sae space and/or ime-inhomogeneous model paramerizaion, boh of which creae difficulies for hedging pracices. In his paper, we propose an m + n model srucure ha bridges he gap beween he wo exising approaches by successfully pricing boh ineres raes and ineres rae opions wihin a finie-dimensional framework. Under his framework, he firs m facors capure he sysemaic variaion of he yield curve and hence are referred o as he yield curve facors. The laer n facors capure he ineres rae opions movemens ha canno be effecively idenified from he yield curve. We label hem as he ineres rae opions facors. We elaborae on he model srucure hrough a simple Gaussian affine example. Firs, we use hree facors o price he yield curve. Then, we capure he remaining ineres rae opions movemens wih anoher hree facors. We esimae he model using eigh years of daa on U.S. dollar London Inerbank Offered Rae (LIBOR), swap raes, and implied volailiies for he ineres rae caps. The esimaion is performed wih a quasi-maximum likelihood mehod joinly wih exended Kalman filer via a wo-sage sequenial procedure. In he firs sage, we esimae he dynamics of he yield curve facors using he LIBOR and swap raes. In he second sage, we esimae he dynamics of he opions facors based on he marke quoes on he implied volailiies for he ineres rae caps. Despie is simple srucure, our model performs well in pricing boh ineres raes and ineres rae caps. The hree yield curve facors explain over 99% of he variaion in he yield curve bu less han 50% of he variaion in he implied volailiy. By incorporaing hree addiional opions facors, he model also explains over 99% of he variaion in he implied volailiy. We perform an ou-of-sample analysis by reesimaing he model using he firs six years of daa and hen comparing he model s performance boh in sample during he firs six years and ou of sample during he las wo years. The subsample parameer esimaes are similar o hose from he whole sample, and he performance comparison shows no deerioraion for he ou-of-sample performance, suggesing ha he model srucure is sable. For furher robusness analysis, we also esimae an alernaive specificaion for he hree yield curve facors ha allow for boh sochasic cenral endency and sochasic volailiy. The esimaion resuls show ha incorporaing sochasic volailiy in he yield curve facors does no replace he need for he addiional opions facors in pricing he ineres rae caps. Wihou he opions facors, he hree yield curve facors explain only 40% of he variaion in he ineres rae caps. Only afer we incorporae he hree opions facors can he model explain over 99% of he ineres rae cap variaion. 1 Examples include he forward rae models of Ho and Lee (1986), Hull and Whie (1993), and Heah, Jarrow, and Moron (1992); he marke rae models of Brace, Gaarek, and Musiela (1997), Jamshidian (1997), Milersen, Sandmann, and Sondermann (1997), and Musiela and Rukowski (1997a); he sring models of Goldsein (2000), Sana-Clara and Sornee (2001), and Longsaff, Sana-Clara, and Schwarz (2001b); and he nonparameric pricing approach of Aï-Sahalia (1996).

3 Heidari and Wu 519 The fac ha ineres rae facors idenified from he yield curve canno accoun for he movemen of ineres rae opions has been documened by several recen sudies. In a join saisical analysis on LIBOR, swap raes, and implied volailiies for swapions, Heidari and Wu (2003) find ha hree principal componens exraced from he yield curve explain over 99% of he ineres rae movemens, bu hey only explain 60% of he variaion in he implied volailiy surface. They furher find ha hree addiional principal componens exraced from he implied volailiies are needed o explain he movemen in he implied volailiy surface. Collin-Dufresne and Goldsein (2002) documen similar evidence on implied volailiies for ineres rae caps and refer o he evidence as unspanned sochasic volailiy. Several possible reasons conribue o he empirical findings. Firs, as suggesed by Collin-Dufresne and Goldsein (2002), here are possibly ineres rae sochasic volailiy facors ha are no spanned by he yield curve. Wihin he affine family of dynamic erm srucural models, hey idenify a se of parameer consrains so ha he sochasic volailiy of ineres raes is no insananeously correlaed wih he value of ineres raes. 2 Neverheless, Bikbov and Chernov (2009) show ha hese parameric consrains do no fully resolve he ension beween he pricing of ineres rae fuures and opions. Generally speaking, ineres rae volailiy affecs he curvaure of he yield curve and hence is linked o he erm srucure, excep under very special parameric resricions as, for example, hose idenified in Collin-Dufresne and Goldsein (2002). Thus, heir unspanned volailiy specificaions can only be regarded as special cases raher han he general rule. In paricular, if a general dynamic erm srucure model has difficuly pricing boh ineres raes and ineres rae opions, i is unlikely ha a consrained version of he model can do he job well. A more plausible inerpreaion comes from misidenificaion and/or misspecificaion. 3 In pracice, even if sochasic volailiy facors affec he yield curve and are spanned by he yield curve in heory, we may sill have difficulies in idenifying hese facors saisically from he yield curve if he dependence is weak. In his case, i is no a maer of idenifying parameric consrains so ha he volailiy facors are indeed unspanned in heory, bu a maer of designing an efficien facor srucure and esimaion procedure so ha facors ha affec he yield curve and he opions can boh be effecively idenified. Ye anoher possible source of failure for dynamic erm srucure models is ha he models price ineres rae opions based on he model-implied fair values of he yield curve, hus ignoring any poenial impacs of he residuals on he yield curve. A small misalignmen beween he observed ineres raes and he modelimplied values can lead o large variaions in he implied volailiies on opions. Such small and emporal misalignmens occur naurally in a finie-dimensional framework, eiher due o model approximaions or marke imperfecions. To avoid 2 They also show ha incorporaing sochasic volailiy under he Heah e al. (1992) framework auomaically generaes sochasic volailiy unspanned by he erm srucure. However, as we have argued earlier, his class of models akes he yield curve as given and hence does no ake a sance on he fair valuaion of he underlying ineres raes. 3 We hank he referee for suggesing his inerpreaion.

4 520 Journal of Financial and Quaniaive Analysis he bias inroduced by such misalignmens, mos exising opion pricing pracices ake he observed yield curve as given, eiher by acceping an infinie-dimensional sae space or hrough ime-inhomogeneous model paramerizaion. Our m + n model accommodaes all hese differen scenarios. Under our model, we decompose he dynamics of each ineres rae series ino a sysemaic yield curve componen and an orhogonal residual componen ha has lile impac on he ineres rae erm srucure bu can have significan impacs on ineres rae opions. The residual componen can proxy for i) he unspanned volailiy of Collin-Dufresne and Goldsein (2002); ii) ineres rae facors ha canno be effecively idenified from he yield curve, even if hey are spanned in heory by he yield curve; and iii) emporary misalignmens beween he observed ineres raes and he fair value implied by a finie-dimensional dynamic erm srucure model. The orhogonal decomposiion also allows us o design an efficien sequenial esimaion procedure, which esimaes he yield curve componen from he ime series of ineres raes in he firs sep and idenifies he remaining opions variaions from he ime series of opions in a second sep. The remainder of he paper is srucured as follows. Secion II elaboraes on he model srucure hrough a Gaussian affine example. Secion III discusses he daa and he esimaion procedure. Secion IV discusses he esimaion resuls and model performance. Secion V considers an alernaive erm srucure specificaion. Secion VI concludes. II. The m + n Model We propose an m + n model srucure o price boh ineres raes and ineres rae opions. We use he firs m facors o capure he sysemaic movemen of he yield curve and he addiional n facors o capure ineres rae dynamics ha canno be effecively idenified from he yield curve bu neverheless have imporan impacs on opion pricing. A. The Basic Model Srucure We fix a filered complee probabiliy space {Ω,F, P, (F ) 0 T } saisfying he usual echnical condiions wih T being some finie, fixed ime. Le F and E denoe wo vecor Markov processes in some sae space D m R m and D n R n, respecively. We assume ha he observed marke quoes on ineres raes, y, are subjec o he following heurisic orhogonal decomposiion: y = Y(F ) + ε(e ), where Y(F ) denoes he fair value of he ineres rae, he dynamics of which are conrolled by he sae vecor F, and ε(e ) denoes he remaining movemens of he ineres rae ha have lile impac on he yield curve variaion bu have significan impacs on ineres rae opions. We assume ha his residual componen is also governed by a finie-dimensional sae vecor E. We chrisen F as he yield curve facors and E as he ineres rae opions facors. We assume ha he

5 Heidari and Wu 521 wo sae vecors are orhogonal o each oher and saisfy he following sochasic differenial equaions under he saisical measure P: (1) df = μ(f )d + Σ(F )dw, de = μ(e )d + Σ(E )dz, where μ(f ) is an m 1 vecor defining he drif and Σ(F ) is an m m marix defining he diffusion of he F process. Similarly, μ(e ) is an n 1 vecor and Σ(E ) is an n n marix defining he drif and diffusion of he E process. W and Z are independen Brownian moions wih dimension m and n, respecively. For any ime [0, T ] and ime of mauriy T [, T ], le P(F, T) denoe he fair value a ime of a zero-coupon bond wih mauriy τ = T, which is only a funcion of he yield curve facors bu independen of he opions facors. The fair values of he spo raes are defined as 1 (2) Y(F, T) T ln P(F, T), and he fair value of he insananeous ineres rae, or he shor rae, r, is defined by coninuiy, ln P (F, T) r (F ) lim. T T We assume ha here exiss a risk-neural measure Q, which is absoluely coninuous wih respec o he saisical measure P, such ha he ime- fair value of a claim o a erminal payoff Π T a ime T > can be wrien as [ ( T ) ] (3) V(F, E, T) = E Q exp r(f s )ds Π T, where E Q [ ] denoes he expecaion operaor condiional on filraion F and under measure Q. Thus, he fair value of a zero-coupon bond can be compued from equaion (3) by seing Π T = 1 for all saes. Since he payoff of a zero-coupon bond is a consan and hence sae independen, he fair value of he zero-coupon bond is only a funcion of he fair value of he shor raes during he life of he bond and hus independen of he opions facors E, consisen wih he original assumpion. Neverheless, for sae-coningen claims such as caps, he payoff a ime T is deermined by he marke observed ineres raes a ha ime, or one period earlier if paid in arrears. Thus, he payoff funcion Π T depends on boh he dynamics of he yield curve facors F and ha of he residual opions facors E. As a resul, he value of he sae-coningen claim will become a funcion of E as well, in addiion o is dependence on he yield curve facors F. Therefore, E can be used o capure movemens in ineres rae opions ha canno be idenified saisically from he yield curve. B. A Gaussian Affine Example To illusrae he idea of our modeling srucure, we consruc and esimae a concree model wihin he analyically racable Gaussian affine family. We assume ha boh he yield curve facors F and he opions facors E have dimensions of hree: m = n = 3. A yield curve dimension of hree is consisen wih he

6 522 Journal of Financial and Quaniaive Analysis empirical evidence of Lierman and Scheinkman (1991), Knez, Lierman, and Scheinkman (1994), and Heidari and Wu (2003), as well as he curren saus quo in affine model design (Dai and Singleon (2000) and Duffee (2002)). The choice of hree addiional opions facors is mainly moivaed by he evidence in Heidari and Wu (2003) and Wadhwa (1999) on implied volailiies for swapions. We assume ha under he saisical measure P, he yield curve facors and he opions facors are boh governed by Ornsein-Uhlenbeck (OU) processes, (4) df = κ F F d + dw, de = κ E E d + dz, where κ F R 3 3 and κ E R 3 3 conrol he mean-revering propery of he wo vecor processes. For idenificaion reasons, we normalize boh sae vecors o have zero long-run means and ideniy diffusion marices. We furher assume ha he fair value of he shor rae r is affine in he yield curve facor, (5) r(f ) = a r + b r F, where a r R is a scalar and b r R 3+ is a vecor. Thus, he fair value of he yield curve only depends on he yield curve facors. Finally, we close he model by assuming a flexible affine marke price of risk for boh ses of facors, (6) γ(f ) = b γ + κ γ F, λ(e ) = b λ + κ λ E, wih b γ, b λ R 3 and κ γ,κ λ R 3 3. Given hese marke price of risk specificaions, he wo ypes of facors remain OU under he risk-neural measure Q, bu wih adjusmens o he drif erms: (7) df = ( b γ κ Q F F ) d + dw Q, de = ( ) b λ κ Q E E d + dz Q, wih κ Q F = κ F + κ γ and κ Q E = κ E + κ λ. Our yield curve facor specificaion belongs o he affine class of Duffie and Kan (1996). The fair value of he zero-coupon bond wih mauriy τ is exponenially affine in he yield curve facor, (8) P(F, + τ) = exp ( a(τ) b(τ) F ), where he coefficiens a(τ) and b(τ) are deermined by he following ordinary differenial equaions: (9) a (τ) = a r b(τ) b γ b(τ) b(τ)/2, b (τ) = ( ) b r κ Q F b(τ), subjec o he boundary condiions a(0) = 0 and b(0) = 0. The ordinary differenial equaions can be solved analyically in erms of he eigenvalues and eigenvecors of he κ Q F marix. The fair value of he spo rae is affine in he yield curve facors, (10) Y(F, + τ) = 1 ( a(τ) + b(τ) ) F. τ

7 Heidari and Wu 523 The differences beween he observed ineres raes and he fair values are wha we call residuals. They arise eiher due o he approximae (misspecified) naure of a model or due o emporal marke imperfecions. Regardless of he source, given well-specified yield curve facor dynamics, he residual on he ineres rae erm srucure should be small. Thus, we can specify he E facor dynamics o capure movemens in ineres rae opions while ignoring heir impacs on he yield curve. In our esimaion exercise, we price U.S. dollar caps, which are porfolios of opions wrien on hree-monh LIBOR. Thus, we need o model he dynamics of he residuals on he hree-monh LIBOR. For his purpose, we assume ha he residuals on he hree-monh LIBOR are a funcion of hree remaining opions facors. In paricular, we assume he following funcional form for he loading of he hree opions facors on he observed hree-monh rae: (11) y(, + h) = 1 ( a(h) + b(h) F + c ) h E, h where h denoes he exac mauriy for he hree-monh LIBOR, c h R 3 is a vecor of consan parameers ha deermine he loading of he opions facors, and y(, + h) is he coninuously compounded ineres rae derived from he observed LIBOR quoe, based on he following relaion: LIBOR(h) = 100 h ( ) e hy(,+h) 1. The linear loading specificaion in equaion (11) of he facors E on he spo rae, insead of on he LIBOR iself, is moivaed parly by an analogy o he linear loading of he yield curve facors and parly by analyical racabiliy in pricing caps. C. Caple Pricing We illusrae he impac of he opions facor dynamics on ineres rae opions by pricing a caple. Each cap conrac consiss of a series of caples. The payoff of he ih caple wih uni dollar noional amoun can be wrien as Π i T = h (LIBOR(h) T K) +, where h denoes he mauriy of he LIBOR and also he paymen inerval (enor) of he cap conrac, T = + ih is he selemen ime (mauriy) of he caple, and K is he srike rae. For U.S. dollar caps, he paymen is made in arrears, i.e., he paymen of he ih caple is deermined a ime T bu paid one period laer a T +h. Based on equaion (3), he ime- fair value of he ih caple is given by CAPLET i = E Q [ ( exp T+h ) ] r (F s ) ds h (LIBOR T (h) K) +, where he observed LIBOR rae depends on boh he yield curve facors F and he remaining opions facors E. Wriing he simply compounded LIBOR rae in

8 524 Journal of Financial and Quaniaive Analysis erms of he coninuously compounded spo rae as in equaion (11), and by he rule of ieraed expecaions, we have [ (12) exp CAPLET i = E Q ( T ) r (F s ) ds (e c h ET (1+hK) e a(h) b(h) F T ) + ]. Absen from he opions facor (E =0 for all ), he caple is equivalen o a pu opion on a zero-coupon bond. In he presence of independen opions movemens E, he caple can be regarded as an exchange opion: The opion holder has he righ o exchange he fair value of a zero-coupon bond for a payoff ha is a funcion of he opions facor E. The approach of dynamic erm srucure models in pricing opions is akin o seing E = 0 for all, even if hey are presen. On he oher hand, he opion pricing lieraure ofen incorporaes ime- and mauriy-dependen parameers in pricing ineres raes so ha he observed yield curve is forced o mach he fair value. If he observed yield curve differs from he fair value implied by a dynamic erm srucure model, he difference will be accommodaed by imeinhomogeneous parameers and will be carried over permanenly ino he fuure. This pracice amouns o he following modificaion of equaion (12): (13) CAPLET i = E Q [ exp ( T [r (F s ) + μ(s)] ds ) (1 (1+hK) e a(,h) b(h) F T ) + ], where μ(s) denoes a ime-inhomogeneous parameer ha accommodaes he observed yield curve. This adjusmen no only affecs he discouning bu also influences he payoff funcion, because now a(, h) becomes ime-inhomogeneous. The adjusmen leads o a dramaic increase in dimensionaliy, since we need a new parameer μ() for all. Furhermore, alhough μ() is ime varying and re-calibraed frequenly, i is reaed as a consan in pricing and hedging derivaives. The uncerainy, and hence risk, associaed wih his adjusmen over ime is ignored. By comparison, under our specificaion in (12), we recognize he exisence of poenial model errors, marke imperfecions, or residual dynamics such as unspanned volailiy ha are difficul o idenify from he yield curve bu neverheless have imporan impacs on opion pricing. We explicily accoun for he impacs of hese residual dynamics on fuure erminal payoffs. Meanwhile, he discouning is sill based on he fair value of he yield curve as in he pracice of dynamic erm srucure models. This reamen makes he valuaion of ineres rae opions consisen wih he valuaion of he underlying ineres raes. Carrying ou he expecaion operaion in equaion (12) leads o he following caple pricing formula: (14) CAPLET i = P (F, T + h)[(1+hr ) N (d 1 ) (1+hK) N (d 2 )],

9 Heidari and Wu 525 where N ( ) denoes he cumulaive densiy of a sandard normal variable and (15) d 1 = ln (1+hR ) / (1+hK) Σ Σ, d 2 = d 1 Σ are sandardized variables. In equaion (15), R denoes he residual-adjused value of he forward hree-monh LIBOR, defined by P (F, T) (1+hR ) = (c P (F, T + h) exp h E T [E T ] + 12 ) c h Var T [E T ] c h, Σ is he ime- condiional variance of hy T = a (h) + (h) F T + c h E T under a forward measure T, and y T is he fuure observed value of he hree-monh coninuously compounded spo rae. The condiional variance Σ can be evaluaed as (16) Σ = b (h) Var T [F T ] b (h) + c h Var T [E T ] c h. Appendix A provides deails on he derivaion of he opion pricing formula. Appendix A also derives he condiional mean and variance of F T and E T under measures Q and T. Equaion (14) shows how he dynamics of he yield curve residuals influence he pricing of an ineres rae caple. The impacs come from wo sources, boh due o he fac ha he erminal payoff of he caple is compued based on he observed marke rae, no on some model-implied fair value. Firs, he forward rae (R ) is adjused for he expeced impac of he residual dynamics. The adjusmen shows up boh proporionally o he caple price and also nonlinearly in he definiion of he sandardized variables d 1 and d 2. Second, he condiional variance of he underlying hree-monh LIBOR rae in he opion pricing formula Σ is he condiional variance of he observed marke rae (hy T ), no ha of he fair value. Thus, Σ capures he aggregae conribuion from boh he yield curve facors F and he independen opions facors E, as illusraed in (16). III. Daa and Esimaion To invesigae he performance of our m + n model, we propose a sequenial esimaion procedure and esimae he affine example using daa on LIBOR, swap raes, and implied volailiies for ineres rae caps. A. Daa Descripion The daa, obained from Lehman Brohers, include weekly (Wednesday) closing mid quoes on i) U.S. dollar LIBOR raes a mauriies of one, wo, hree, six, and 12 monhs; ii) swap raes a mauriies of wo, hree, five, seven, 10, 15, and 30 years; and iii) a-he-money implied volailiies for ineres rae caps a opion mauriies of one, wo, hree, four, five, seven, and 10 years. The sample spans eigh years from April 6, 1994 o April 17, 2002 (420 weekly observaions for each series).

10 526 Journal of Financial and Quaniaive Analysis The U.S. dollar LIBOR raes are simply compounded ineres raes. The mauriies are compued following acual/360 day-coun convenion, saring wo business days forward. The U.S. dollar swap raes have paymen inervals of half years and are relaed o he zero-coupon bond prices (discoun facors) by 1 p(, + nh) SWAP(, Nh) = 200 N i=1 p(, + ih), where h = 0.5 is he paymen inerval of he swap conrac, and N is he number of paymens over he mauriy of he swap conrac. The a-he-money cap conracs are on hree-monh LIBOR raes, wih a paymen inerval of hree monhs. The paymen is made in arrears. The srike price is se o he swap rae of he corresponding mauriy. The implied volailiy quoes are obained under he framework of he Black (1976) model, where he LIBOR rae is assumed o follow a geomeric Brownian moion. Given an implied volailiy (IV) quoe, he invoice price of he cap conrac is compued according o he Black formula: (17) N 1 CAP(, Nh) = Lh p(, + (i +1)h) i=1 (R(, + ih, + (i +1)h)N (d 1i ) KN (d 2i )), where L denoes he noional value of he cap conrac, h=0.25 denoes he hreemonh paymen inerval, R(, + ih, + (i +1)h) denoes he forward LIBOR rae, K denoes he srike rae, and N ( ) denoes he cumulaive normal funcion wih (18) d 1i = ln(r(, + ih, + (i +1)h)/K) +IV2 ih/2 IV, ih d 2i = d 1i IV ih. Table 1 repors he summary saisics on LIBOR, swap raes, and implied volailiies. The average ineres raes exhibi an upward sloping erm srucure. The sandard deviaion of ineres raes shows a hump-shaped erm srucure ha peaks around six monhs. The ineres raes are all highly persisen, wih he weekly auocorrelaion esimaes ranging from o LIBOR raes also show some moderae excess kurosis. The mean implied volailiy exhibis a humpshaped erm srucure ha peaks a hree-year mauriy. The sandard deviaion of he implied volailiy declines wih increasing mauriy. The implied volailiies are also persisen, wih he weekly auocorrelaion esimaes ranging from o B. A Sequenial Esimaion Procedure for he m + n Model We propose a sequenial wo-sage procedure for esimaing he m + n model. The firs-sage esimaes he yield curve facors using he LIBOR and swap raes. The second-sage esimaes he opions facors using cap opion prices. A each

11 Heidari and Wu 527 TABLE 1 Summary Saisics of Ineres Raes and Implied Volailiies Enries are summary saisics of ineres raes and implied volailiies. Mean, Sd Dev, Skewness, Kurosis, and Auo denoe he sample esimaes of he mean, sandard deviaion, skewness, excess kurosis, and firs-order auocorrelaion, respecively. In he mauriy column, LIBOR mauriies are in monhs (m), swap and cap mauriies are in years (y). The daa are weekly closing mid quoes from Lehman Brohers from April 6, 1994 o April 17, 2002 (420 observaions per series). Mauriy Mean Sd Dev Skewness Kurosis Auo Panel A. LIBOR and Swap Raes (%) 1 m m m m m y y y y y y y Panel B. Cap Implied Volailiy (%) 1 y y y y y y y sage, we cas he model ino a sae-space form, obain efficien forecass on he condiional mean and variance of he observed series using an exended Kalman filer, and build he likelihood funcion on he forecasing errors of he observed series, assuming ha he forecasing errors are normally disribued. The model parameers are esimaed by maximizing he likelihood funcion. To esimae he Gaussian affine model, in he firs sage, we regard he yield curve facors as he unobservable saes and specify he sae propagaion equaion using an Euler approximaion of he yield curve facor dynamics in equaion (1), (19) F = Φ F F 1 + Qε, where Φ F = exp( κ F Δ) denoes he auocorrelaion marix wih Δ = 1/52 being he lengh of he weekly discree-ime inerval, Q = IΔ denoes he insananeous covariance marix wih I being an ideniy marix of dimension hree, and ε denoes an independen and idenically disribued rivariae sandard normal innovaion vecor. For noaional clariy, we normalize he discree-ime inerval o one. The measuremen equaions for he firs-sage esimaion are consruced based on he observed LIBOR and swap raes by assuming addiive, normally disribued pricing errors, (20) m = [ ] LIBOR(F, i) + Ve, SWAP(F, j) i = 1, 2, 3, 6, 12 monhs j = 2, 3, 5, 7, 10, 15, 30 years,

12 528 Journal of Financial and Quaniaive Analysis where m denoes he measuremen series generically, V denoes he covariance marix of he measuremen errors, and e denoes he sandardized error vecor, which has a sandard normal disribuion. For he second sage, we regard he opions facors as he unobservable saes and specify he sae propagaion equaion based on a discree-ime version of he opions facor dynamics: (21) E = Φ E E 1 + Qε, where Φ E = exp( κ E Δ) and Q = IΔ. We consruc he measuremen equaions based on he firs-sage pricing error on he hree-monh LIBOR and he seven cap series, again assuming addiive, normally disribued pricing errors, [ ] LIBOR (i) LIBOR(F, i) (22) m = + Ve, CAP(F, E, j) i = 3 monhs j = 1, 2, 3, 4, 5, 7, 10 years, where we conver he implied volailiy quoes ino dollar prices based on he Black formula in equaion (17) and $100 noaional value. For boh sages of esimaion, given iniial guess of model parameers, we use he exended Kalman filer o updae he condiional mean and condiional covariance marix of he saes and measuremen variables. We furher assume ha he forecasing errors on he measuremen series are normally disribued and define he weekly log likelihood funcion as (23) l (Θ i, m ) = 1 [ ln A + (m m ) (A ) 1 (m m ) ], i = 1, 2, 2 where Θ i denoes he parameer se for sage-i esimaion, and m and A denoe he condiional mean and covariance marix of he forecass of he measuremen series, respecively. The model parameers are esimaed by maximizing he sum of he weekly log likelihood defined in (23). Many economeric sudies of affine models follow some variaion of maximum likelihood esimaion. While he sae variables are in general no observable, hey are ofen direcly invered from he observed discoun bonds by assuming ha m of hese bonds are priced perfecly by he m facors. The oher bonds are hen assumed o be priced wih errors, and he likelihood funcion can be consruced based on he condiional densiy of he laen variables and he pricing errors. 4 However, in pracice, i is arbirary o decide which raes are priced exacly and which are priced wih errors. A more reasonable assumpion is ha all ineres raes or bond prices conain errors. An efficien esimaion sraegy should demand ha he model values go hrough all he observed daa poins in a leas 4 This exac pricing assumpion, or some varian of i, is mainained in, among ohers, Chen and Sco (1993), Duffee (1999), (2002), Duffie and Singleon (1997), Longsaff and Schwarz (1992), and Pearson and Sun (1994).

13 Heidari and Wu 529 square sense, raher han forcing he model o mach an arbirary se of poins and ignoring he ohers. A convenien approach o dealing wih observaion errors is o cas he model in sae-space form augmened by measuremen equaions ha relae he observed ineres raes or bond prices o he underlying sae variables. When he sae variables are Gaussian and he measuremen equaions are linear, he Kalman filer yields he efficien sae updaes in he leas square sense (Pennacchi (1991)). In our applicaion, he sae variables are Gaussian, bu he measuremen equaions are nonlinear in he sae variables. The exended Kalman filer deals wih he nonlineariy in he measuremen equaion via a Taylor expansion. Examples of exended Kalman filering in erm srucure model esimaions include Duffee and Sanon (2003) and Leippold and Wu (2007). The filering echnique fis naurally in our modeling framework. We firs apply Kalman filering o LIBOR and swap raes. The measuremen errors on hese ineres raes are essenially wha we call residuals on he yield curve. We make use of he residuals on he hree-monh LIBOR and he ineres rae cap daa in idenifying he opions facor dynamics in a second-sage esimaion. We assume ha he measuremen errors on each series are independen bu exhibi disinc variance σ 2 e. Thus, we have 31 parameers for he firs-sage esimaion: Θ 1 [κ F R 6,κ Q F R6, b r R 3+, b γ R 3, a r R,σ 2 e R 12+ ]. A his sage, we idenify he yield curve facor dynamics from he 12 ineres rae series. For he second-sage esimaion, we have 26 parameers: Θ 2 [κ E R 6,κ Q E R6, c h R 3, b λ R 3,σ R 8+ ]. A his sage, we ake he yield curve facors exraced from he firs sage as given and idenify he opions facor dynamics mainly from he ineres rae opions. IV. Empirical Performance of he Gaussian Affine Model Based on he esimaion resuls on he 3+3 Gaussian affine model, we discuss he differen roles played by he yield curve facors and he opions facors in pricing he ineres rae erm srucure and ineres rae opions. A. Pricing he Yield Curve wih Three Yield Curve Facors Panel A of Table 2 repors he summary properies of pricing errors on he yield curve obained from he firs-sage esimaion. The pricing errors are defined as he difference in basis poins (bps) beween he marke quoes on LIBOR and swap raes and he model-implied fair values as a funcion of he hree yield curve facors. The las row repors he sample averages of he saisics over he 12 ineres rae series. Overall, he pricing errors are very small. The average pricing error is less han 1 bp. The average sandard deviaion and mean absolue pricing errors are boh less han 5 bps. The las column repors he percenage explained variaion, defined as one minus he raio of he pricing error variance o he variance of he original series, represened in percenage poins. The explained variaion esimaes are over 99% for all bu one series. Hence, he hree yield curve facors capure he erm srucure of ineres raes well.

14 530 Journal of Financial and Quaniaive Analysis TABLE 2 Summary Saisics of Pricing Errors from he Gaussian Affine Model Enries repor he summary saisics of he pricing errors on LIBOR and swap raes (Panel A of Table 2), obained from he firs-sage esimaion, and on cap implied volailiies (Panel B), obained from he second-sage esimaion on he Gaussian affine model. The pricing error is defined as he difference beween he observed marke quoes and he model-implied fair values. The columns iled Mean, Median, Sd Dev, MAE, Auo, Max, and Min denoe he mean, median, sandard deviaion, mean absolue error, firs order auocorrelaion, maximum, and minimum of he measuremen errors a each mauriy, respecively. The las column (VR) repors he percenage variance explained for each series by he hree yield curve facors in Panel A and by he Gaussian affine model in Panel B, defined as one minus he raio of pricing error variance o he variance of he original series, in percenage poins. The las row of each panel repors average saisics. Mauriy Mean Median Sd Dev MAE Auo Max Min VR Panel A. Errors on he Yield Curve (basis poins) 1 m m m m m y y y y y y y Average Panel B. Errors on Cap Implied Volailiies (%) 1y y y y y y y Average Furher inspecion shows ha he measuremen errors are very small for swap raes a moderae mauriies (wo o 10 years), bu are larger for LIBOR and long-mauriy swap raes. The mean absolue errors are only abou 1 bp for wo-, hree-, and five-year swap raes, and he model prices he seven-year swap rae almos perfecly. On he oher hand, he mean absolue errors for he 12-monh LIBOR and he 30-year swap rae are abou 10 bps. The difference beween shorand long-erm swap raes may represen liquidiy differences. The overall larger measuremen errors on he LIBOR marke may indicae some srucural differences beween he LIBOR and swap markes ha canno be accommodaed by our model. The segmenaion beween he LIBOR and swap markes is well known in he indusry (James and Webber (2000)). Longsaff, Sana-Clara, and Schwarz (2001a) find similar marke segmenaions beween he cap marke, which is based on LIBOR, and he swapion marke, which is based on swap conracs. Our resuls indicae ha such inconsisencies in he opions marke may acually sar in he underlying ineres rae marke. B. Pricing Caps Wih and Wihou Addiional Opions Facors Given he esimaed yield curve facors, we firs price ineres rae caps based on he model-implied yield curve, ignoring he poenial impac of he yield curve

15 Heidari and Wu 531 residuals and unspanned volailiy. The pricing resuls show ha alhough he hree yield curve facors can explain over 99% of variaion on he yield curve, hey are far from sufficien in capuring he movemen of implied volailiies. The hree yield curve facors only explain 48.35% of he aggregae variaion in implied volailiies. Li and Zhao (2006) repor similar poor opion pricing and hedging performance for hree-facor quadraic erm srucure models. Bikbov and Chernov (2009) show ha mos hree-facor dynamic erm srucure models perform well in explaining he yield curve, bu poorly in explaining he eurodollar fuures opions, even if hey apply he parameric specificaion proposed by Collin-Dufresne and Goldsein (2002) o accoun for unspanned sochasic volailiy. Thus, he poor performance of he yield curve facors in opion pricing is no specific o our model design, bu generic o dynamic erm srucure models ha ignore he poenial impacs of he residuals and unspanned or unidenified volailiy movemens. The performance in pricing ineres rae caps improves dramaically when we include he hree addiional opions facors. Panel B of Table 2 repors he summary saisics of he pricing errors on implied volailiies based on he esimaed model. Here, he pricing errors are defined as he differences in percenage poins beween marke quoes and model-implied values on he implied volailiy. The saisics show negligible mean pricing errors. The sandard deviaions of he pricing errors are all wihin 40 bps, and he mean absolue pricing errors are all wihin 30 bps. Boh numbers are below he half-a-percenage-poin benchmark for average bid-ask spreads on implied volailiies for ineres rae caps. Finally, he las column shows ha he model explains all cap series over 99%. The four graphs in Figure 1 conras he pricing of he implied volailiies wih and wihou he opions facors a four sample daes. The circles denoe he marke quoes on implied volailiy. The solid lines represen he fair value compued from he esimaed model, which almos always go hrough he daa circles. In conras, he dashed lines, which represen he pricing from only he hree yield curve facors while ignoring he opions facors, deviae significanly from he daa poins. Wihou he help of he opions facors, he yield curve facors alone eiher underprice or overprice he ineres rae caps. C. Yield Curve Facor Dynamics and he Term Srucure of Ineres Raes Panel A of Table 3 repors he parameer esimaes and he absolue magniudes of he -saisics (in parenheses) from he firs-sage esimaion on he yield curve facor dynamics. The small diagonal elemens for κ F and κ Q F show ha he yield curve facors are highly persisen under boh he saisical measure P and he risk-neural measure Q. Furhermore, he significan and large esimaes of he off-diagonal elemens sugges ha he hree yield curve facors have srong dynamic ineracions wih one anoher. The esimaes for b r show ha he loadings of he firs wo facors on he shor rae are small, bu he loading of he hird facor is large and saisically significan. The esimaes on b γ reveal negaive marke price for yield curve facors, significanly so for he firs facor. The esimae for a r represens he long-run mean for he insananeous shor rae.

16 532 Journal of Financial and Quaniaive Analysis FIGURE 1 Pricing Cap Implied Volailiies Wih and Wihou Addiional Opions Facors Circles denoe marke quoes on cap implied volailiies. Solid lines denoe he fair value compued from he esimaed Gaussian affine model. Dashed lines represen he pricing from only he hree Gaussian affine yield curve facors while ignoring he addiional opions facors. Graph A. December 24, 1996 Graph B. March 25, 1998 Graph C. Sepember 23, 1998 Graph D. Sepember 26, 2001 According o equaion (8), he coefficiens [a(τ), b(τ)] deermine he erm srucure of ineres raes. The fair values of coninuously compounded spo raes are linked o he yield curve facors F by (24) Y(F, + τ) = [ ] [ ] a(τ) b(τ) + F. τ τ Since we normalize he long-run mean of he sae vecor o zero, he inercep a(τ)/τ defines he mean erm srucure of he spo rae. The slope coefficien b(τ)/τ capures he insananeous response of he spo rae o uni shocks in he yield curve facors and can hus be regarded as he loading of he yield curve facors on he erm srucure. The ordinary differenial equaions in (9) show ha he risk-neural dynamics of he yield curve facors (κ Q F and b γ) inerac wih he shor rae loading funcion (a r and b r ) o deermine he coefficiens [a(τ), b(τ)] and hence whole erm srucure of ineres raes.

17 Heidari and Wu 533 TABLE 3 Full Sample Parameer Esimaes of he Gaussian Affine Model Enries repor he maximum likelihood esimaes of he 3+3 Gaussian affine model parameers and he absolue magniudes of he -saisics (in parenheses). The esimaion uses he full sample of over eigh years of weekly daa from April 6, 1994 o April 17, 2002 (420 weekly observaions for each series). Panel A of Table 3 repors parameers (Θ 1 ) ha are relaed o he hree yield curve facors and are esimaed using 12 LIBOR and swap raes. Panel B repors he parameers (Θ 2 ) ha are relaed o he hree opions facors and are esimaed using seven cap series. Panel A. Θ 1 κf κ Q F (0.07) (0.51) (0.19) (5.84) (2.75) (0.41) (114.9) (4.56) (9.18) (13.7) (20.7) (10.0) b r b γ a r (1.09) (0.08) (32.2) (105.3) (9.41) (1.28) [ (4.54) ] Panel B. Θ 2 κ E κ Q E (0.02) (0.60) (0.06) (0.12) (0.91) (0.15) (0.68) (0.02) (0.13) (2.68) (1.88) (0.66) c h (0.24) (1.60) (11.6) b λ (0.57) (0.50) (0.40) The hree lines in Graph A of Figure 2 plo he coefficiens b(τ)/τ, which measure he conemporaneous response of he spo rae curve o uni shocks from each of he hree yield curve facors. The coefficiens are compued based on he parameer esimaes in Panel A of Table 3. The hree lines illusrae how he hree facors conrol he variaion of ineres raes a differen mauriies. The firs facor (he solid line) loads up mos heavily on long-erm ineres raes, he loading of he second facor (he dashed line) peaks around hree-year mauriy, and he las facor (he dashed-doed line) mainly loads up on he shor end of he yield curve. FIGURE 2 Facor Conribuions o Ineres Raes and Condiional Volailiies The hree lines in Graph A of Figure 2 represen he conemporaneous response of he spo rae curve o uni shocks on he hree yield curve facors under he esimaed 3+3 Gaussian affine model. The wo lines in Graph B plo he conribuion of he yield curve facors (solid line, wih scale on he lef y-axis) and opions facors (dashed line, wih scale on he righ y-axis) o he condiional volailiy of he hree-monh LIBOR a differen condiioning horizons, in annualized percenages. Graph A. Ineres Rae Term Srucure Graph B. Condiional Volailiy Term Srucure

18 534 Journal of Financial and Quaniaive Analysis D. Opions Facor Dynamics and he Term Srucure of Condiional Variance Since he hree yield curve facors explain over 99% of he variaion in ineres raes, he remaining opions facors have minimal impacs on he erm srucure of ineres rae levels. Neverheless, hey can have imporan impacs on cap pricing, mainly hrough heir conribuion o he condiional variance of he observed hree-monh ineres raes. As shown in equaion (14), caple pricing depends crucially on he condiional variance dynamics of he observed hree-monh ineres rae. According o equaion (16), his condiional variance can be decomposed ino wo componens: he conribuion from he yield curve facors (F) and he conribuion from he opions facors (E): (25) Σ = b (h) Var T [F T ] b (h) + c h Var T [E T ] c h. Appendix A shows ha he condiional variance marix of F T and E T are conrolled by heir respecive risk-neural dynamics κ Q F and κq E : (26) Var T [F T ] = T e κq F s e κq F s ds, Var T [E T ] = T e κq E s e κq E s ds. We have discussed he esimaes on κ Q F and heir impacs on he erm srucure of ineres rae levels in he previous subsecion. In equaion (25), b(h) denoes he loading coefficien of he hree-monh spo rae on he hree yield curve facors, which are compued a [0.006, 0.006, ]. Hence, he condiional variance of he hird yield curve facor conribues more significanly o he condiional variance of he hree-monh rae. Neverheless, he large and significan off-diagonal elemens of κ Q F sugges ha he firs- and second-yield curve facors also have srong impacs on he condiional variance of he hird facor. Panel B of Table 3 repors he second-sage esimaes and absolue magniudes of he -saisics (in parenheses) on he opions facor dynamics. The wo off-diagonal elemens in he las row of κ Q E are large and significan. Thus, he firs and second opions facors srongly impac he dynamics of he hird opions facor. The esimaes on he loading coefficiens c h are close o zero for he firs wo elemens bu significanly posiive for he hird elemen, suggesing ha only he condiional variance of he hird opions facor conribues significanly o Σ. Neverheless, he firs wo opions facors influence he condiional variance dynamics of he hird opions facor hrough he wo off-diagonal elemens in he las row of κ Q E. Based on he esimaed model parameers in Table 3, we compue he conribuion o he condiional variance Σ from boh he yield curve facors and he opions facors according o equaions (25) and (26). Graph B of Figure 2 plos he conribuions in annualized percenage volailiy a differen opion mauriies. The solid line, wih scale on he lef side, denoes he conribuion from he yield curve facors, b(h) Var T [F T ]b(h)/(t ). The conribuion generaes a hump shape ha peaks around one-and-a-half-year mauriy. The dashed line, wih scale on he righ side, denoes he conribuion from he opions facors,

19 Heidari and Wu 535 c h VarT [E T ]c h /(T ), which also generaes a hump-shaped erm srucure bu peaks a four-year mauriy. No only does he hump shape from he opions facors beer mach he sample mean erm srucure of he implied volailiy (Table 1), bu he differen scales of he wo lines also show ha he condiional volailiy conribuion from he opions facors is several imes larger han he condiional volailiy conribuion from he yield curve facors. Therefore, alhough he opions facors have minimal impac on he ineres rae erm srucure, hey have large impacs on he erm srucure of condiional variance and hence on opion pricing. E. Ou-of-Sample Performance To sudy he ou-of-sample performance, we reesimae he model using he firs six years of daa from April 6, 1994 o April 5, 2000 (314 weekly observaions for each series). Then, we use hese esimaed model parameers o compare he model performance boh in sample during he firs six years and ou of sample during he las wo years from April 12, 2000 o April 17, 2002 (106 weekly observaions for each series). If he model is well specified and our esimaion generaes sable model parameers, we would expec he model s ou-of-sample performance o be similar o is in-sample performance. Table 4 repors he model parameer esimaes and absolue magniudes of he -saisics (in parenheses) using he subsample of six years of daa. The esimaes are close o wha we have obained from he full sample in Table 3, indicaing ha he esimaion generaes sable model parameers and ha he ineres rae behavior has no experienced dramaic changes over he las wo years of our daa sample. Table 5 compares he summary saisics of he in-sample pricing errors during he firs six years (on he lef side) and ou-of-sample pricing errors during he las wo years (on he righ side). The saisics show ha he model performs well boh in sample and ou of sample. There is no visible deerioraion for he ou-of-sample performance, showing ha our model is well specified and robus. V. Alernaive Term Srucure Specificaions Wihin he generic m + n facor srucure, he choice of he number of facors and he specificaions of he facor dynamics are governed by balanced consideraions for parsimony, analyical racabiliy, and empirical performance. In previous secions, we have chosen a Gaussian affine specificaion o illusrae he heoreical idea and is empirical performance. The choice of he facor srucure is moivaed by previous facor analysis on he ineres rae and implied volailiy erm srucures. The choice of he Gaussian affine specificaion is mainly moivaed by analyical racabiliy. Under his specificaion, we can derive boh he ineres raes and cap prices in analyical forms. The analyical forms faciliae our illusraion on he differen roles played by he yield curve facors and he opions facors. Our empirical analysis furher shows ha he Gaussian affine specificaion generaes saisfacory performance for boh yield curve and cap pricing.

20 536 Journal of Financial and Quaniaive Analysis TABLE 4 Subsample Parameer Esimaes of he Gaussian Affine Model Enries repor he maximum likelihood esimaes of he 3+3 Gaussian affine model parameers and he absolue magniudes of he -saisics (in parenheses). The esimaion uses he firs six years of weekly daa from April 6, 1994 o April 5, 2000 (314 weekly observaions for each series). Panel A of Table 4 repors parameers (Θ 1) ha are relaed o he hree yield curve facors and are esimaed using 12 LIBOR and swap raes. Panel B repors he parameers (Θ 2) ha are relaed o he hree opions facors and are esimaed using seven cap series. κ Q κ F F (0.03) (58.8) (0.03) (0.06) (7.66) (49.2) (3.77) (2.55) (0.27) (7.79) (10.4) (43.9) b r b γ a r (3.42) [ (0.00) (0.00) (0.01) (0.55) (0.08) (16.7) ] Panel B. Θ 2 κ E κ Q E (0.02) (1.02) (0.07) (0.24) (0.05) (0.07) (1.54) (0.84) (0.15) (3.93) (1.61) (1.63) c h (0.39) (0.57) (6.41) b λ (0.63) (0.22) (0.27) TABLE 5 In-Sample and Ou-of-Sample Performance of he Gaussian Affine Model Enries repor he summary saisics of boh in-sample and ou-of-sample pricing errors on LIBOR and swap raes in Panel A of Table 5, and on cap implied volailiies in Panel B under he Gaussian affine model. The pricing error is defined as he difference beween he observed marke quoes and he model-implied fair values. The columns iled Mean, Median, Sd Dev, MAE, Auo, Max, and Min denoe, respecively, he mean, median, sandard deviaion, mean absolue error, firs-order auocorrelaion, maximum, and minimum of he measuremen errors a each mauriy. Under VR, we repor he percenage variance explained for each series by he hree yield curve facors in Panel A and by he Gaussian affine model in Panel B. The esimaion is based on he firs six years of daa. In-Sample Errors Ou-of-Sample Errors ( ) ( ) Mauriy Mean Median Sd Dev MAE Auo Max Min VR Mean Median Sd Dev MAE Auo Max Min VR Panel A. Errors on he Yield Curve (basis poins) 1 m m m m m y y y y y y y Avg Panel B. Errors on Cap Implied Volailiies (%) 1 y y y y y y y Avg