Do Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility

Transcription

1 Do Bonds Span the Fied Income Makets? Theoy and Evidence fo Unspanned Stochastic olatility PIERRE COLLIN-DUFRESNE and ROBERT S. GOLDSTEIN July, 00 ABSTRACT Most tem stuctue models assume bond makets ae complete, i.e., that all fied income deivatives can be pefectly eplicated using solely bonds. Howeve, we find that, in pactice, swap ates have limited eplanatoy powe fo etuns on at-the-money staddles potfolios mainly eposed to volatility isk. We tem this empiical featue unspanned stochastic volatility (US). While US can be captued within an HJM famewok, we demonstate that bivaiate models cannot ehibit US. We detemine necessay and sufficient conditions fo tivaiate Makov affine systems to ehibit US. Fo such USmodels, bonds alone may not be sufficient to identify all paametes. Rathe, deivatives ae needed. Piee Collin-Dufesne is fom the Gaduate School of Industial Administation, Canegie Mellon Univesity, GSIA 35A, 5000 Fobes Ave., Pittsbugh PA 53, and Robet S. Goldstein is fom Washington Univesity, St Louis, campus bo 33, Bookings Dive, St Louis, MO 6330, We thank Jespe Andeasen, Dave Backus, Dave Chapman, Daell Duffie, Fancis Longstaff, Claus Munk, Pedo Santa-Claa, Ken Singleton, Chis Telme, Len Umantsev, Stan Zin and semina paticipants at: the Ameican Finance Association meetings in Atlanta 00, the Teas Finance Festival, the 00 Fied Income Winte Confeence at Stanfod Univesity, The Euopean Finance Association Meetings in Bacelona 00, The Univesity of Connecticut, Columbia Univesity, The Univesity of Illinois, Penn State Univesity, The Univesity of Rocheste, The Univesity of Wisconsin, and Stanfod Univesity fo helpful comments. All emaining eos ae ou own.

2 Most time-homogeneous models of the tem stuctue pedict that bond pices ae sufficient to complete the fied-income makets. One implication of this pediction is that fied income deivatives ae edundant secuities. Anothe (elated) implication is that bonds can be used to hedge volatility isk. These implications ae in contast to the equity-deivative liteatue, whee it is common to assume that volatility isk cannot be hedged by tading in the undelying stock alone (e.g., Heston (993)). In such a case, stock options ae not edundant secuities. In this pape we pesent empiical evidence suggesting that inteest ate volatility isk cannot be hedged by a potfolio consisting solely of bonds. Using data on swap ates, caps, and floos fo thee diffeent cuencies, we find that thee is a athe weak coelation between changes in swap ates and etuns on at-the-money staddles. In paticula, egession analysis indicates that in some cases as little as 0 pecent of staddle etuns can be eplained by changes in the tem-stuctue of swap ates. Howeve, the esiduals of these egessions ae highly coss-coelated acoss staddle matuities. Indeed, pincipal components analysis indicates that a single additional state vaiable can eplain moe than 85 pecent of the emaining vaiation. These findings stongly suggest that thee is at least one state vaiable which dives innovations in inteest ate deivatives, but does not affect innovations in the swap ates (and thus, bond pices) themselves. In othe wods, these findings suggest that the bond maket by itself is incomplete. 3 We note that it is staightfowad to captue this featue, which we tem unspanned stochastic volatility (US), by diectly specifying the joint dynamics of fowad ates (o equivalently, bond pices) and the state vaiables that dive fowad ate volatility. (see e.g., Andeasen, Collin-Dufesne and Shi (997), Kimmel (00), and Collin-Dufesne and Goldstein (00)). This appoach is analogous to the picing of equity deivatives (e.g., Heston (993)) by diectly specifying the joint dynamics of a taded asset (i.e., a stock) and its volatility. One disadvantage of such an appoach, howeve, is that in such a famewok bond pices become inputs to the model, athe than pedictions of the model. Hence, such an appoach povides no testable implications fo the coss-section of bond pices. In contast to modeling fowad ate dynamics diectly, most models that attempt to investigate the coss-sectional and time-seies behavio of bond pices choose a set of latent vaiables to seve as the state vecto, and then define the spot ate as a function of these state vaiables. The state vecto dynamics ae typically assumed to be Makov and time-homogeneous. Inteestingly, most of these models pedict that bonds alone ae sufficient to complete the fied-income makets. Indeed, most tem stuctue models fall within the so-called affine class of Duffie and Kan (996, heeafte DK), whee bond yields ae linea in the entie set of state vaiables. Because all state vaiables show up in bond pices, these models pedict that all souces of isk affecting fied income deivatives can be completely hedged by a potfolio consisting solely of bonds. Fo eample, the stochastic volatility models of Fong and asicek (99) and Longstaff and Schwatz (99) geneate bond yields that ae

3 linea in both the spot ate and the volatility state vaiables. Hence, these models pedict that bonds can be used to hedge volatility isk. Below, we identify a class of time-homogeneous Makov models with a finite state vaiable epesentation that povides testable implications fo both the time seies and coss-sectional behavio of bond pices, yet pemits fied income deivatives to be non-edundant secuities. In paticula, we identify a class of affine models that can ehibit US. The affine famewok is convenient because it povides closed-fom solutions fo bond yields that ae linea in the state vaiables. Howeve, we identify paamete estictions such that bond yields do not depend on the volatility state vaiable. As such, bonds cannot be used to hedge volatility isk, in tun implying that bonds do not span the fied income makets. Afte poviding a fomal definition of US, we show that it is not possible fo bivaiate Makov affine models to ehibit such behavio, thus uling out the models of Fong and asicek (99), Longstaff and Schwatz (99), and Chen and Scott (993) as potential candidates. Moe geneally, we demonstate that even non-affine bivaiate models of the shot ate cannot geneate US. We then identify necessay and sufficient conditions fo a tivaiate Makov system to ehibit US. While such models as Balduzzi, Das and Foesi (996), and Chen (996) cannot satisfy these estictions, we demonstate that the maimal A (3) model poposed by Dai and Singleton (000, heafte DS) can ehibit US. Focusing on tivaiate models, DS analyze the maimal numbe of paametes that can be identified given a seies of bond pices. Below, we ague that thei analysis is even moe geneal in that maimality efes to the maimum numbe of paametes that can be identified given all fied income secuities. As we demonstate, the distinction is impotant because, fo the class of models which ehibit US, bond pices alone ae not sufficient to detemine all of the identifiable paametes of the model. Rathe, both bonds and fied income deivatives ae needed to identify the system. The absence of the volatility state vaiable in bond pices implies that bond innovations ae not contempoaneously affected by volatility innovations, and theefoe cannot be used to hedge volatility isk instantaneously. Ove a longe hoizon, howeve, bond pices ae affected by changes in volatility. This non-contempoaneous effect begs the question whethe ou models, which geneate US in continuous-time, will geneate US if data ae obseved at only discete time intevals. To investigate whethe the poposed class of models can eplicate ou empiical findings, we simulate a monthly time seies of staddle pices and swap ates in both a taditional stochastic volatility model (we use the A (3) model of DS (000)), and in a simila thee-facto model which ehibits US. We then epeat the egession analysis descibed above fo both simulated economies. The esults confim that if a taditional model is used to geneate the data sample, then ove 95 pecent of the vaiation in staddle pices can be eplained by changes in swap ates. Howeve, if a US model is used to geneate the data, then as little as 5 pecent of the vaiation may be eplained by changes in the swap ates, simila to the esults obtained when actual histoical data is used. 3

4 To ou knowledge, unspanned stochastic volatility models wee fist investigated by Andeasen, Collin-Dufesne and Shi (997, heeafte ACS). They note that, while the HJM famewok uniquely specifies the dift of the fowad ates in tems of its volatility stuctue unde the isk-neutal measue, this specification is not sufficient to pice inteest ate contingent claims if the volatility stuctue evolves stochastically. Indeed, one also needs to specify the abitage-fee pocess fo these volatilityspecific factos unde the isk-neutal measue. Since the HJM estiction does not povide any guidance in that espect, ACS popose that contacts such as futues on yields should be used to identify and calibate the model. 4 Below, we do not eplicitly conside the backing-out of the abitage-fee dynamics of the volatility-specific state vaiables. Rathe, we take the dynamics unde the isk-neutal measue as given, and focus on the picing implications of such a model fo fied income deivatives. Futhe, we identify necessay and sufficient conditions fo a model to display US within a time-homogeneous setting. In contast to ACS, who investigate a two-facto HJM model with a deteministic (Gaussian) volatility stuctue and pice deivatives using a non-ecombining lattice appoach, we develop a moe geneal famewok. Futhe, ou models possess closed-fom solutions fo deivatives pices, as shown in, fo eample, Duffie, Pan, and Singleton (000, heeafte DPS). Recently, in independent wok, Kimmel (00a, 00b) investigates a HJM-type andom field model 5 of the tem stuctue whee volatility of bond pices is diven by latent vaiables that possess an affine stuctue. Futhe, he identifies the patial diffeential equation that deivative secuities with homogeneous payoff stuctues satisfy. 6 In his empiical pape, Kimmel (00b) focuses on the dynamics of bond pice volatility and coelation stuctues, since andom field models offe no pedictions fo the coss section of bond pices. In contast, we identify a class of affine models that possesses a finite dimensional Makov epesentation fo bond and bond-option pices. While most of ou wok is esticted to the time-homogenous setting, we also discuss how to simultaneously calibate the tem stuctues of both inteest ates and volatilities. The est of the pape is as follows. In Section I we povide empiical evidence that suggests the bond maket is incomplete. In paticula, we pesent evidence that thee ae souces of isk which dive innovations in staddles but do not dive innovations in the swap ates themselves. In Section II we identify unde what conditions affine models can ehibit US. In paticula, we fist show that no two-dimensional system can ehibit US. We then identify necessay and sufficient estictions fo a thee-dimensional affine system to ehibit US. To demonstate that the poposed models ehibit US even ove finite time intevals, we epeat the empiical pocedue of Section I using data geneated fom ou models, but sampled at only monthly time intevals. In Section III we show that models ehibiting US ae geneated natually when fowad ates, athe than the spot ate and othe state vaiables, ae taken as pimitives. Some special cases ae investigated. We conclude in Section I. 4

5 I. Empiical Suppot fo Unspanned Stochastic olatility Empiical evidence suggests that multiple factos ae needed to adequately captue the dynamics of the coss-section of bond pices. 7 Model-independent facto-analysis finds that thee factos eplain almost all yield cuve vaiation (Litteman and Scheinkman (99)). Model-dependent investigations within the affine famewok (Chen and Scott (993), DS (000)) o quadatic models (Ahn, Dittma, and Gallant (00)) similaly find that at least thee factos ae necessay to adequately captue the dynamics of the tem stuctue of inteest ates. Recently, the focus has shifted to estimating the numbe of factos necessay to pice fied-income deivatives. Boudoukh et al. (997) find that appoimately 90 pecent of the vaiation in picing of motgage-backed secuities can be eplained by a few inteest ate level factos. Howeve, they ae unable to eplain the final 0 pecent. Longstaff, Santa-Claa, and Schwatz (00a, 00b) find that, when picing Ameican-style swaptions, seveal state vaiables ae needed. They also find that the implied volatility computed fom cap pices eflect fou state vaiables. Thei findings diffe fom those of Andesen and Andeasen (00), who ague that a one-facto model with level dependence of the volatility stuctue of fowad ates might be as effective in fitting and hedging caps and floos. Fan, Gupta, and Ritchken (00) test vaious multi-facto models of the tem stuctue and find that they systematically mis-pice caps and floos. Hee we eploe a elated question. In paticula, we investigate how many bonds ae needed to hedge inteest ate volatility-isk. Using vaious poies fo the shot ate, pevious empiical studies have found that stochastic volatility is a obust featue of shot ate dynamics (Benne, Hajes, and Kone (996), and Andesen and Lund (997)). Hee, we eamine how much of the vaiation of etuns in staddles (potfolios of at-the-money caps and floos) can be eplained by the vaiations in swap ates. We focus on staddles because they ae highly sensitive to bond-pice volatility isk. I.A. The Data We use monthly data on swap ates, caps, and floos fo U.S., U.K., and Japan fom Datasteam fo the peiod anging fom Febuay 995 to Decembe 000. The available swap ate data includes matuities of one, two, thee, fou, five, si, seven, eight, nine and ten yeas. 8 The si-month LIBOR ate is used as a poy fo the si-month swap ate. 9 We use the available swap ate data to constuct the zeo-coupon bond yield cuve, which in tun is used to detemine the discount facto fo the cashflows in the CAP and FLOOR maket. This appoach implicitly assumes that (i) the floating leg of the swap contact is valued at pa and hence that the quoted swap ate is equivalent to a pa-bond ate fo an issue with LIBOR-cedit quality, 0 and (ii) that CAP, FLOOR, and SWAP makets have homogeneous cedit quality. We intepolate zeo-coupon bond yields with intemediate o nonavailable matuities fom the closest available yields to maimize smoothness (i.e., minimize squaed cuvatue) of the obtained yield cuves. 5

6 The cap and floo data ae quoted in tems of implied volatility fo at-the-money (ATM) caps and floos. The implied volatilities ae obtained using the Black (976) fomula. (see Hull (000) p. 540). The phase at-the-money implies that both caps and floos have the same stike, which is set to equal the fowad swap ate (see Musiela and Rutkowski (997) p. 393), implying that caps and floos have the same initial value. Using ou computed zeo-coupon cuve we tansfom the implied volatility data into cap and floo pices. Thus, fo seveal diffeent matuities we obtain a time seies of constant matuity, at-the-money cap and floo pices. Howeve, since ou goal is to analyze the hedging pefomance of diffeent tem stuctue models, we need monthly changes in pices fo a given cap o floo contact. Unfotunately, since the CAP/FLOOR maket is mostly a boke/intebank maket, we ae unable to obtain tansaction data on eisting cap and floo contacts. We thus esot to intepolation in ode to estimate monthly changes in cap and floo pices. The pocedue we adopt is the following. In month n we have an implied volatility n fo an at-the-money cap with time-to-matuity T and stike K. In month (n +)we use data on implied volatility fo at-the-money caps fo seveal matuities to intepolate and thus estimate the implied volatility coesponding to a cap with stike K and time-tomatuity T ;. We use this estimated implied volatility as an input to the Black fomula with stike K and appopiately intepolated fowad ates and zeo-coupon yields. We thus compute a mati of one-month changes in pices of at-the-money cap pices. Note that we do not follow a contact ove its whole life. Rathe, each month we stat with ATM cap pices and then use data fom the net month to compute one-month pice changes. This appoach has two advantages. Fist, this method minimizes the noise intoduced by the intepolation pocedue used to estimate the implied volatility of the non-obseved cap pices. Second, it consides only potfolios that ae delta-neutal at inception, making them moe sensitive to changes in volatility (elative to changes in inteest ate levels). We poceed similaly with floos. As a simple consistency check of the intepolation pocedue, we test the cap-floo paity condition by computing coesponding monthly changes in the fowad swap contact. We find that the intepolated values satisfy (cap est ; floo est = fowad swap est ) etemely well. 3 I.B. Methodology and Results We then poceed to analyze monthly etuns of staddles fo diffeent matuities. Ceating staddle potfolios allows us to focus on the pesence of unspanned stochastic volatility (US) because staddles ae athe insensitive to (small) changes in the level of inteest ates, but ae etemely sensitive to changes in volatility. 4 Such an analysis is impotant fo those financial institutions which tend to be delta-neutal, that is, those fims which hedge away inteest ate isk. We un sepaate egessions of changes in staddle pices fo matuities of one, two, thee, fou, five, seven, and ten yeas on changes in swap ates. Since we would like to obtain an estimate of the best possible hedging of volatility isk we can achieve by using swaps, we conside as independent 6

7 vaiables as many swap ates as ae available. Fo the U.K. and Japan data we conside an -facto model and fo the U.S. an eight-facto model. 5 The R and adjusted R ae epoted in Table I. 6 Clealy thee is multicollineaity in the independent vaiables, but the R povides an uppe bound to the amount of vaiation in staddle etuns that can be hedged with swaps. The R ae quite low, implying that potfolios of bonds (o equivalently, swaps) have vey limited ability to hedge volatility isk. We emphasize that these findings ae inconsistent with the pedictions of the taditional tem stuctue models with stochastic volatility (such as Fong and asicek (99), Longstaff and Schwatz (99), Chen and Scott (993), and DK (996)). Indeed, as we show in Section II.D., eplicating the above egession in a simulated taditional affine economy would esult in an R well above 90 pecent. Inset Table I about hee Afte unning the multi-facto egessions, we estimate (sepaately fo each county) the covaiance mati of the esiduals acoss staddle matuities. We then pefom a pincipal components analysis on each covaiance mati. The coesponding eigenvalues ae epoted in Table II. We find that the fist eigenvalue captues ove 80 pecent of the emaining vaiation fo each of the thee counties. 7 This finding implies that the low R obtained fo the egession of staddles on swaps is not due to noisy data but athe to model mis-specification. Futhemoe, these esults suggest that one, o at most two, additional unspanned stochastic volatility state vaiables ae sufficient to eplain almost all of the vaiation in staddle etuns acoss matuities. That is, the tem stuctue of volatilities is mostly diven by one o two state vaiables whose dynamics ae mostly independent of those factos that dive swap ate innovations. Inset Table II about hee In Table III we epot the esults of egessing the etuns of an equally weighted potfolio of staddles on changes in thee potfolios of swap ates. These thee swap ate potfolios, which captue ove 98 pecent of the vaiation of swap ate changes in all thee cuency makets, ae the fist thee pincipal components of the swap tem stuctue, and coespond oughly to estimates of changes in level, slope, and cuvatue of the yield cuve. We pefom this egession fo thee easons. Fist, by investigating the etuns of an equally weighted potfolio of staddles athe than staddles of individual matuities, we povide futhe evidence that ou esults ae not due to noisy data. Second, by looking at the fist thee pincipal components of swap innovations athe than innovations in all available swap matuities, we eliminate multi-collineaity. Finally, the facto loadings on these pincipal components ae of inteest. Inset Table III about hee 7

8 The esults confim ou pevious findings in that the factos diving the tem stuctue of swap ates can baely eplain 0 pecent of the vaiation of staddle etuns (in fact, as little as.3 pecent fo the U.K. data 8 ). Inteestingly, staddle etuns appea to be negatively coelated with changes in level, but positively coelated with changes in slope, at least fo the U.S. and Japan makets. The thid facto, often associated with cuvatue, is statistically insignificant in all thee makets. The finding that volatility is negatively elated to level (holding slope constant) is somewhat counteintuitive fom the standad esult (e.g., Chan et al. (99)) that volatility ises with the spot ate level. A possible eplanation is that the time seies we investigate includes flight-to-quality events, whee inteest ates plummeted while volatility ose. 9 As an independent check on the poposed intepolation scheme, we also un egessions whee the dependent vaiables ae the changes in the implied volatilities of caps and floos, athe than ou constucted staddle etuns. As noted by Ledoit and Santa-Claa (999), staddle etuns and changes in implied volatilities ae likely to have simila infomation content, since at-the-money options ae appoimately linea in volatility. Nealy identical esults ae obtained (and thus not epoted), suggesting that the intepolation scheme is not diving ou esults. 0 These empiical findings suggest that bonds do not span the fied income makets. In paticula, caps and floos seem to be sensitive to stochastic volatility that cannot be hedged by a position solely in bonds. In the net section, we povide a time-homogeneous famewok that is consistent with these findings. II. Affine Models of Incomplete Bond Makets We assume that uncetainty is descibed by a standad filteed pobability space. The innovations that dive fied-income secuities ae descibed by a d-dimensional vecto of Bownian motions z Q. The filtation is the natual filtation associated with the Bownian motion. Slightly genealizing equation (4.) of DK (996), we conside the class of models whose N-state-vaiable, d-facto dynamics (with N d) possess an affine stuctue as follows: dx =(ax + b) dt p + vdz Q () whee a R NN, b R N, and R Nd has ank d. The components of the (d d) diagonal mati v ae affine in the state vaiables X. A well-known esult fom linea algeba guaantees that thee eists an (N;d)-dimensional space, the kenel of the mati >, such that each vecto in this space satisfies > =0. It is convenient to otate the initial set of state vaiables fx ::: X N g to a new set fx ::: X d X 0 ::: X0 g d+ N so that the last (N ; d) of them ae defined via X 0 i = N X j= i j X j i (d + N) () 8

9 o, in mati notation, X 0 = > X. Fom thei definition, it follows that these state vaiables ae i i locally deteministic. We thus popose the following: DEFINITION : Afte suitable otation, an N-vaiate, d-facto affine model of the tem stuctue possesses the following popeties: () Thee is a set of N d state vaiables X i i = :::N that ae jointly Makov, whee each element of the dift and covaiance mati of the vecto pocess X(t) is affine in the N state P vaiables. N () The instantaneous isk-fee ate is an affine function of these state vaiables: t = 0 + i= X (t), i i with at least one non-zeo i. Futhemoe, thee is no smalle subset of the N state vaiables that is both jointly Makov and sufficient to descibe the dynamics of t. (3) The fist d state vaiables have a diffusion mati that is full ank. The last (N ; d) state vaiables ae locally deteministic. We now povide a fomal definition of incomplete bond makets, consistent with the tetbook definition (e.g., Duffie (996), and Kaatzas and Sheve (998)). DEFINITION : Define H as the set of all matices obtained by stacking any finite collection of bond-pice diffusion (ow) vectos. Define d H as the lagest ank of any of these matices. A tem stuctue model geneates incomplete bond makets if d H <d. Within the affine class of models, bond pices can be witten in the fom: P T (t) = ep A(T ; t) + NX i= B i (T ; t) X i (t)! (3) whee A( ) B ( ) ::: B N ( ) ae continuous deteministic functions that ae solutions to a system of odinay diffeential equations (see DK (996)). Noting item 3 of definition, Itô s lemma implies that incomplete bond makets can also be chaacteized by: PROPOSITION : An N-vaiate, d-facto affine model geneates incomplete bond makets if and only if thee eists a set of paametes f ::: d g not all zeo such that dx i= i B i ( )=0 8 > 0 : (4) The numbe of linealy independent sets of paamete fg that satisfy this condition equals the numbe of state vaiables that ae not spanned by the bond makets, and thus, the numbe of additional non-bond secuities needed to complete the fied-income makets. 9

10 The intuition fo this esult is the following. If equation (4) holds, then without loss of geneality, we can take d 6=0. Then, we can wite Plugging this into equation (3), we find P T (t) = ep A(T ; t) + Xd; i= Xd; B d ( )=; i B i ( ) : (5) d B i (T ; t) i= X i (t) ; i X d (t) + d NX i=d+ B i (T ; t) X i (t)! : (6) It is convenient to change vaiables fom (X ::: X N ) to (Y ::: Y X ::: X d; d N ), whee the fy i g ae defined via: Y i (t) X i (t) ; i d X d (t) 8 i = :::d; : (7) Unde this change of vaiables, bond pices ae independent of the state vaiable X d : P T (t) = ep A(T ; t) + Xd; i= B i (T ; t) Y i (t) + NX i=d+ B i (T ; t) X i (t)! : (8) Hence, no potfolio that is composed solely of bond can complete the fied-income makets, because X d -isk cannot be hedged by bonds. Moe fomally, equation (4) implies that the ank of the diffusion mati of the etun on any potfolio compising bonds of diffeent matuities is less than d. Below, we detemine the necessay conditions fo affine models to geneate incomplete bond makets. This basically amounts to identifying paamete estictions on the dynamics of the state vaiables X i (i = ::: d) so that the functions B i () (i = ::: d) satisfy equation (4). The method we use to identify these paamete P estictions effectively educes to pefoming a Taylo seies epansion on the functions B i B ( j i )= (0) j=0 j, whee B j (0) efes to the j th time-deivative j! of the function, evaluated at =0. It then follows that equation (4) can be witten: 0 = = dx i= X j=0 i j j! X j=0 B j (0) i ( ) j 8 > 0 (9) j!! i B j (0) 8 > 0 : (0) i dx i= Equation (0) implies that affine models geneate incomplete bond makets if and only if the model satisfies: 0= dx i= i B j (0) 8j =(0 ::: ) : () i 0

11 II.A. Bivaiate Affine US Models The bivaiate affine models of Fong and asicek (99) and Longstaff and Schwatz (99) wee the fist tem stuctue models to incopoate stochastic volatility into a fied-income famewok. Hence, it seems natual to investigate unde what conditions, if any, bivaiate affine tem stuctue models can geneate unspanned stochastic volatility (US). As a point of efeence, we note that the model of Heston (993), which seves as a benchmak fo stochastic volatility models of equity options, is a bivaiate affine model of equity etuns ehibiting US. Below, howeve, we demonstate that: PROPOSITION : Bivaiate affine models of the tem stuctue cannot geneate incomplete bond makets, and thus, cannot ehibit US. PROOF: See Appendi A. The intuition fo why bivaiate affine models cannot ehibit US can be povided in tems of duation (P ) and conveity (P ). By definition, a bivaiate model ehibiting US would imply bond pices ae functions of only the time-to-matuity and the spot ate, and independent of the spot ate volatility : P T (t t t )=P T (t t ). This in tun implies that bond pices must satisfy P T (t ) (t )+P T (t ) (t )=PT (t ) ; P T (t ) 8 T: () t Note that the ight-hand side of equation () is a function only of, while the left-hand side is a function of both and. Since it is not possible fo the atio of duation and conveity to be constant acoss matuities, thee is no way fo the left hand side to be independent of, unless the spot ate pocess itself is one-facto Makov ( (t ) = (t ), (t ) = (t )), which also pecludes US. We emphasize that this heuistic agument is not limited to the affine famewok. 3 Indeed, we can show moe geneally that: PROPOSITION 3: Bivaiate Makov model of the tem stuctue, affine o othewise, cannot geneate incomplete bond makets, and thus, cannot ehibit US. PROOF: See Appendi B. The implication of Poposition 3: is that at least thee state vaiables ae necessay fo an affine model to geneate incomplete bond makets. II.B. Tivaiate Affine US Models Hee we deive necessay conditions fo thee-dimensional affine models to geneate incomplete bond makets. Fist we pove that one can always conveniently otate the undelying state vaiables

12 to make them economically meaningful: PROPOSITION 4: Evey affine model of thee state vaiables that geneates incomplete bond makets can be witten so that the thee state vaiables ae: ) the spot ate,, ) the dift of the spot ate, = dt EQ [d], and 3) the vaiance of the spot ate = dt EQ [(d) ]. PROOF: See Appendi C. We emphasize that this poposition is not tivial. Fist, this poposition implies: COROLLARY : Tivaiate Gaussian models (such as Langetieg (980)) cannot geneate incomplete bond makets. This follows because in a Gaussian famewok, the vaiance of the spot ate is a constant, and thus cannot be a state vaiable. Futhe, Poposition 4: implies that models such as and d t = ( t ; t ) dt + p t dz Q (3) ; p d t = ; t dt + f t dz Q + gp t dz Q (4) ; p d t = ; t dt + t dz p Q + ; dz Q (5) 3 d t = ( ; t ) dt + p t dz Q (6) ; p d t = ; t dt + t dz Q p + t dz Q (7) ; p d t = ; t dt + t dz Q + p ; dz Q 3 ; cannot geneate incomplete bond makets. In the fist eample, dt ; a Q [d] is not linealy independent of. In the second eample, dt E Q [d] is not linealy independent of. Poposition 4: allows us to limit ou seach fo tivaiate models that geneate incomplete bond makets to models whose bond pices take the fom 4 (8) P T (s s s s )=e A(T ;s)+b (T ;s) s +B (T ;s) s + s : (9) It is convenient to investigate sepaately those models that have =0, which we efe to as models that ehibit US, fom those models that have 6= 0. We eamine this second case fist. With

13 6= 0, we can change vaiables fom f s s g to f s s ; s + s g. This implies the bond pice can be witten P T (s s s s )=P T (s s s )=e A(T ;s)+b (T ;s) s +B (T ;s) s (0) independent of s. We claim: PROPOSITION 5: Thee ae no tivaiate models with state vaiables f t t dt EQ [d t ] t g whose bond-pices can be witten in the fom of equation (0). Hence, all tivaiate affine models that geneate incomplete bond makets have bond pices of the fom: P T (s s s s )=P T (s s s )=e A(T ;s)+b (T ;s) s +B (T ;s) s : () PROOF: See Appendi D. The implication of this poposition is that all tivaiate affine models that geneate incomplete bond makets ae also models that ehibit US. We emphasize, howeve, that the fome class is lage if we look at models with moe than thee state vaiables. Given Poposition 5:, it is convenient to define dt EQ [d] = m 0 + m + m + m () dt EQ [(d) ] = (3) dt EQ [d d] = c 0 + c + c + c : (4) By applying Itô s lemma to equation (), and then collecting tems of ode constant,, and, we find that the time-dependent coefficients ae defined though A 0 = m 0 B + 0 B + c 0 B B (5) B 0 B 0 = m B + B + c B B ; (6) = m B + B + c B B + B (7) and satisfy the bounday conditions A(0) = 0 B (0) = 0 B (0) = 0 : (8) Futhemoe, by collecting tems of ode, we find that this model ehibits US if and only if fo all dates the following condition holds: We claim: 0=m B ( )+ B ( )+c B ( )B ( )+ B ( ) : (9) 3

14 PROPOSITION 6: With the functions A( ), B ( ), B ( ) defined implicitly though equations (5)- (7), the necessay and sufficient conditions fo the model to ehibit US (i.e., fo equation (9) to hold) ae that one of the following two sets of paamete estictions holds: 8 8 m = ; (c ) + c m = ;(c +(c ) + c =c ) m >< = 3c m >< = 3c m = m = o = ;c (c +c c ) = ;c (c +(c ) + c =c ) >: = 4c +6c c >: = 4c c +c = (c ) = c +(c ) + c =c (30) PROOF: See Appendi E. A few points ae woth noting. Fist, the eason that thee ae two sets of paamete estictions that geneate US is because equation (9) is a quadatic equation in B () o B (). This in tun geneates two possible solutions fo B () in tems of B (). Second, these two sets of estictions educe to the same set if and only if c 6= 0 and c + c c = 0. This condition obtains, fo eample, when the covaiance between the shot ate and its dift depends only on the volatility, an impotant special case (basically, the so-called A (3) models) which we eamine below. Finally, we note that seveal of the estictions noted in Poposition 6: occu natually once we limit the class of models to those which ae admissible: That is, those which estict the squae-oot state vaiables to be non-negative. To povide some intuition fo the poof of Poposition 6:, sufficiency obtains because the ighthand side of equation (9) can be shown to be identically zeo when eithe of the two sets of paamete conditions holds. Necessity obtains because if any one of the conditions is not satisfied, then we can show, by taking epeated time deivatives of the system of ODE s (Ricatti equations) evaluated at =0, that equation () cannot hold. We note that the models of both Chen (996) and Balduzzi et al. (996) cannot satisfy these necessay estictions, and thus cannot display US. Also, as noted in Coollay :, the A 0 (3) class of models of DS (000) cannot ehibit US. Howeve, DS s maimal A (3) A (3), and A 3 (3) models do have the fleibility to ehibit US. 5 Fo simplicity, we only conside the so called A (3) family of affine models given by: p dv = v (^v ; v) dt + v vdz Q (3) v h i p d = (^ ; ) + (^ ; ) + v (^v ; v) dt + + vdz q Q + + v dzq + p v vdz Q (3) v h i p q d = (^ ; ) + (^ ; ) + v (^v ; v) dt + + vdz Q + + v dzq + p v vdz Q (33) v whee z z z v ae independent standad Bownian motions. These equations coespond to the most geneal A (3) admissible model. Admissibility equies that all pocesses be well defined and, in the 4

15 paticula case at hand, equies that v be a standad squae oot pocess. Intuitively, if any othe state vaiable appeas in the dift o diffusion of v, then the positivity of v cannot be guaanteed, because both and can take on both positive and negative values. We claim: PROPOSITION 7: Necessay and sufficient paamete estictions fo the A (3) model given in equations (3) to (33) to display US ae: v ( v ; )+ v = + v + (34) 9( ; ) = ( + ) (35) 3 ( + ) = + ( + + v v )+ v v v (36) ( 9 + ) = ( + ) +( + ) +( v + v + v v ) : (37) If these estictions ae imposed, then bond pices take the fom: P T (t) = ep fa(t ; t) +B (T ; t) t + B (T ; t) t + B v (T ; t) v t g (38) whee the functions A( ) B ( ) B ( ) ae: h i B ( ) = ;9 ( + ) +6( ; )e ; 3 ( + ) + 3( ; )e ; 3 ( + ) B ( ) = A( ) = 9 h ( + ) Z 0 ds i ; e ; 3 ( + ) (40) " + B (s) + + B (s) +( + )B (s)b (s)+ B (s)( ^ + ^ + v ^v) +B (s)( ^ + ^ + v ^v) +B v (s) v ^v i (39) : (4) US obtains because: B v ( v )= B ( ) 8 0 : (4) PROOF: To pove that these paamete estictions ae necessay and sufficient, we use Poposition 6:, and then change vaiables fom ( ) to ( v) given by: = ^ + v ^v + ^ ; ; ; v v (43) = ( + )+( + +)v: (44) v Fom DK (996), we know bond pices take the fom of equation (38). This pemits us to detemine the set of Ricatti equations satisfied by the A() and the B() functions. In paticula, the Ricatti equations fo B () and B () ae B 0 ( ) = ; B ( ) ; B ( ) ; (45) B 0 ( ) = ; B ( ) ; B ( ) (46) 5

16 the solutions to which ae povided in equations (39) and (40). It is then a matte of staightfowad (but tedious) veification that B v ( )= v B ( ) solves the ODE if the US paamete estictions ae satisfied. It is inteesting to note that by imposing the US paamete estictions on the A (3) model, we obtain a closed-fom solution fo the B v function. This is in contast to the geneal A (3) model, whee B v does not possess an analytic solution. Futhemoe, this closed-fom solution possesses the typical Gaussian eponential time-decay stuctue. Clealy though, even unde the US paamete estictions, this A (3) model does not degeneate to a Gaussian model. Thus we obtain a model with a tem stuctue simila to that of a Gaussian two-facto model, but whee the shot ate volatility follows an autonomous squae oot (CIR) pocess. Futhe, one can show that fo the US model with v =0the poposition above holds fo any autonomous one-facto Makov volatility pocess. In othe wods, we may have an affine bond pice fomula fo a state vecto which is not necessaily affine! 6 II.C. US and Maimality DS identify the maimum numbe of paametes that can be identified within diffeent classes of thee-state vaiable affine models conditional on obseving only bond pices. In paticula, they find that the maimal A (3) model obtains when the following ove-identifying estictions: ^ = v = =0and = ;, ae applied to equations (3) to (33). In that case the emaining 4 paametes should be identifiable fom bond pices. Howeve, fo the A (3) model ehibiting US, it can be seen by looking at the closed-fom solution fo bond pices that at most eight paametes (o combinations theeof) ae identifiable fom the coss-section of bond pices. 7 Howeve, the notion of maimality of DS appaently genealizes to the numbe of state vaiables and paametes that can be identified by obseving panel data on all fied income secuities (i.e., not just bond pices). Indeed, the invaiant otations poposed by DS depend only on the fom of the fundamental patial diffeential equation of fied-income secuities, which is independent of the bounday conditions specific to bond pices. While we have yet to detemine the maimum numbe of paametes which ae identifiable fo the A (3) model ehibiting US, hee we give an eample of a model which ehibits US, guaantees admissibility, and demonstates that it is not possible to identify all paametes fom bond pices alone. PROPOSITION 8: Conside the following model: dv = ( v ; Q v v) dt + v p vdz Q v (47) d = ( ; + v) dt + dz Q (48) d = ( ; ) dt + p + vdz Q + dzq (49) 6

17 with the added estictions to guaantee that the model is admissible: v > 0 0 : (50) Futhe assume that the isk pemia ae such that dz Q v = dz v + p vdt (5) dz Q dz Q = dz (5) = dz (53) then bond pices take the fom ((T ; t) ): whee P T (t t t )=epfa( ) ; B ( ) t ; B ( ) t g (54) B ( ) = ; ; e ; (55) B ( ) = A( ) = ; ; e ; Z 0 ds " (B (s)) + + (B (s)) + B (s) B (s) ; B (s) # (56) : (57) Futhe, Q cannot be identified fom bond pices alone. Rathe, fied income deivative pices ae v needed to identify Q. v PROOF: Since the state vecto dynamics ae affine, we know fom DK (996) that bond pices take the fom: P T (t t t )=epfa( ) ; B ( ) t ; B ( ) t ; B v ( ) v t g (58) whee the bond pice satisfies the PDE: P = P t + P ( ; ) + P ; + v + P v v ; Q v v + P + v + + P + v P vvv + P : (59) Collecting tems that ae linea in,, v, and constant geneate the system of ODE s: B 0 ( ) = ; B (60) B 0 ( ) = B ; B (6) B 0 ( ) = B v ; Q B v v ; B ; v B (6) v A 0 ( ) = ; B ; v B v + B + + B + B B (63) 7

18 with the initial conditions B (0) = B (0) = B v (0) = A(0) = 0. The solutions to equations (60) to (6) ae those given in equations (55) to (56). Note that these equations satisfy B ; B = 0, implying that equation (6) educes to B 0 v ( )=;Q v B v ; v B v : (64) Given the initial condition B v (0) = 0, it is clea that B v ( ) = 0 US. Note that since B v ( )=0, equation (63) educes to 8. Hence, this model ehibits A 0 ( )=; B + B + + B + B B (65) independent of v. Moe geneally, note that bond pices ae completely independent of all of the paametes ( Q, v v and v ) that dive volatility dynamics. This implies the model can be etended to allow fo a vey simple two-step calibation pocedue to fied-income deivatives (such as atthe-money Caps/Floos). Fist, as in Hull and White (990), the paamete can be made timedependent to fit the tem stuctue of fowad ates. Second, some of the paametes f v Q v vg can be made time-dependent to fit the tem stuctue of volatilities, without affecting the initial calibation of the tem stuctue of fowad ates. Unde the histoical measue, the volatility dynamics follow dv = ( v ; Q v v) dt + v p v ; dzv + p vdt ( v ; v v) dt + v p vdzv (66) whee v Q ; v v. With this specification of the isk pemia, a time seies of bond data can identify the state vaiable v, along with the paametes which show up unde the histoical measue, namely, v, v, and v. Howeve, given only bond pices, it is not possible to identify Q. It is v staightfowad to demonstate, though, that Q can be identified if othe fied income deivatives ae v available. II.D. Simulation of US Model Using Monthly Sampling The models poposed above geneate bond pices that ae independent of the cuent volatility state vaiable. Hence, these models pedict that instantaneous bond etuns cannot hedge instantaneous changes in volatility, and theefoe cannot hedge staddles. Note, howeve, that the empiical suppot fo this class of models comes fom data that ae sampled monthly. To demonstate that the poposed class of models is consistent with ou empiical findings, we pefom the following epeiment. We fist simulate a time seies of monthly swap ate, cap and floo pices fom a paticula A (3) economy whee the paametes govening the state vecto do not satisfy the US estictions. 8 We then egess staddle etuns on changes in swap ates. Ou esults indicate that only thee swap 8

19 ates ae necessay fo the thee-facto model to obtain an R above 90 pecent. That is, even with only monthly sampling, and esticting the OLS egession to constant coefficients, the obseved staddle etuns ae almost pefectly eplained by changes in swap ates. We then epeat the same epeiment in a simila economy, ecept this time we adjust the paamete values so that the necessay estictions fo the A (3) model to ehibit US ae satisfied. The egession analysis in this simulated economy eveals that, even though data is sampled monthly, one still obtains the continuous-time esult that (i) only two diffeent swap matuities can be used as egessos, o else the invesion of the covaiance mati becomes nealy singula, and (ii) only about 30 pecent of the vaiation in staddles can be eplained by these egessos. These findings imply that ou poposed model can geneate US even if data is sampled only monthly. III. US within a HJM Famewok Ou empiical findings stongly suggest that thee ae souces of isk that dive innovations in staddle etuns, but do not (instantaneously) affect the undelying swap ates. Within a tivaiate affine setting, we ae able to identify paamete estictions that geneate a class of models consistent with these empiical findings. A potential citicism of this appoach, howeve, is that the imposed knife-edge paameteization gives the appeaance that the constuction of models ehibiting US is contived. In this section, howeve, we demonstate that US is geneated natually within a HJM famewok. Indeed, we demonstate below that almost all HJM stochastic volatility models geneate US. Futhemoe, by specializing to models that possess Makov epesentations, we demonstate that within a HJM envionment the estictions found in the pevious section aise natually. Within the HJM famewok, fowad ates f T (s) ae taken as inputs. A simple model in this class has the fom: df T (s) = T (s) ds + B T (s s ) dz Q (s) (67) d s = m s ds + s dz Q (s) (68) The dift T (s) is detemined by the volatility stuctue, as shown by Heath, Jaow, and Moton (99). Fom Itô s lemma and the definition of fowad ates, f T (s) log P T (s), we obtain the bond pice dynamics dp T s P T = s ds ; B T (s s ) dz Q (s) (69) s R whee we have defined B T T (s s ) s B (s u s ) du. We note that eithe the set equations (67) and (68), o equations (69) and (68), can be used to chaacteize the system. Thus, fom a HJM pespective, we ae effectively modeling the dynamics of a set of taded assets (i.e., bonds), and the state vaiable diving the volatility of these assets. 9

20 Define as the coelation between the Bownian motion that geneates bond pice innovations (dz Q (s)) and the Bownian motion that geneate volatility (dzq (s)). Note that, ecluding the cases =, all HJM stochastic volatility models ehibit US. In this sense, US is a vey natual phenomenon when one diectly models fowad ates (o equivalently, bond pices) and thei volatility dynamics, athe than modeling spot ate and its volatility dynamics. This esult is analogous to US models of equity pices, such as Heston (993). That is, when one diectly models the dynamics of a taded asset, (equity o bond/fowad pices) and that asset s volatility, US is (almost) always geneated. In contast, in a standad affine model one typically specifies the dynamics of state vaiables (e.g., the shot ate) that ae not taded assets. In geneal, these models will not ehibit US. 9 III.A. A Two-Facto HJM Model Ehibiting US Conside the two-facto model: df T (s) = a(s T ) A(s T )(s) ds + a(s T ) (s) dz Q (s) (70) Q d(s) = (s) ds + b p(s) dz p Q (s) + ; dz Q (s) (7) p whee dz Q (s) and dzq (s) ae independent Bownian motions, and we have defined Q (s) Q 0 + Q (s) + Q (s) (7) (s)! 0 +! (s) +! (s) (73) A(s T ) Z T s dv a(s v) : (74) As fist noted by HJM, the dift of the fowad ate dynamics unde the isk-neutal measue ; a(s T ) A(s T )(s) is uniquely specified by the volatility stuctue on Q (s), the dift of. 30 a(s T ) p (s). Howeve, thee ae no estictions As specified, the model leads to vey geneal dynamics fo the tem stuctue of fowad ates, and hence also fo the isk-fee ate. In paticula, fo abitay functions a(s T ), the dynamics of the the system ; ff T (s)g s will in geneal be non-makov. Howeve, as demonstated by Cheyette (995), a Makov epesentation can be found if we assume the functional fom a(s T )=a(t )=a(s) and! =0. Below, we genealize these findings. 3 III.B. Makov Repesentation and Eistence As mentioned peviously, fo geneal functions a(s T ) it is not possible to obtain a Makov epesentation fo the model poposed above. Although in a companion pape (Collin-Dufesne and Goldstein (00)) we povide closed-fom solutions fo a lage numbe of deivatives fo geneal functional fom of a(s T ), we cannot in geneal deive simple algoithms to pice path-dependent instuments such as Ameican options. In this section we show that if a(s T ) is modeled as sepaable: 0

21 a(s T ) = a(t )=a(s) fo some function a(), then a Makov epesentation of the model obtains. 3 We claim: PROPOSITION 9: Assume a(s T ) takes the fom: a(s T )=a(t )=a(s). Define Y (t) = Z t 0 ds (s) a(t)a(t) a (s) (75) R whee A(t) = t ds a(s). Then the model poposed in equations (70) to (7) possesses a Makov epesentation in the thee state vaiables f(t) (t) Y(t)g. The state vecto is affine, and bond pices ae eponentially affine functions of the subset f(t) Y(t)g of the state vecto. All fiedincome deivatives ae solutions to a patial diffeential equation, subject to appopiate bounday conditions. PROOF: Integating the fowad ate dynamics we obtain: whee we have defined Z t a(t)a(s) X(t) ; ds (s) a (s) 0 Applying Itô s lemma we obtain the dynamics of X(t), Y (t): dx(t) = (t) =f t (t) =f t (0) + Y (t) +X(t) (76) ;(t) A(t) a(t) + a0 (t) a(t) X(t) Z t a(t) + dz Q p(s) (s) 0 a(s) dt + dy (t) = (t) A(t) a 0 (t) a(t) + Y (t) a(t) + a(t) A(t) Using equation (76) we obtain the dynamics of the shot-tem ate: t (0) ; a(t) d(t) = + Y (t) + (t) ; f t a 0 (t) A(t) a(t) : (77) p (t) dz Q (t) (78) # dt : (79) p dt + (t) dz Q (t) : (80) Recalling the definition of (t), it is clea that f(t) Y(t) (t)g fom a Makov system. Moe geneally, the fowad ates may be witten as: a(v) a(v)a(v) f v (t) =f v (0) + X(t) + Y (t) : (8) a(t) a(t)a(t) Thus bond pices satisfy: Z T P T (t) = ep ; dv f v (t) (8) = ep = ep ; ; t Z T t Z T t dv f v (0) ; M (t T ) X(t) ; N (t T ) Y (t) dv f v (0) ; M (t T ) ; (t) ; f t (0) ; Y (t) ; N (t T ) Y (t) (83) (84)

22 whee we have defined M (t T ) A(T ) ; A(t) a(t) A (T ) ; A (t) N (t T ) a(t)a(t) (85) : (86) Finally, conside a path-independent Euopean contingent claim that has a payoff at time T that is a function of the entie tem stuctue at time T, i.e., (T ) T fp v (T )g T vt. The pice of that secuity is (t) =E hep(; R i Q T (s)ds)(t )jf t t = F (t (t) (t) Y(t)) whee the second equality follows fom the Makov-popety. Moeove, a standad agument (which equies some egulaity conditions on F and its deivatives, see Duffie Appendi E, p. 96) shows that R ; t ep ; ds 0 (s) F t (t) (t) Y(t) is a Matingale and that its dift must vanish, o equivalently ; ; E Q t df t t dt t Y t = t F t t t Y t : Using Itô s lemma we obtain the patial diffeential equation fo the pice of the Euopean contingentclaim: 0 = F t +(t) F + F b + F b + F Y (t) A(t) a 0 (t) a(t) + Y (t) a(t) + a(t) A(t) t (0) ; # a(t) +F + Y (t) + (t) ; f t a 0 (t) Q (0) + A(t) a(t) (t) ; F: (87) Although they do not note the elevance to incomplete bond makets, a simila model appeas in de Jong and Santa Claa (999), whee they investigate the special case a(t) =e ;t. This model clealy ehibits unspanned stochastic volatility. In paticula, note that bond pices (equation (84)) ae eponential-affine functions of and Y alone, and hence cannot hedge changes in. Since Y is locally deteministic, the innovations of any bond can be hedged by a position in any othe bond and the money maket fund. Howeve, the set fy g ae not jointly Makov. As a consequence, the dynamics of bond pices ove a finite time peiod depend on the dynamics of the additional state vaiable as well. In geneal, it is not possible to guaantee that the above stochastic diffeential equations fo and ae well-defined. Indeed, fo geneal initial tem stuctues and paamete choices, (t) may take on negative values. 33 The following lemma demonstates that thee eists a feasible set of paametes such that emains stictly positive (almost suely) and the SDE s ae well-defined. Fo simplicity, we conside the special case a(t) =e ;t.

23 PROPOSITION 0: If the paametes and the initial fowad ate cuve satisfy:. 0,! 0.! Q ;! = Q! 3. (0) > 0, and 4.! f t (0) +! f t (0) +! Q 0 ;! 0 Q then t ;! +! b +b!! 8t 0 > 0 8t 0 a.s. and the SDE s fo the fowad ates f v (t) 8v and the stochastic volatility (t) ae well-defined. PROOF: Note that unde condition of the poposition ; d(t) =! f t (0) +! f t (0) +! Z(t) +! Q ;! Q + Q (t) p 0 0 dt + (t) dw (t) (88) dz(t) = ((t) ; Z(t)) dt (89) p p whee! +! b +b!! and dw t = (! +! b) dz (t) + ;! bdz (t) is a standad Bownian motion and Z(0) = 0. A mino adaptation of the poof of the SDE Theoem in DK (996) (which etends Felle (95) to a vecto of affine pocesses) to account fo deteministic coefficients in the dift of, allows us to conclude that the SDE fo fowad ates and stochastic volatility state vaiables ae well-defined. Note that the above poposition puts joint estictions on both the feasible set of paametes and the initial cuve of fowad ates. Also note that fo this special choice of volatility stuctue the model admits a Makov epesentation of the tem stuctue such that it has an affine stuctue in the sense of DK (996) o DPS (000), but with thee distinct featues: (i) it is consistent with the initial tem stuctue, (ii) it is not time homogeneous, and (iii) it esults in only a subset of the state vaiables enteing the bond pices eponentially (i.e., the loading of the log-bond pice is zeo fo the state vaiable ). Ou appoach povides a staightfowad and efficient method to constuct HJM affine models with unspanned stochastic volatility. 34 I. Conclusion Most time homogeneous models of the tem stuctue ae estictive in that they assume all souces of isk inheent in the pices of deivative secuities can be completely hedged by a potfolio consisting solely of bonds. Ou empiical evidence suggests that this assumption is countefactual. Indeed, using data fom thee diffeent counties (U.S., U.K. and Japan), we find that changes in the tem stuctue of swap ates have vey limited eplanatoy powe fo etuns on at-the-money staddles. We tem this featue unspanned stochastic volatility (US). Futhemoe, innovations in at-the-money staddle 3

24 etuns ae highly coelated. Pincipal component analysis suggests that a single common facto independent of etuns on swap ates eplains most of the vaiation in staddles. We study conditions unde which the affine models can be made consistent with this empiical obsevation. We find that bivaiate Makov models, special cases of which include Fong and asicek (99) and Longstaff and Schwatz (99), cannot ehibit US. In othe wods, two-facto Makov shot ate models necessaily lead to complete bond makets povided sufficient diffeent matuity bonds ae taded. Futhe, we identify necessay and sufficient paamete estictions fo tivaiate Makov affine systems to display US. While such estictions may appea somewhat contived, we ague this occus because the standad affine famewok takes as pimitives the specification of a lowdimensional Makov vecto of state vaiables which ae not taded assets. In contast, we show that US occus natually when fowad ate dynamics (o equivalently, bond pice dynamics) ae taken as pimitives of the model. Simulated economies of the poposed models suggest that US can be geneated even if data is sampled only monthly. Futhe, ou esults suggest that when estimating isk-neutal paametes of a model, it is essential to use as inputs both swaps and fied income deivative secuities. Indeed, it appeas that thee ae some paametes whose estimates have minimal impact on fitting the moments of swap ate data, yet have significant picing implications fo fied-income deivatives such as staddles. Moeove, some isk-neutal paametes ae not even in theoy identifiable given only bond pices but athe equie that fied-income deivatives be obseved. The empiical evidence we povide suppoting the concept of US aises othe impotant testable empiical issues. Fist, how much additional eplanatoy powe do latent US state vaiables possess fo eplaining the time-seies and coss-sectional behavio of bond pices? We note that all pevious empiical studies within a time homogeneous setting implicitly assume that all factos diving the tem stuctue can be inveted fom bond pices alone. Second, an implication of US is that pices of both bonds and fied income deivatives ae needed to detemine paamete values. Hence, US may offe a potential avenue to impove ecent attempts at captuing the joint dynamics of the tem stuctue and fied income deivatives. Finally, US challenges standad appoaches to hedging fied income deivatives as it equies the use of at least one efeence deivative to hedge othe fied income deivatives. 4

25 Appendi: Poofs A. Poof of Poposition By definition, all bivaiate affine models can be epesented in tems of the spot ate and some othe state vaiable, whee: 4 d 3 8 < 5 N 4 : ; dt ; dt P T (s s s )=E Q s ; dt (c 0 + c + c ) dt ; d (c 0 + c + c ) dt : dt (A) That is, the difts and vaiances of the two state vaiables, along with the covaiance between the two state vaiables, ae linea in these two state vaiables. At date-s, the pice of a discount bond matuing at date T is defined though i he ; R T s du u 39 = 5 : (A) It is well known that if the system is well-defined, then the bond pice satisfies the PDE 0 = ;P + P s + P ; P ; + P ; P and that the solution takes the fom: ; P (c 0 + c + c ) (A3) P (s s s )=e A(T ;s)+b (T ;s) s +B (T ;s) s (A4) with bounday conditions A(0) = 0 B (0) = 0 B (0) = 0 : (A5) This implies that the time-dependent coefficients A(), B (), and B () satisfy A 0 = 0 B + 0 B + 0 B + 0 B + c 0B B (A6) B 0 B 0 = B + B + B + B + c B B ; (A7) = B + B + B + B + c B B : (A8) Fo this system to geneate incomplete bond makets, we have shown in Poposition : above that thee must eist a set of coefficients f g such that B ( )+ B ( )=0, with at least one of the coefficients non-zeo. Howeve, we see fom equations (A7), (A8) and the bounday conditions that B 0 (0) = ;, B0 (0) = 0. This implies that (i) B ( ) cannot be identically zeo and that (i) B ( ) cannot be a multiple of B ( ). Theefoe, the only possibility fo this model to ehibit US is fo = 0, 6= 0, and B ( ) = 0 8. Howeve, this case is also not possible. Indeed, fom equation (A8), the condition B ( )=0implies that 0= B (T ; t) + B (T ; t) 8 T: (A9) 5

26 Since the dynamics of B is uniquely specified by equation (A7) (with B () set to zeo) and B ( ) is not identically zeo, the only way equation (A9) can be satisfied is if = 0, = 0. In such a case, howeve, the spot ate pocess itself is one-facto Makov, implying that all fied-income secuities can be epessed as functions of the spot ate only. It may appea at fist that equation A9 can be satisfied by pemitting and to take on a paticula time dependence. Howeve, this time dependence would only allow a bond with a single matuity T to have its bond pice be independent of : All othe matuities U would have B (U ; t) 6= 0. B. Poof of Poposition 3 Conside the following famewok: d = ( t) dt + ( t) dz (B) d = ( t) dt + ( t) dz + ( t) dz (B) whee z, z ae two independent Bownian motions unde the isk-neutal measue, and ae functions satisfying standad egulaity conditions fo the SDEs to fom a well-defined two-facto Makov system (see, fo eample, the Appendi of Duffie (996)). In this case, the diffusion mati of the state vecto, S is a.e. invetible. Note that by the Makov popety the pice h of a zeo coupon i bond depends only upon the cuent value of the state vaiables: P T (t) = E Q e ; R T t ds s jft = P T (t t t ). Theefoe, fo bond makets to be incomplete, thee must eist coefficients f g, both not zeo, such that: P T (t )+ P T (t )=0 8 T t : (B3) Hee, we have applied Itô s lemma and have used the invetibility of the mati of the state vecto S t. This is analogous to equation (4) fo the affine case. Since fo small matuities () wehavep t+ (t ) e ; we deduce that (i) thee eists T t such that P T (t ) 6= 0, and (ii) P T (T )=0. Result (i) implies that to obtain incomplete makets (i.e., fo equation (B3) to hold), cannot be zeo. Togethe with esult (ii) this implies =0. These esults can be poved moe igoously. Wite bond pice as a Taylo seies epansion in time-to-matuity: P T (t t t )= X j=0 g j (t t t )(T ; t) j : The final condition and the definition of the bond pice P T (t t t )=E Q t that g 0 (t t t )=, g (t t t )=;. This in tun guaantees that (B4) i he ; R T t ds s guaantee P T (t t t ) lim T!t P T (t =0: (B5) t t ) Using this in equation B3 leads to the esult. Thus incomplete bond makets obtain only if P T (t ) = P T (t ) 8t T. This implies that, if bond makets ae incomplete, the fundamental PDE solved by bond pices is: P T (t ) (t )+P T (t ) (t )=PT (t ) ; P T (t ) : (B6) t 6

27 We now demonstate that this implies that must be independent of and thus that the shot ate must be one-facto Makov, in tun implying that bivaiate models cannot geneate incomplete bond makets. Note that fo any two matuities ft T g we can wite equation (B6) in mati-fom as " # P T (t ) P T (t ) ( t) P T (t ) ; P T P T (t ) P T t (t ) = (t ) ( t) P T (t ) ; P T (B7) t (t ) Fist, suppose thee eists T T (possibly dependent on (t )), such that the mati P T (t ) P T (t ) P T (t ) P is invetible. By pe-multiplying equation (B7) by this inveted mati, T (t ) we see that the ight-hand side is a function only of, implying that both (t ) = (t ) and (t ) = (t ), which implies that the spot ate is one-facto Makov, in tun implying that bivaiate models cannot geneate incomplete bond makets. Second, suppose that it is not possible to find two matuities such that the above mati is invetible. Then fo all T T the mati is not full ank and its deteminant must be zeo: (t ) P T (t ) ; P T (t ) P T (t ) =0: P T (B8) Howeve, we know that fo sufficiently small, P ( ) ) e ;, demonstating that equation B8 cannot hold in geneal. (Again, this can be made moe igoous by pefoming a Taylo seies epansion). Hence, ou claim follows. C. Poof of Poposition 4 By definition, evey affine model of thee state vaiables can be witten in tems of the spot ate and two othe (non-degeneate) state vaiables and y such that d, d, and dy have a dift vecto and instantaneous covaiance mati of the fom: = dt E[d] dt E[d] dt E[dy] y y c 0 c y 0 + c + c y = c m 0 + m + m + m y y m 0 + m + m + m y y m y 0 + my + my + my y y + c + c + cy y cy + c + cy y y y cy + cy + cyy y cy 0 + c y (C) + c y + c y + cy + cy + cyy y + cyy y + cy + cyy y y + 0 y + y + yy y (C) Note that if each of the fou coefficients m, m, y, is zeo, then the spot ate pocess is onefacto Makov. Hence, at least one of these coefficients must be non-zeo fo the system to display y US. We fist conside the case whee eithe o is non-zeo, implying that we can take the y vaiance of d to be a second state vaiable. We call this case. Late, we conside the case whee eithe m o m is non-zeo, implying that we can take the dift of d to be a second state vaiable. y :

28 We call this case. Case It follows fom the definition of a tivaiate affine US model that we can descibe the system as dt E[d] m + 0 m + m + m = 6 4 c 0 c 0 + c + c 6 4 dt E[d] dt E[d ] 7 5 = 6 4 m 0 + m + m + m m 0 c 0 + m + m + m + c + c + c c 0 + c + c c 0 + c + c c 0 + c 7 5 (C3) + c + c + c + c + c + c + c + c (C4) Now, fo ou poposition to be incoect, it must be that the system can display US and m =0, fo only then would we not be able to choose E[d] as a thid state vaiable to eplace. Futhe, fo the system to display US, at least one of the paametes fm c g must be non-zeo, o else the system educes to a bivaiate Makov tem-stuctue model in f g. To show that imposing the condition m =0would peclude US, we conside two cases. The fist case eamines whethe the bond pice can take the fom: : Case a: P (T ; t t t )=e A(T ;t)+b (T ;t) t +B (T ;t) t : (C5) The second case eamines whethe the bond pice can take the fom: Case b: P (T ; t t t )=e A(T ;t)+b (T ;t) t +B 3 (T ;t) t : (C6) Given that is an abitay state vaiable, we claim that these two scenaios incopoate all possibilities fo equation (4) to hold. The poof is as follows: All affine models have bond pices that can be witten in the fom: P (T ; t t t t )=e A(T ;t)+b (T ;t) t +B (T ;t) t +B 3 (T ;t) t : (C7) Now, it can be shown that B ( ) cannot vanish. Futhe, it can be shown that B 0 (0) = ;, B 0 (0) = 0, B0 (0) = 0. The intuition fo this esult is that, fo vey shot times to matuity, 3 the bond pice must go like e ;. Scenaio a investigates the possibility B 3 ( ) = 0 8, fo then, we set ( = 0 = 0) to satisfy equation (4). Analogously, scenaio b investigates the possibility B ( ) = 0 8, fo then, we set ( = 0 3 = 0). Both scenaios include as a special case B ( )=08 and B 3 ( ) =08. Finally, we claim that the case whee none of the B i ( ) vanish is also investigated by a, because in such a case equation (4) can only hold if B ( )=B 3 ( ), fo some constant. But this in tun implies that, by change of vaiables fom the abitay vaiable to anothe abitay 0 +, we ae back to investigating scenaio a. 8

29 Case a Assuming that bonds ae only functions of f g as in equation (C5), and imposing the condition m =0, the bond pice P (T ; t t t ) satisfies the pde 0 = ;P + P t + P ; m 0 + m + m + P ; m 0 + m + m + m + P + P ; ; P c 0 + c + c + c : (C8) Futhemoe, the affine stuctue of equation (C5) pemits us to identify the ODE s satisfied by B ( ) and B ( ) by collecting tems linea in,, and. We find: B 0 ( ) = B m + B m + B + B B c ; (C9) B 0 ( ) = B m + B + B B c (C0) 0 = B m + B m + B + B B c + B (C) with the bounday conditions B (0) = 0, B (0) = 0. Fo ou poposition to be incoect, thee must eist a solution to equations (C9)-(C), and at least one of the paametes fm cvg being non-zeo. The poposed method of solution is to note that equations (C9) to (C) ae tue fo all times to matuity, and hence can be diffeentiated an abitaily lage numbe of times. Each diffeentiation potentially adds anothe estiction via equation (C). Effectively, we ae pefoming a Taylo-seies epansion in time-to-matuity. Fo eample, the lowest ode imposes the estictions: B 0 (0) = ; (C) B 0 (0) = 0 (C3) 0 = 0 : (C4) Diffeentiation of equations (C9) to (C) geneates the following system of equations: B 00 ( ) = B0 m + B0 m + B B 0 +(B0 B + B B 0 ) c B 00 ( ) = B0 m + B B0 +(B0 B + B B0 ) c (C5) (C6) 0 = B 0 m + B0 m + B B0 +(B0 B + B B0 ) c + B B0 : (C7) These equations allow us to identify B 00 (0) = ;m (C8) B 00 (0) = 0 (C9) 0 = ;m : (C0) Hence, fo this system to display US, a necessay condition is that the paamete m zeo. Continuing in this manne, we find fo the net ode of diffeentiation: must be set to B 000 (0) = ;; m (C) B 000 (0) = 0 (C) 0 = : (C3) 9

30 Clealy, equation C3 demonstates this model cannot display US. Case b Assuming that bonds ae only functions of f g as in equation (C6), and imposing the condition m =0, the bond pice P (T ; t t t ) satisfies the pde 0 = ;P + P t + P ; m 0 + m + m + P ; m 0 + m + m + m + P + P ; ; P c 0 + c + c + c : (C4) Futhemoe, the affine stuctue of equation (C5) pemits us to identify the ODE s satisfied by B ( ) and B ( ) by collecting tems linea in,, and. We find: B 0 ( ) = B m + B m + B + B B c ; (C5) 0 = B m + B + B B c B 0 ( ) = B m + B m + B + B B c (C6) + B (C7) with the bounday conditions B (0) = 0, B (0) = 0. Fo ou poposition to be incoect, thee must eist a solution to equations (C5)-(C7), and at least one of the paametes fm c g being non-zeo. Following the same appoach as used above, we diffeentiate the set of equations (C5) to (C7) to see if the model can display US. The lowest ode imposes the estictions: B 0 (0) = ; (C8) 0 = 0 (C9) B 0 (0) = 0 : (C30) Howeve, successive diffeentiation of equations (C5) to (C7) foces all thee coefficients fm c g to be zeo, the poof of which is available upon equest. Recall that this implies that the fied-income maket is then bivaiate-makov, which has been shown peviously unable to ehibit US. Hence, this scenaio also cannot display US. Thus, bond pices of the fom in equation (C5) cannot ehibit US. Case It follows by definition of a US model that we can descibe the system as 6 4 dt E[d] dt E[d] dt E[d] = 6 4 m 0 + m + m + m m 0 + m + m + m (C3) 30

31 c + c + 0 c + c c + c + c + 0 c = c + c + 0 c + c c + c + c + c 0 5 : c + c + c + 0 c c + c + c + c (C3) Now, fo ou poposition to be incoect, it must be that the system can display US and =0, fo only then would we not be able to choose a[d] as a thid state vaiable to eplace. Futhe, fo the system to display US, at least one of the paametes fm c g must be non-zeo, o else the system educes to a bivaiate Makov tem-stuctue model in f g. Following the stategy taken above, to show that imposing the condition =0would peclude the model fom geneating incomplete bond makets, we conside two cases. The fist case eamines whethe the bond pice can take the fom: Case a: P (T ; t t t )=e A(T ;t)+b (T ;t) t +B (T ;t) t : (C33) The second case eamines whethe the bond pice can take the fom: Case b: P (T ; t t t )=e A(T ;t)+b (T ;t) t +B (T ;t) t : (C34) Following the same agument as in Case, given that is an abitay state-vaiable, it follows that these two scenaios incopoate all possibilities fo equation (4) to hold. 3 7 Case a Assuming bond pices ae functions of only f g as in equation (C33), and imposing the condition =0, bond pices P (T ; t t t ) satisfy the PDE 0 = ;P + P t + P () +P m + 0 m + m + m + P P + c + c + c : (C35) P c 0 Futhe assuming that bond pices have an affine stuctue as in equation (C5), and then collecting tems linea in,, and, we find that B 0 ( ) = B m + B + B B c + B ; (C36) B 0 ( ) = B m + B + B B c (C37) 0 = B m + B + B B c + B + B (C38) with the bounday conditions B (0) = 0, B (0) = 0. Fo ou poposition to be incoect, thee must eist solutions to equations (C36) to (C38), with at least one of the paametes fm cg not equal to zeo. Following the same appoach as used above, we note that equations (C36) to (C38) ae tue fo all times to matuity, and hence can be diffeentiated an abitaily lage numbe of times. Each 3

32 diffeentiation potentially adds anothe estiction via equation (C38). Fo eample, the lowest ode imposes the estictions: B 0 (0) = ; (C39) B 0 (0) = 0 (C40) 0 = 0 : (C4) Diffeentiation of equations (C36) to (C38) geneates the following system of equations: B 00 m + B B0 +(B0 B + B B0 ) c + B B 0 (C4) B 00 m + B B0 +(B0 B + B B0 ) c (C43) These equations allow us to identify 0 = B 0 m + B B0 +(B0 B + B B0 ) c + B B0 + B0 : (C44) B 00 (0) = 0 (C45) B 00 (0) = 0 (C46) 0 = ; : (C47) Note that equation (C47) is inconsistent with equation (C39). Hence, this model cannot display US. Case b Assuming bond pices ae functions of only f g as in equation (C34), and imposing the condition =0, bond pices P (T ; t t t ) satisfy 0 = ;P + P t + P () +P m + 0 m + m + m + P P + c + c + c : (C48) P c 0 Assuming the bond pice is of the fom in equation C34, by collecting tems linea in,, and, we find that B 0 ( ) = B m + B + B B c + B ; (C49) 0 = B m + B + B B c (C50) B 0 ( ) = B m + B + B B c + B + B (C5) with the bounday conditions B (0) = 0, B (0) = 0. Below, we pove that these equations ae inconsistent, thus poving that this system cannot geneate incomplete bond makets. By setting =0, equations (C49) to (C5) imply: B 0 (0) = ; (C5) 0 = 0 (C53) B 0 (0) = 0 : (C54) 3

33 Howeve, successive diffeentiation of equations (C49) to (C5) foces all thee coefficients fm cg to be zeo, the poof of which is available upon equest. Recall that this implies that the fied-income maket is then bivaiate-makov, which has been shown peviously unable to ehibit US. Hence, this scenaio also cannot display US. Since both scenaios a and b cannot ehibit US, ou claim is poved. D. Poof of Poposition 5 Assuming that bond pices ae functions only of time-to-matuity, the spot ate, and some geneic state vaiable as in equation (0), and that the thid state vaiable of the model is t dt EQ [d t ], then bond pices satisfy the patial diffeential equation: 0 = ;P + P t + P () +P m + 0 m + m + m + P P + c + c + c : (D) P c 0 Futhe assuming that the bond pice is of the fom in equation 0, by collecting tems linea in,, and, we find that B 0 ( ) = B m + B + B B c + B ; (D) B 0 ( ) = B m + B + B B c + B (D3) 0 = B m + B + B B c + B + B (D4) with the bounday conditions B (0) = 0, B (0) = 0. This implies: B 0 (0) = ; B0 (0) = 0 : (D5) Diffeentiating equation (D4), and using equation (D5) to evaluate this equation at =0, we find the contadiction 0=; : (D6) Hence, models such as that poposed in equation (0), whee the affine model is tivaiate Makov in f g cannot eist. E. Poof of Poposition 6 Necessity Using the initial conditions B (0) = 0 B (0) = 0, equations (6), (7), and (9) imply: B 0 (0) = ; (E) B 0 (0) = 0 (E) 0 = 0 : (E3) 33

34 Diffeentiating equations (6), (7), and (9), and using the conditions in equations (E) to (E3), we find: One moe diffeentiation poduces: B 00 (0) = 0 (E4) B 00 (0) = ; (E5) 0 = 0 : (E6) B 000 = ;m (E7) B 000 = ;m (E8) 0 = ;m + (E9) implying that a necessay condition fo this model to ehibit US is: m =, which is one of the conditions fo both sets of paamete estictions given in equation (30). Continuing in this manne, we eventually un into a banch, whee we can choose one of two diffeent conditions to satisfy 0 = RHS. Afte epeating the Taylo seies epansion fo seveal moe steps along each of these banches, the thid equation eventually poduces 0 = 0 fo the net 5 iteations aftewads, stongly suggesting that these necessay estictions ae in fact also sufficient. We now pove this is indeed the case. 34

35 Sufficiency Define, F ( )=m B ( )+ B ( )+c B ( )B ( )+ B ( ) : (E0) which is the ight-hand side of equation (9). It is sufficient to show that if eithe set of paamete estictions given in equation (30) holds then F ( ) = 0 8 if B ( ) B ( ) satisfy the system of ODE given by equations 5 7. Indeed, in that case B 3 ( ) = 0 8 and B ( ) B ( ) satisfying the system of ODE given by equations 5 7 is a solution of the initial system of ODE veified by B B B 3 (given in DK (996) fo eample). Futhe, eistence and uniqueness of the solution to such a system is poved by Duffie, Filipovic and Schachemaye (00). If eithe set of paamete estictions given in equation (30) holds, then substituting into equation (E0), taking deivatives and substituting epessions fo B 0 and B0 fom equations (5) (7) we obtain: F 0 ( )= c + B ( )(c + c c ) F ( ) : (E) The solution to this equation is: R 0 F ( )=F (0)e ds c +B (s)(c +c Given the initial condition F (0) = 0, it is clea that, F ( ) = 0 8. c ) : (E) 35

36 Notes Buaschi and Jackweth (00) povide empiical evidence in this diection. An at-the-money staddle is a potfolio composed of an at-the-money cap and floo. As constucted, this potfolio is hedged against small changes in the inteest ate level. Hence, this potfolio is mainly eposed to volatility isk. 3 A ecent pape by Heidai and Wu (00) confims ou findings. The authos pefom an analysis of the factos diving swaption implied volatilities and document the eistence of volatility specific factos. 4 Futues on yields have chaacteistics simila to those of the log contacts poposed by Neubege (994) to hedge foeign echange volatility isk. Howeve, futues on yields have the additional featue that pemits inteest ate level isk to also be hedged. 5 See, fo eample, Kennedy (994, 997), Goldstein (000), and Santa-Claa and Sonette (00). 6 The patial diffeential equation he deives is only valid fo secuities with payoffs that ae homogeneous in zeo-coupon bond pices. 7 See, fo eample, Litteman, Scheinkman and Weiss (99), Litteman and Scheinkman (99), Knez, Litteman and Scheinkman (994), Dybvig (997), Duffie and Singleton (997), and DS (000). 8 Fo the U.S. we have one, two, thee, fou, five, seven, and ten fo the whole sample, and si, eight, and nine yeas stating Febuay 997. Fo Japan the cap data has a clea epoting flaw in the yea 000, so fo Japan we only used data fom Febuay 995 to Decembe We also used an etapolation of the available swap ates, with negligible impact on the esults. 0 This is a standad tetbook assumption, but see Duffie and Singleton (997) and Collin-Dufesne and Solnik (00) fo a discussion of this assumption. Using an agument simila to that of Duffie and Singleton (999), this assumption essentially allows us to use the same instantaneous default and liquidity isk-adjusted ate to discount cashflows unde the isk-neutal measue. Indeed the diffeence between a cap and a floo is by definition equal to a fowad swap contact with fist payment date equal to the matuity of the shotest caplet/floolet. 3 We note that, due to inteest ate fluctuations, cap and floo pices ae no longe equal one month afte inception. That is, the fowad swap contact does not in geneal have zeo value one month afte inception. 4 Typically a staddle is long an ATM call and ATM put and thus delta neutal and thus sensitive to volatility changes to a fist ode. We note that caps and floos ae actually potfolios of caplets and floolets each with the same stike. Thus each caplet and floolet is actually not ATM, and the staddle is not stictly speaking delta-neutal. Howeve, to a fist-ode, staddles of caps and floos ae still volatility sensitive. 5 Matuities 0.5 to fo U.K. and Japan, matuities 0.5, one, two, thee, fou, seven, ten fo the 36

37 U.S. 6 One potential citicism with ou appoach is that, in geneal, tem stuctue models pedict that the facto loadings (i.e., the s) ae inteest-level dependent, athe than constant, as we assume. To test this concen, we pefom a second egession on the U.S. data whee, in addition to the changes in swap levels, we include tems that ae poducts of the level and the change in swap level. If the model that we ae testing is seiously mis-specified due to the inteest ate level dependence of the s, then this second egession should pick this up. Instead, the adjusted R of this second egession is almost identical to that obtained fo the oiginal egession. This finding is not too supising, since in the time peiod we investigate inteest ate levels emained in a faily naow ange of (5 to 8 pecent). We also included othe non-linea tems (squaed, cubed, and coss-multiplied changes in swap ates) as independent vaiables. Simila esults ae obtained, and thus not epoted. 7 We also pefom the same analysis on the coelation mati of the esiduals. Simila esults ae obtained, and thus ae not epoted. 8 The eason fo the much lowe eplanatoy powe of ou egessions fo U.K. data is that ou sample peiod incopoates the devaluation of the pound in 998. At that time inteest ates plummeted and implied volatilities on cap and floos spiked. The esults ae simila if we look at the peiod pio to the devaluation: It seems that changes in swap ates and staddle pices espond diffeently to factos diving the devaluation. The peiod afte the devaluation appeas to be much moe in line with the U.S. and Japan esults. 9 Futhe, as discussed in note 4, ou staddles ae not stictly speaking delta-neutal which may eplain some of these facto loadings. 0 Futhe, in a ecent pape Heidai and Wu (00) pefom a simila study on swaption volatilities. They find a slightly highe eplanatoy powe of tem stuctue factos fo swaption volatilities. This may patly be due to thei unning egessions on levels and not changes. Howeve, they also find significant evidence of US factos in the swaption maket. DK only conside the case whee the mati is N N and non-degeneate. Without loss of geneality, it is assumed hee that the the fist d state vaiables of the oiginal set (and thus also the otated set ) ae chosen so that thei volatility mati is full ank, and the i i = d +:::N vectos (with element i j ) used fo the otation fom a basis of the (N ; d)- dimensional kenel of >. Below, we will simply efe to this otated system as X. 3 We thank Jespe Andeasen fo suggesting this. 4 One may wonde why the tem popotional to is not consideed in satisfying equation (4). The answe is that when is one of the state vaiables, its coefficient B (T ; s) is the only one whose st - ode Taylo seies epansion coefficient is non-zeo. The intuition again is that, fo small matuities, the bond pice goes like P = e ;, that is, eponentially linea in. 5 DS focus on the A (3) and A (3) models, both of which have thee factos and thee state vaiables. These models diffe by the numbe of state vaiables appeaing unde the squae oot : one fo the A (3) model and two fo the A (3) model. Below, we eamine in depth the A (3) model because DS find that it is somewhat supeio at fitting the dynamics of swap ates. 6 In the paticula case whee v = 0 and the US estictions 34 to 37 hold, one can show (by substituting the solution into the fundamental PDE) that the bond pices have the fom given in 37

38 equations 38 to 4 with B v ( ) = 0 8 even if v follows an abitay one facto Makov pocess dv t = v (v t t)dt + v (v t t) dz Q v. 7 Only thee paametes ae sepaately identifiable fom B, B since equation (35) holds, and five paametes ae identifiable fom A since equation (4) holds. 8 We chose paametes based on DS Table II, p To compute the cap and floo pices we use the closed fom solution appoach poposed by Heston (993) and etended by DPS. 9 Similaly, if one wee to constuct a geneal equilibium model fo (multiple) stock pices stating fom some fundamental low-dimensional state vaiable vecto (such as in Co, Ingesoll and Ross (985a) o DPS (000)), the model would necessitate some estictions on the paametes of the pocess of the state vaiables fo the stock pice pocesses to ehibit US. Indeed, in geneal, all souces of isk would be spanned by the stock pices (as long as the numbe of state vaiables is smalle than the numbe of stock pices). 30 As noted in Andeasen, Collin-Dufesne, and Shi (997), the HJM estiction alone does not identify the pocess of unde the isk-neutal measue, since the Gisanov facto associated with z cannot be identified fom changes in bond pices alone. To detemine the maket pice of isk associated with volatility-specific isk z, eithe the pices of othe inteest ate sensitive secuities in addition to bond pices must be taken as input to the model, o some equilibium agument must be made. 3 When! = 0, Jeffey (995) demonstates that fo the shot ate to be one-facto Makov the functions a(s T ) must satisfy a vey specific functional fom (his equation (8), p. 63). 3 Fo simila sepaability assumptions made to obtain Makov epesentation in standad HJM models, see fo eample: Cavehill (994), Cheyette (995), and Ritchken and Sankaasubamaniam (995). 33 Moeove, the squae oot diffusion coefficients does not veify the standad Lipschitz conditions at zeo, but see Duffie (996) Appendi E p. 9 and DK (996). 34 We note that by appopiately choosing the initial fowad cuve, this model educes to a special case of the time-homogeneous affine models pesented in the pevious section. Indeed, it is a twofacto, thee-state vaiable affine model with US. The additional locally deteministic state vaiable is the lowest cost to pay in ode to obtain a model which ehibits US. 35 Hee, the state vaiable is not necessaily the same state vaiable in the fist system, but athe just some abitay state vaiable that is linealy independent of both and. 38

39 Refeences Ahn, Dong-Hyun, Robet Dittma, and Ronald Gallant, 00, Quadatic tem stuctue models: Theoy and evidence, The Review of Financial Studies 5 no, Andesen, Leif, and Jespe Andeasen, 00, Facto dependence of bemudan swaption pices: Fact o fiction?, Fothcoming The Jounal of Financial Economics. Andesen, Toben, and Jespe Lund, 997, Estimating continuous-time stochastic volatility models of the shot-tem inteest ate, Jounal of Econometics v77 n, Andeasen, Jespe, Piee Collin-Dufesne, and Wei Shi, 997, An abitage model of the tem stuctue of inteest ates with stochastic volatility, Woking pape. Poceedings of the Fench Finance Association AFFI 97 in Genoble. Balduzzi, Pieluigi, Sanjiv Das, and Silveio Foesi, 996, A simple appoach to thee facto affine tem stuctue models, Jounal of Fied Income 6, Black, Fishe, 976, The picing of commodity contacts, Jounal of Financial Economics 3, Boudoukh, Jacob, Matthew Richadson, Richad Stanton, and Robet Whitelaw, 997, Picing motgage-backed secuities in a multifacto inteest ate envionment: A multivaiate density estimation appoach, The Review of Financial Studies v0 n Summe, Benne, Robin J., Richad H. Hajes, and Kenneth F. Kone, 996, Anothe look at altenative models of the shot-tem inteest ate, Jounal of Financial and Quantitative Analysis, Buaschi, Andea, and Jens Jackweth, 00, The pice of a smile: Hedging and spanning in option makets, The Review of Financial Studies 4(), Cavehill, Andew, 994, When is the shot ate makovian, Mathematical Finance 4 (Oct), Chan, K.C., G. Andew Kaolyi, Fancis A. Longstaff, and Anthony B. Sandes, 99, An empiical compaison of altenative models of the shot-tem inteest ate, Jounal of Finance 47, Chen, Lin, 996, Stochastic mean ans stochastic volatility a thee facto model of the tem stuctue of inteest ates and its application to picing of inteest ate deivatives, Blackwell Publishes, Ofod, U.K. Chen, Ren R., and Louis Scott, 993, Maimum likelihood estimation fo a multifacto equilibium model of the tem stuctue of inteest ates, Jounal of Fied Income decembe,vol.3 no 3, 4 3. Cheyette, Oen, 995, Makov epesentation of the heath-jaow-moton model, BARRA Inc. woking pape. 39

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43 Table I The R and adjusted R of the egession of staddle etuns with matuities fone, two, thee, fou, five, seven, teng yeas on the changes in swap ates fo all available matuities (f0.5, one, two, thee, fou, five, seven, teng fo U.S. data and f0.5,one, two, thee, fou, five, si, seven, eight, nine teng fo U.K. and Japan data). Although multi-collineaity is evident in the egessos, the R epesents an uppe bound on the popotion of the vaiance of staddle etuns that can be hedged by tading in swaps. U.S. Staddles U.K. Staddles Japan Staddles Matuity R Adjusted R R Adjusted R R Adjusted R Table II Eigenvalues of pincipal component decomposition of the covaiance mati of esiduals, odeed by magnitude of the eigenvalue. Note that ove 80 pecent of the vaiation is captued by the fist pincipal component fo each county. U.S. Residuals U.K. Residuals Japan Residuals Eigenvecto Eigenvalue % Eplained Eigenvalue % Eplained Eigenvalue % Eplained Table III Coefficient and adjusted R obtained egessing the etun of an equally weighted potfolio of staddles on thee potfolios of swap ates eplicating the fist thee factos of a pincipal analysis of swap ate changes. t-statistics ae in paenthesis. The same analysis is conducted fo each county. The vey low R epoted fo the U.K. data is appaently due to the devaluation of the pound in 998. Fac Fac Fac3 adj. R U.S % (-3.89) (.30) (.30) U.K % (0.) (.70) (-0.9) Japan % (-.48) (4.40) (-0.7) 43