Estimating Shadow-Rate Term Structure Models with Near-Zero Yields

Transcription

1 Journal of Financial Economerics Advance Access published April 9, 2014 Journal of Financial Economerics, 2014, Vol. 0, No. 0, Esimaing Shadow-Rae Term Srucure Models wih Near-Zero Yields JENS H. E. CHRISTENSEN and GLENN D. RUDEBUSCH Federal Reserve Bank of San Francisco ABSTRACT Sandard Gaussian affine dynamic erm srucure models do no rule ou negaive nominal ineres raes a conspicuous defec wih yields near zero in many counries. Alernaive shadow-rae models, which respec he nonlineariy a he zero lower bound, have been rarely used because of he exreme compuaional burden of heir esimaion. However, by valuing he call opion on negaive shadow yields, we provide esimaes of a hree-facor shadow-rae model of Japanese yields. We validae our opion-based resuls by closely maching hem using a simulaion-based approach. We also show ha he shadow shor rae is sensiive o model fi and specificaion. (JEL: G12, E43, E52, E58) KEYWORDS: affine dynamic erm srucure models, zero lower bound, moneary policy Nominal yields on governmen deb in several counries have fallen very near heir zero lower bound (ZLB). Noably, yields on Japanese governmen bonds of various mauriies have been near zero since Similarly, many U.S. Treasury raes edged down quie close o zero in he years following he financial crisis in lae Accordingly, undersanding how o model he erm srucure of ineres raes when some of hose ineres raes are near he ZLB commands aenion for bond porfolio pricing, risk managemen, for macroeconomic and moneary policy The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem. We hank Peer Feldhüer, Don Kim, Leo Krippner, and David Lando for helpful commens on previous drafs of his aricle. We also hank paricipans a he FDIC s 23rd Annual Derivaives Securiies & Risk Managemen Conference and he NBER Summer Insiue 2013 as well as seminar paricipans a he Copenhagen Business School, he Swiss Naional Bank, he Office of Financial Research, he Federal Reserve Board, and he Federal Reserve Bank of San Francisco for helpful commens. Address correspondence o Jens H. E. Chrisensen, Federal Reserve Bank of San Francisco, 101 Marke Sree MS 1130, San Francisco,CA 94105, or jens.chrisensen@sf.frb.org doi: /jjfinec/nbu010 The Auhor, Published by Oxford Universiy Press. All righs reserved. For Permissions, please journals.permissions@oup.com

2 2 Journal of Financial Economerics analysis. Unforunaely, he workhorse represenaion in finance for bond pricing he affine Gaussian dynamic erm srucure model ignores he ZLB and rouinely places posiive probabiliies on fuure negaive ineres raes. This counerfacual flaw sems from ignoring he exisence of currency, which is a readily available sore of value. In he real world, an invesor always has he opion of holding cash, and he zero nominal yield of cash will dominae any securiy wih a negaive yield. 1 To recognize he opion value of currency in bond pricing, Black s (1995) inroduced he noion of a shadow shor rae, which is driven by fundamenals and can be posiive or negaive. The observed shor rae equals he shadow shor rae excep ha he former is bounded below by zero. While Black s (1995) use of a shadow shor rae o accoun for he presence of currency holds much inuiive appeal, i has rarely been used. In par, his infrequency reflecs he fac ha ineres raes in many counries have long been some disance above zero, so he Gaussian models posiive probabiliies on negaive fuure ineres raes are negligible and unlikely o be an imporan deerminan in bond pricing. In recen years, wih yields around he world a hisoric lows, his raionale no longer applies. However, a second facor limiing he adopion of he shadow-rae srucure has been he difficuly in esimaing hese nonlinear models. Gorovoi and Linesky (2004) derive quasi-analyical bond price formulas for he case of one-facor Gaussian and square-roo shadow-rae models. 2 Unforunaely, heir resuls do no exend o mulidimensional models. Insead, he small se of previous research on shadow-rae models has relied on numerical mehods for pricing. 3 However, in ligh of he compuaional burden of hese mehods, previous esimaions of shadow-rae models have focused on models ha use only one or wo facors. For example, Ichiue and Ueno (2007) and Kim and Singleon (2012) underake a full maximum-likelihood esimaion of a wo-facor Gaussian shadow-rae model on Japanese bond yield daa using he exended Kalman filer and numerical opimizaion. These analyses were limied o only wo pricing facors because he numerical mehods required for shadow-rae models wih more han wo facors were compuaionally oo onerous. This pracical shorcoming is poenially quie serious given he prevalence of higher-dimensional bond pricing models in research and indusry. 4 Indeed, o overcome he pracical difficulies of empirical implemenaion, Ichiue and Ueno (2013) simplify he srucure by ignoring bond 1 Acually, he ZLB can be a somewha sof floor. The nonnegligible coss of ransacing in and holding large amouns of currency have allowed yields o push slighly below zero in a few counries, noably in Denmark recenly. To accoun for insiuional currency fricions in our analysis, we could replace he zero lower bound on yields wih some appropriae, possibly ime-varying, negaive epsilon as deailed in Secion Ueno e al. (2006) use hese formulas when calibraing a one-facor Gaussian model o a sample of Japanese governmen bond yields. 3 Kim and Singleon (2012) and Bomfim (2003) use finie-difference mehods o calculae bond prices, while Ichiue and Ueno (2007) employ ineres rae laices. 4 Indeed, Kim and Singleon (2012) sugges ha he shadow-rae model resuls of Ueno e al. (2006) are influenced by heir use of a one-facor shadow-rae model ha may no be flexible enough o fi heir sample of Japanese daa. Similarly, he Kim and Singleon (2012) wo-facor resuls may no generalize o

3 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 3 convexiy effecs, so he magniude of he resuling deviaions from arbirage-free pricing is unclear. An alernaive opion-based approach o reduce he compuaional burden associaed wih he ZLB, suggesed by Krippner (2012), appears o allow for racable esimaion of dynamic erm srucure shadow-rae models wih more han wo facors. The inuiion for he opion-based approach is ha he price of a sandard observed bond (which is consrained by he ZLB) should equal he price of a shadow-rae bond (which is no consrained by he ZLB) minus he price of a call opion peraining o he possibiliy ha he unconsrained shadow raes may go negaive. Tha is, he owner of a shadow bond would have o sell off he probabiliy mass associaed wih he shadow (zero-coupon) bond rading above par in order o mach he value of he observed bond. Unforunaely, his call opion is difficul o value, so Krippner (2012) provides only an approximae soluion o he correc one. Krippner suggess ha he approximaion error is likely small, bu lile is known in pracice abou is size and properies. In his aricle, we implemen his new opion-based approach o esimae he firs hree-facor shadow-rae model on Japanese yield daa. 5 Specifically, we use he opion-based mehod o esimae a shadow-rae version of he Gaussian arbirage-free Nelson-Siegel (AFNS) model inroduced in Chrisensen, Diebold, and Rudebusch (2011), henceforh CDR. The AFNS model class provides a flexible and robus srucure for dynamic erm srucure modeling ha has performed well on a variey of yield samples by combining good fi wih racable esimaion. Furhermore, as we show in his aricle, wih an opionbased esimaion approach, he AFNS specificaion of he pricing facor dynamics leads o analyical formulas for he insananeous shadow forward raes. These new closed-form expressions faciliae sraighforward empirical implemenaion of higher-order shadow-rae models. We demonsrae his wih an esimaion of shadow-rae AFNS models using Japanese erm srucure daa, which are of special ineres because hey include a long period of near-zero yields. In paricular, we esimae one-, wo-, and hree-facor versions of he shadow-rae AFNS model and compare hese o one-, wo-, and hree-facor versions of he sandard Gaussian AFNS model. We find ha shadow-rae models can provide beer fi as measured by in-sample merics such as he RMSEs of fied yields and he likelihood values. Sill, i is eviden from hese in-sample resuls ha a sandard hree-facor Gaussian dynamic erm srucure model like our Gaussian hree-facor AFNS model has enough flexibiliy o fi he cross-secion of yields fairly well a each poin in ime even when he shorer end of he yield curve is flaened ou a he ZLB. However, higher-order models. Finally, noe ha Bauer and Rudebusch (2013) argue ha addiional macroeconomic facors will be especially useful a he ZLB o augmen he sandard yields-only model. 5 Wu and Xia (2013) derive a discree-ime version of he Krippner framework and implemen a hreefacor specificaion using U.S. Treasury daa. In relaed research, Priebsch (2013) derives a second-order approximaion o he Black (1995) shadow-rae model and esimaes a hree-facor version hereof, bu i requires he calculaion of a double inegral in conras o he single inegral needed o fi he yield curve in he Krippner framework.

4 4 Journal of Financial Economerics i is no he case ha he Gaussian model can accoun for all aspecs of he erm srucure a he ZLB. Indeed, we show ha our esimaed hree-facor Gaussian model clearly fails along wo dimensions. Firs, despie fiing he yield curve, he model canno capure he dynamics of yields a he ZLB. One sark indicaion of his is he high probabiliy he model assigns o negaive fuure shor raes obviously a poor predicion. Second, he sandard model misses he compression of yield volailiy ha occurs a he ZLB as expeced fuure shor raes are pinned near zero, longer-erm raes flucuae less. The shadow-rae model, even wihou incorporaing sochasic volailiy, can capure his effec. We hen examine hree feaures of he shadow-rae model in deail. As noed above, he opion-based approach provides only an approximaion o a fully consisen arbirage-free dynamic erm srucure model. For our hreefacor shadow-rae AFNS model, we compare he opion-based approximaion o simulaion-based resuls and find ha hey are very close. Indeed, he opionbased approximaion errors are ypically an order of magniude smaller han he in-sample fied errors, so he poenial loss from using an opion-based approach in a realisic seing like ours appears o be minimal. Second, we assess he efficiency of he exended Kalman filer we use in esimaing shadow-rae models by comparing he resuls o hose obained wih he unscened Kalman filer. The resuls indicae ha exended versus unscened filering makes very lile difference, even for our sample of near-zero Japanese yields, which is very promising as he exended Kalman filer is less compuaionally inensive. Third, we examine he robusness o model specificaion of he shadow shor rae, which has been recommended by some o be a useful measure of he sance of moneary policy a he ZLB (e.g., Krippner 2012, 2013b; Bullard, 2012). We find ha here is noable disagreemen abou he value of he shadow shor rae across models wih differen numbers of facors. This sensiiviy o model specificaion suggess ha conclusions based on he shadow shor rae near he zero boundary are likely o be fragile. Finally, we should menion wo alernaive frameworks o modeling yields near he ZLB ha guaranee posiive ineres raes: sochasic-volailiy models wih square-roo processes and Gaussian quadraic models. Boh of hese approaches suffer from he heoreical weakness ha hey rea he ZLB as a reflecing barrier and no as an absorbing one as in he shadow-rae model. Empirically, of course, he recen prolonged periods of very low ineres raes seem more consisen wih an absorbing sae. In addiion, Dai and Singleon (2002) disparage he fi of sochasicvolailiy models, while Kim and Singleon (2012) compare quadraic and shadowrae empirical represenaions and find a sligh preference for he laer. Sill, we consider all hree modeling approaches o be worhy of furher invesigaion, bu we view he shadow-rae model o be of paricular ineres because away from he ZLB i reduces exacly o he sandard Gaussian affine model, which is by far he mos popular dynamic erm srucure model. Therefore, he enire voluminous lieraure on affine models remains compleely applicable and relevan when given a modes shadow-rae weak o handle he ZLB.

5 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 5 The res of he aricle is srucured as follows. Secion 2 inroduces he shadowrae framework and he opion-based approach. Secion 3 deails our shadow-rae AFNS model. Secion 4 describes our Japanese yield daa. Secion 5 presens our empirical findings for one-, wo-, and hree-facor shadow-rae models. Finally, Secion 6 concludes. Two appendices provide echnical deails on model esimaion and deailed model esimaion resuls. 1 SHADOW-RATE MODELS In his secion, we inroduce wo ypes of shadow-rae erm srucure models. The firs is he original approach offered byblack (1995). The second is he opion-based approach inroduced in Krippner (2012). 1.1 The Black Shadow-Rae Model The concep of a shadow ineres rae as a modeling ool o accoun for he ZLB can be aribued o Black (1995). He noed ha he observed nominal shor rae will be nonnegaive because currency is a readily available asse o invesors ha carries a nominal ineres rae of zero. Therefore, he exisence of currency ses a zero lower bound on yields. To accoun for his ZLB, Black posulaed as a modeling ool a shadow shor rae, s, ha is unconsrained by he ZLB. The usual observed insananeous riskfree rae, r, which is used for discouning cash flows when valuing securiies, is hen given by he greaer of he shadow rae or zero: r =max{0,s }. (1) Accordingly, as s falls below zero, he observed r simply remains a he zero bound. While Black (1995) described circumsances under which he zero bound on nominal yields migh be relevan, he did no provide specifics for implemenaion. Gorovoi and Linesky (2004) derive one-facor shadow-rae model bond price formulas, which Ueno e al. (2006) use o calibrae a one-facor Gaussian shadow-rae model o Japanese yield daa, bu hese formulas do no generalize o mulifacor models. Insead, previous researchers have employed numerical mehods for pricing. Bomfim (2003) use finie-difference mehods o calculae bond prices, while Ichiue and Ueno (2007) employ ineres rae laices. Kim and Singleon (2012) provide a comprehensive analysis of his ype and implemen wo-facor affine Gaussian and quadraic Gaussian shadow-rae models. Kim and Singleon (2012) derive he parial differenial equaion (PDE) ha bond prices mus saisfy under he resricion ha he risk-free rae used for discouning is he greaer of he shadow rae or zero, P τ 1 ( 2 2 r P x x ) P x KQ (θ Q x)+max{0,s(x)}p=0, P(0,x)=1. (2)

6 6 Journal of Financial Economerics They solve his PDE using a finie-difference mehod. Unforunaely, for more han wo facors, such numerical mehods render i very difficul o solve he associaed higher-dimensional PDE sysems wihin a reasonable ime. 6 This is a severe limiaion o esimaing shadow-rae models since he bond pricing lieraure has focused on models wih a leas hree facors driving bond yields. 1.2 Opion-Based Shadow-Rae Models To overcome he curse of dimensionaliy ha limis numerical-based esimaion of shadow-rae models, Krippner (2012) suggesed an alernaive opion-based approach ha could make shadow-rae models almos as easy o esimae as he corresponding non-shadow-rae model. In paricular, esimaion of opion-based shadow-rae models wih more han wo sae variables could be racable. To illusrae his new approach, consider wo bond-pricing siuaions ha differ only because one has a currency in circulaion ha has a consan nominal value and no ransacion coss, while he oher has no currency. In he world wihou currency, he price of a shadow-rae zero-coupon bond, P(,T), may rade above par, as is risk-neural expeced insananeous reurn equals he risk-free shadow shor rae, s, which may be negaive. 7 In conras, in he world wih currency, he price a ime for a zero-coupon bond ha pays $1 when i maures a ime T is given by P(,T). This price will never rise above par, so nonnegaive yields will never be observed. Consider he relaionship beween he wo bond prices a ime for he shores (say, overnigh) mauriy available, δ. In he presence of currency, invesors can eiher buy he zero-coupon bond a price P(,+δ) and receive one uni of currency he following day or jus hold he currency. As a consequence, his bond price, which would equal he shadow bond price, mus be capped a 1: P(,+δ) = min{1,p(,+δ)} = P(,+δ) max{p(,+δ) 1,0}. Tha is, he availabiliy of currency implies ha he overnigh claim has a value equal o he zero-coupon shadow bond price minus he value of a call opion on he zero-coupon shadow bond wih a srike price of 1. More generally, we can express he price of a bond in he presence of currency as he price of a shadow bond minus he call opion on values of he bond above par: P(,T)=P(,T) C A (,T,T;1), (3) 6 Richard (2013) goes beyond Kim and Singleon (2012) and presens a second-order approximaion o a hree-facor Black (1995) model using a four-dimensional laice grid wih more han 10 million nodes, and even hen any insananeous correlaion beween he sae variables has o be ignored o calculae bond prices. 7 The modeling approach wih unobserved, or shadow, componens has an analogy in he corporae credi lieraure. There, i is frequenly assumed ha he asse value process of a firm exiss bu is unobserved. Insead, prices of he firm s equiy and corporae deb, which can be inerpreed as derivaives wrien on he firm s asses (see Meron, 1974), are used o draw inferences abou he asse value process.

7 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 7 where C A (,,T;1) is he value of an American call opion a ime wih mauriy T and srike price 1 wrien on he shadow bond mauring a T. In essence, in a world wih currency, he bond invesor has had o sell off he possible gain from he bond rising above par a any ime prior o mauriy. Unforunaely, analyically valuing his American opion is complicaed by he difficuly in deermining he early exercise premium. However, Krippner (2012) argues ha here is an analyically close approximaion based on racable European opions. 8 Specifically, he argues ha he above discussion suggess ha he las incremenal forward rae of any bond will be nonnegaive due o he fuure availabiliy of currency in he immediae ime prior o is mauriy. As a consequence, he inroduces he following auxiliary bond price equaion P a (,T,T +δ)=p(,t +δ) C E (,T,T +δ;1), (4) where C E (,T,T +δ;1) is he value of a European call opion a ime wih mauriy T and srike price 1 wrien on he shadow discoun bond mauring a T +δ. I should be sressed ha P a (,T,T +δ) is no idenical o he bond price P(,T) in Equaion (3) whose yield observes he zero lower bound. The key insigh is ha he las incremenal forward rae of any bond will be nonnegaive due o he fuure availabiliy of currency in he immediae ime prior o is mauriy. By leing δ 0, his idea is aken o is coninuous limi, which idenifies he corresponding nonnegaive insananeous forward rae: [ f (,T)= lim d ] δ 0 dδ P a(,t,t +δ). (5) Now, he discoun bond prices whose yields observe he zero lower bound are approximaed by P app. (,T)=e T f (,s)ds. (6) The auxiliary bond price drops ou of he calculaions, and we are lef wih formulas for he nonnegaive forward rae, f (,T), ha are solely deermined by he properies of he shadow rae process s. Specifically, Krippner (2012) shows ha f (,T)=f (,T)+z(,T), where f (,T) is he insananeous forward rae on he shadow bond, which may go negaive, while z(,t) is given by [ d { C E (,T,T +δ;1) } ] z(,t)= lim. δ 0 dδ P(,T) In addiion, i holds ha he observed insananeous risk-free rae respecs he nonnegaiviy equaion (1) asinheblack (1995) model. 8 Krippner (2012, 2013a) describes he opion-based shadow-rae framework as a porfolio of a coninuum of European opions, unlike our descripion based on a single American opion, bu he ulimae pricing formula is he same.

8 8 Journal of Financial Economerics Finally, yield-o-mauriy is defined he usual way as y(,t) = 1 T = 1 T T T f (,s)ds = y(,t)+ 1 T f (,s)ds+ 1 T T lim δ 0 [ δ T lim δ 0 [ δ C E (,s,s+δ;1) P(,s) C E (,s,s+δ;1) ] ds P(,s) ] ds. I follows ha bond yields consrained a he ZLB can be viewed as he sum of he yield on he unconsrained shadow bond, denoed y(,t), which is modeled using sandard ools, and an add-on correcion erm derived from he price formula for he opion wrien on he shadow bond ha provides an upward push o deliver he higher nonnegaive yields acually observed. Imporanly, he resul above is general and applies o any assumpions made abou he dynamics of he shadowrae process. However, in realiy, as implemenaion requires he calculaion of he limi erm under he inegral, he opion-based shadow-rae models are limied o he Gaussian model class. I is imporan o sress ha since he observed discoun bond prices defined in Equaion (6) differ from he auxiliary bond price defined in Equaion (4) and used in he consrucion of he nonnegaive forward rae in Equaion (5), he opionbased framework should be viewed as no fully inernally consisen and simply an approximaion o an arbirage-free model. 9 Of course, away from he ZLB, wih a negligible call opion, he model will mach he sandard arbirage-free erm srucure represenaion. Some may find he lack of a heoreically airigh opion-based arbiragefree formulaion disconcering. However, his feaure should be pu in conex of he res of he shadow-rae modeling lieraure, which is invariably plagued by approximaion. Alhough many empirical shadow-rae erm srucure papers sar wih a heoreically consisen model, various simplificaions are made o faciliae empirical implemenaion. For example, Ichiue and Ueno (2013) sar wih a rigorous framework, bu in heir esimaion, hey omi Jensen s inequaliy erms o obain a soluion. Alernaively, Kim and Singleon (2012) rigorously solve a PDE using a finie-difference mehod, bu he numerical burden resrics heir resuls o a wo-facor model, which is widely considered oo parsimonious o be realisic. In implemening he opion-based approach, we keep in mind he adage: There are no rue models only useful ones. Thus, he quesion becomes how good 9 In paricular, here is no explici PDE ha bond prices mus saisfy, including boundary condiions, for he absence of arbirage as in Kim and Singleon (2012) and shown in Equaion (2). An addiional source of discrepancy beween he opion-based framework and Black s shadow-rae model is he fac ha all discouning in he former is done wih he shadow rae, while, in he laer, i is done wih he consrained shor rae in equaion (1), see Krippner (2013a) for a discussion.

9 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 9 he opion-based shadow-rae approximaion is near he ZLB. Krippner (2012) compares he opion-based resuls o analyical ones for a calibraed Gaussian onefacor model, and suggess ha he approximaion can be quie good. We go furher and examine his issue in he conex of an esimaed hree-facor model below. While analyical resuls are no available for a hree-facor model comparison, we use simulaion-based resuls as a benchmark and find ha he approximaion error is quie small. 2 THE SHADOW-RATE AFNS MODEL In his secion, we consider a Gaussian model ha leads o racable formulas for bond yields in he opion-based shadow-rae framework. To model he riskfree shadow rae, we employ he affine arbirage-free class of Nelson Siegel erm srucure models derived in CDR. This class of models is very racable o esimae and has good in-sample fi and ou-of-sample forecas accuracy. 10 Here, we exend he AFNS model o incorporae a nonnegaiviy consrain on observed yields. 2.1 The Sandard AFNS(3) Model We firs briefly describe he sandard hree-facor AFNS(3) model, which ignores he ZLB on yields. In his class of models, he risk-free rae, which we ake o be he poenially unobserved shadow rae, is given by s =X 1 +X2, while he dynamics of he sae variables (X 1,X2,X3 ) used for pricing under he Q-measure have he following srucure: 11 dx X 1 dx 2 σ dw 1,Q = 0 λ λ X 2 dx 3 d+ σ 21 σ 22 0 dx 2,Q 00 λ X 3. (7) σ 31 σ 32 σ 33 dx 3,Q The AFNS model dynamics under he Q-measure may appear resricive, bu CDR show his srucure coupled wih general risk pricing provides a very flexible 10 See, for example, he discussion and references in Diebold and Rudebusch (2013). 11 We have fixed he mean under he Q-measure a zero and assumed a lower riangular srucure for he volailiy marix, which comes a no loss of generaliy, as described by CDR. As discussed in CDR, wih a uni roo in he level facor under he pricing probabiliy measure, he model is no arbirage-free wih an unbounded horizon; herefore, as is ofen done in heoreical discussions, an arbirary maximum horizon is imposed.

10 10 Journal of Financial Economerics modeling srucure. Indeed, CDR demonsrae ha his specificaion implies zerocoupon bond yields ha have he popular Nelson and Siegel (1987) facor loading srucure, y(,t)=x 1 + ( 1 e λ(t ) λ(t ) )X 2 + ( 1 e λ(t ) λ(t ) e λ(t )) X 3 A(,T) T. In his formulaion, he hree facors, X 1, X2, and X3, are idenified by he loadings as level, slope, and curvaure, respecively. The yield funcion also conains a yieldadjusmen erm, A(,T) T, ha is ime invarian and depends only on he mauriy of he bond. CDR provide an analyical formula for his erm, which under our idenificaion scheme is enirely deermined by he volailiy marix. The corresponding insananeous forward raes are given by f (,T)= T lnp(,t)=x1 +e λ(t ) X 2 +λ(t )e λ(t ) X 3 +Af (,T), (8) where he yield-adjusmen erm in he insananeous forward rae funcion is given by A f (,T) = A(,T) T = 1 2 σ 11 2 (T )2 1 ( 1 e 2 (σ σ 22 2 λ(t ) ) λ 1 [ 1 2 (σ σ σ 33 2 ) λ 2 2 λ 2 e λ(t ) 2 (T )e λ(t ) λ + 1 λ 2 e 2λ(T ) + 2 λ (T )e 2λ(T ) +(T ) 2 e 2λ(T )] σ 11 σ 21 (T ) 1 e λ(t ) λ σ 11 σ 31 [ 1 λ (T ) 1 λ (T )e λ(t ) (T ) 2 e λ(t )] [ 1 (σ 21 σ 31 +σ 22 σ 32 ) λ 2 2 λ 2 e λ(t ) 1 λ (T )e λ(t ) + 1 λ 2 e 2λ(T ) + 1 λ (T )e 2λ(T )]. ) Bond Opion Prices To implemen he opion-based approach o he shadow-rae model, we need he analyical formula for he price of he European call opion wrien on he shadow bond described above.

11 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 11 From sandard asse pricing heory i follows ha he value of a European call opion wih mauriy T and srike price K wrien on he zero-coupon bond mauring a T +δ is given by C E (,T,T +δ;k)=e Q ] [e T s u du max{p(t,t +δ) K,0}. Unrepored calculaions show ha he value of he European call opion wihin he AFNS(3) model is given by 12 C E (,T,T +δ;k) = P(,T +δ) (d 1 ) KP(,T) (d 2 ), where ( ) is he cumulaive probabiliy funcion for he sandard normal disribuion and ln d 1 = ( P(,T+δ) P(,T)K ) v(,t,t +δ) and d 2 =d 1 v(,t,t +δ) v(,t,t +δ) wih ( 1 e v(,t,t +δ) = σ11 2 δ2 (T )+(σ21 2 +σ 22 2 λδ ) 2 1 e 2λ(T ) ) λ 2λ [ ( 1 e +(σ31 2 +σ σ 33 2 λδ ) 2 1 e 2λ(T ) ) λ 2λ +e 2λδ[ δ 2 (T +δ ) 2 e 2λ(T ) δ (T +δ )e 2λ(T ) + 2λ 2λ e 2λ(T ) ] 4λ 3 1 2λ (T )2 e 2λ(T ) 1 2λ 2 (T )e 2λ(T ) + 1 e 2λ(T ) 4λ 3 (1 e λδ )e λδ [ λ 2 δ (T +δ )e 2λ(T ) + 1 e 2λ(T ) ] 2λ + 1 e λδ [ 1 e 2λ(T ) λ 2 (T )e 2λ(T )] 2λ + 1 λ δe λδ[ (T )e 2λ(T ) 1 e 2λ(T ) ] 2λ ] + 1 λ e λδ[ (T ) 2 e 2λ(T ) + 1 λ (T )e 2λ(T ) 1 e 2λ(T ) ] 2λ 2 +2σ 11 σ 21 δ(1 e λδ ) 1 e λ(t ) λ 2 +2σ 11 σ 31 δ [ 1 λ (T )e λ(t ) 1 λ e λδ( δ (T +δ )e λ(t )) +2(1 e λδ ) 1 e λ(t ) ] λ 2 12 The calculaions leading o his resul are available from he auhors upon reques. For European opions, he pu-call pariy applies. As a consequence, he value of European pu opions wrien on P(,T +δ) can be similarly calculaed; see Chen (1992) for deails.

12 12 Journal of Financial Economerics [ ( 1 e λδ ) 2 1 e 2λ(T ) +(σ 21 σ 31 +σ 22 σ 32 ) λ λ + 1 λ 2 e 2λδ[ δ (T +δ )e 2λ(T ) + 1 e 2λ(T ) ] 2λ + 1 [ λ 2 (T )e 2λ(T ) + 1 e 2λ(T ) ] 2λ 1 λ 2 e λδ[ δ (2T +δ 2)e 2λ(T ) + 1 e 2λ(T ) ] ]. λ 2.3 The Shadow-Rae B-AFNS(3) Model We refer o he complee hree-facor shadow-rae model as he B-AFNS(3) model. 13 Given he above AFNS(3) shadow-rae process and he price of a shadow bond opion, we are now ready o price bonds ha observe he nonnegaiviy consrain in a B-AFNS(3) model. Krippner (2012) provides a formula for he ZLB insananeous forward rae, f (,T), ha applies o any Gaussian model ( f (,T) ) 1 f (,T)=f (,T) +ω(,t) exp ω(,t) 2π ( 1 2 [ f (,T) ] 2 ), ω(,t) where f (,T) is he shadow forward rae and ω(,t) is relaed o he condiional variance appearing in he shadow bond opion price formula as follows: ω(,t) 2 = 1 2 lim 2 v(,t,t +δ) δ 0 δ 2. Wihin he B-AFNS(3) model, he formula for he shadow forward rae, f (,T), is provided by equaion (8), while ω(,t) akes he following form: 14 ω(,t) 2 = σ11 2 (T )+(σ σ 22 2 ) 1 e 2λ(T ) 2λ [ 1 e +(σ31 2 +σ σ λ(T ) ) 1 4λ 2 (T )e 2λ(T ) 1 2 λ(t )2 e 2λ(T )] 1 e λ(t ) [ +2σ 11 σ 21 +2σ 11 σ 31 (T )e λ(t ) + 1 e λ(t ) ] λ λ +(σ 21 σ 31 +σ 22 σ 32 ) [ (T )e 2λ(T ) + 1 e 2λ(T ) ]. 2λ 13 Following Kim and Singleon (2012), he prefix B- refers o a shadow-rae model in he spiri of Black (1995), while he number shows he number of sae variables. Krippner (2012), 2013b) adops he prefix CAB for currency-adjused bond." 14 The calculaions leading o his resul are available from he auhors upon reques.

13 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 13 Now, he zero-coupon bond yields ha observe he ZLB, denoed y(,t), are easily calculaed as y(,t)= 1 [ T ( f (,s) ) 1 ( f (,s) +ω(,s) exp 1 [ f (,s) ] 2 ) ] ds. (9) T ω(,s) 2π 2 ω(,s) As highlighed by Krippner (2012), wih Gaussian shadow-rae dynamics, he calculaion of zero-coupon bond yields involves only a single inegral independen of he facor dimension of he model, which grealy faciliaes empirical implemenaion. 2.4 Nonzero Lower Bound for he Shor Rae In his secion, we generalize he model and consider a lower bound for he shor rae ha may differ from zero, i.e. r =max{r min,s }. A few papers have used a nonzero lower bound for he shor rae. In he case of U.S. Treasury yields, Wu and Xia (2013) simply fix he lower bound a 25 basis poins. A similar approach is applied o Japanese, UK, and U.S. yields by Ichiue and Ueno (2013). 15 As an alernaive, Kim and Priebsch (2013) rea r min as a free parameer o be esimaed, and using U.S. Treasury yields, hey obain a value of 14 basis poins. In our seing, o derive he implicaions for he yield funcion, we merely change he srike price of he bond opion from 1 o K =e r minδ in he formulas in Secion 1.2. Thus, he general formula for he yield ha respecs he r min lower bound is given by y(,t)=y(,t)+ 1 T T lim δ 0 [ δ C E (,s,s+δ;e rminδ ) ] ds. P(,s) I follows ha he forward rae ha respecs he r min lower bound is 16 ( f (,T) rmin ) 1 ( f (,T)=r min +(f (,T) r min ) +ω(,t) exp 1 ω(,t) 2π 2 [ f (,T) rmin ω(,ts) where he shadow forward rae, f (,T), and ω(,t) remain as before. However, we remain quie scepical abou he use of a non-zero r min. In par, his is because hey have no been well moivaed. This is especially rue for r min values ] 2 ), 15 For Japan, Ichiue and Ueno (2013) impose a lower bound of 9 basis poins from January 2009 o December 2012 and reduce i o 5 basis poins hereafer. For he Unied Saes, hey use a lower bound of 14 basis poins saring in November Finally, for he UK, hey assume he sandard zero lower bound for he shor rae. 16 The calculaions leading o his resul are available from he auhors upon reques.

14 14 Journal of Financial Economerics ha are greaer han he observed yields in he sample (and U.S. Treasury yields have reached a low of 1 basis poin in recen years). In addiion, our own unrepored resuls indicae ha he esimaed value of he shadow rae can be very sensiive o he value of r min. Similarly, using U.S. Treasury yields, Bauer and Rudebusch (2013) find ha he value of he shadow shor rae is quie sensiive across a range of values for r min. Thus, in general, he lower bound for he shor rae, r min, should be chosen wih care. In he conex of our Japanese daa, he daa indeed sugges ha zero is he appropriae lower bound wih he lowes one- and wo-year yields recorded being 0.0 basis poin and 1.3 basis poin, respecively, while he six-monh yield breaches he zero bound on a few occasions, bu is never lower han 2 basis poins. 2.5 Marke Prices of Risk So far, he descripion of he B-AFNS(3) model has relied solely on he dynamics of he sae variables under he Q-measure used for pricing. However, o complee he descripion of he model and o implemen i empirically, we will need o specify he risk premiums ha connec he facor dynamics under he Q- measure o he dynamics under he real-world (or hisorical) P-measure. I is imporan o noe ha here are no resricions on he dynamic drif componens under he empirical P-measure beyond he requiremen of consan volailiy. To faciliae empirical implemenaion, we use he exended affine risk premium developed by Cheridio, Filipović, and Kimmel (2007). In he Gaussian framework, his specificaion implies ha he risk premiums Ɣ depend on he sae variables; ha is, Ɣ =γ 0 +γ 1 X, where γ 0 R 3 and γ 1 R 3 3 conain unresriced parameers. 17 The relaionship beween real-world yield curve dynamics under he P-measure and risk-neural dynamics under he Q-measure is given by dw Q =dw P +Ɣ d. Thus, he P-dynamics of he sae variables are dx =K P (θ P X )d+ dw P, (10) where boh K P and θ P are allowed o vary freely relaive o heir counerpars under he Q-measure. Finally, we noe ha he model esimaion is based on he exended Kalman filer and described in Appendix A. 17 For Gaussian models, his specificaion is equivalen o he essenially affine risk premium specificaion inroduced in Duffee (2002).

15 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 15 Rae in percen year yield 4 year yield 1 year yield 6 monh yield Figure 1 Japanese governmen bond yields. We show ime-series plos of Japanese governmen bond yields a weekly frequency, a mauriies of 6 monhs, 1 year, 4 years, and 10 years. The daa cover he period from January 6, 1995, o May 3, DATA The bulk of our sample of Japanese governmen bond yields is idenical o he daa se examined by Kim and Singleon (2012). 18 Their daa se conains six mauriies: six-monh yields and one-, wo-, four-, seven-, and en-year yields, and all yields are coninuously compounded and measured weekly (Fridays). The Kim and Singleon (2012) sample, however, covers only January 6, 1995, o March 7, 2008, and so ends before he recen global financial crisis episode, which was marked by exremely low bond yields in Japan and in many oher counries. This recen episode is exremely ineresing o consider from a variey of economic and finance perspecives; herefore, we augmen he original Kim and Singleon (2012) sample wih Japanese governmen zero-coupon yields downloaded from Bloomberg hrough May 3, Figure 1 shows he variaion over ime in four of he six yields. During wo periods from 2001 o 2005 and from 2009 o 2013 six-monh and one-year yields are pegged near zero. These episodes are obvious candidaes for possible negaive shadow raes. As noed by Kim and Singleon (2012), hese periods also 18 We hank Don Kim for sharing hese daa. 19 When he wo sources of daa overlap during 2007 and 2008, he wo ses of yields mach almos exacly.

16 16 Journal of Financial Economerics Table 1 Facor loadings for Japanese governmen bond yields Mauriy (monhs) Loading on Firs P.C. Second P.C. Third P.C % explained The firs six rows show how bond yields a various mauriies load on he firs hree principal componens. The boom row shows he proporion of all bond yield variabiliy explained by each principal componen. The daa are weekly Japanese governmen bond yields from January 6, 1995, o May 3, display reduced volailiy of shor- and medium-erm yields due o he zero bound consrain. Researchers have found ha hree facors are ypically needed o model he ime-variaion in cross secions of bond yields (e.g., Lierman and Scheinkman, 1991). Indeed, for our sample of Japanese bond yields, percen of he oal variaion is accouned for by hree facors. As Table 1 repors, he firs principal componen loading s across mauriies (he associaed eigenvecor) is uniformly negaive, so like a level facor, a shock o his componen changes all yields in he same direcion irrespecive of mauriy. The second principal componen is a slope facor, as a shock o his componen seepens or flaens he yield curve. Finally, he hird componen has a U-shaped facor loading as a funcion of mauriy, which is naurally inerpreed as a curvaure facor. This paern of level, slope, and curvaure moivaes our use of he Nelson Siegel level, slope, and curvaure facors for modeling Japanese bond yields, even hough we emphasize ha our esimaed sae variables are no idenical o he principal componens. 4 RESULTS In his secion, we describe and assess one-, wo-, and hree-facor empirical shadow-rae models. We firs compare he shadow-rae model fi o he daa relaive o each oher and o non-shadow-rae dynamic erm srucure models. We also discuss some of he advanages of using Gaussian shadow-rae models over sandard Gaussian models in a near-zlb environmen. Nex, we evaluae he closeness of he opion-based approximaion o a maching simulaed shadow-rae model, before we sudy he efficiency of he exended Kalman filer in esimaing shadow-rae models. Finally, we examine he sensiiviy of he shadow shor rae o he number of facors in he model.

17 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models In-sample Fi of Sandard and Shadow-Rae Models We begin by considering he simples possible case for he shadow-rae dynamics, namely he one-facor Gaussian model of Vasiček (1977). Alhough his model may seem o be oo simple o be of ineres, i has been employed by several previous sudies 20 and is a useful ool for comparison. In his one-facor case, he facor dynamics of he shadow rae s used for pricing under he risk-neural Q-measure are ds =κ Q (θ Q s )d+σ dw Q, wih he risk-free rae given by he greaer of he shadow rae or zero: r =max{0,s }. The insananeous forward rae is given by while f (,T)=e κq (T ) s +θ Q (1 e κq (T ) ) 1 2 σ 2( 1 e κq (T ) κ Q ) 2, ω(,t) 2 =σ 2 1 e 2κQ (T ) 2κ Q. Allowing for ime-varying risk premiums, he dynamics under he objecive P- measure are fully flexible, ds =κ P (θ P s )d+σ dw P. We refer o his represenaion inspired by Black (1995) as he B-V(1) model. We also esimae he sandard Vasiček (1977) model, denoed as he V(1) model, wihou he nonnegaiviy consrain or he shadow-rae inerpreaion. Table 2 repors he summary saisics of he fied errors for he V(1) and B-V(1) models. 21 The beer fi of he B-V(1) model across all yield mauriies is noable, wih an average roo mean-squared error (RMSE) improvemen of 1.7 basis poins. This beer fi can also be seen in he higher likelihood value of he B-V(1) model. To mos closely approximae he wo-facor Gaussian shadow-rae model of Kim and Singleon (2012), 22 we esimae a wo-facor version of he B-AFNS model ha has level and slope facors bu no curvaure facor. This model is characerized by a shadow rae given by s =X 1 +X2. 20 These include Gorovoi and Linesky (2004), Ueno, Baba, and Sakurai (2006), and Krippner (2012). 21 The esimaed parameers of all models in his secion are provided in Appendix B. 22 This is heir B-AG2 model.

18 18 Journal of Financial Economerics Table 2 Summary saisics of model fi Mauriy in monhs RMSE All yields Max logl One-facor models V(1) , B-V(1) , Two-facor models AFNS(2) , B-AFNS(2) , Three-facor models AFNS(3) , B-AFNS(3) , The able presens he roo mean-squared error of he fied bond yields from one-, wo-, and hree-facor models esimaed on he weekly Japanese governmen bond yield daa over he period from January 6, 1995, o May 3, All numbers are measured in basis poins. The las column repors he obained maximum log-likelihood values. The sae variables (X 1,X2 ) used for pricing under he risk-neural Q-measure have he following dynamics: ( ) dx 1 = dx 2 ( ) ( ) 00 X 1 ( σ λ X 2 d+ σ 21 σ 22 ) ( dw 1,Q ) dx 2,Q. As for he P-dynamics, we focus on he mos flexible specificaion wih full K P marix ( ) )[( dx 1 κ P =( 11 κ12 P θ P 1 κ21 P κp 22 dx 2 θ P 2 ) ( )] X 1 X 2 ( σ11 0 d+ σ 21 σ 22 ) ( dw 1,P ) dw 2,P. This model has a oal of en parameers, wo less han he canonical B-AG2 model used by Kim and Singleon (2012). We esimae boh he sandard version of his model wihou any consrains relaed o he ZLB, denoed as he AFNS(2) model, and he corresponding shadow-rae model, denoed as he B-AFNS(2) model. Table 2 also repors summary saisics for he fi of he wo-facor models. The AFNS(2) model performs reasonably well, bu he B-AFNS(2) model has smaller yield RMSEs. The fi of he B-AFNS(2) model is comparable o he B-AG2 model esimaed in Kim and Singleon (2012) even hough he B-AFNS(2) model has fewer parameers under he Q-dynamics used for pricing Our RMSEs are very close o our esimaed error sandard deviaions, σ ε (τ i ), and o he esimaed error deviaions repored by Kim and Singleon (2012).

19 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 19 Finally, we exend he analysis o hree-facor models. In he AFNS(3) model, he risk-neural Q-dynamics used for pricing are as deailed in Secion 2, while we assume fully flexible facor dynamics under he P-measure: dx 1 dx 2 dx 3 = κ P 11 κp 12 κp 13 κ P 21 κp 22 κp 23 κ P 31 κp 32 κp 33 θ P 1 θ P 2 θ P 3 X 1 X 2 X 3 d+ σ σ 21 σ 22 0 σ 31 σ 32 σ 33 dw 1,P dw 2,P dw 3,P. Table 2 repors he summary saisics of he fied errors of he regular AFNS(3) model as well as is shadow-rae version, B-AFNS(3). Similar o wha we observed for he wo-facor models, he shadow-rae model ouperforms is sandard counerpar when i comes o model fi. In comparing model fi across he wo- and hree-facor models, he AFNS(3) model is on par wih he B-AFNS(2) model, while he B-AFNS(3) model has a bi closer fi han eiher of hem. 4.2 Why Use a Shadow-Rae Model? Before urning o an analysis of he shadow rae models, i is useful o reinforce he basic moivaion for our analysis by examining shor rae forecass and volailiy esimaes from he esimaed AFNS(3) model. Wih regard o shor rae forecass, any sandard affine Gaussian dynamic erm srucure model may place posiive probabiliies on fuure negaive ineres raes. Accordingly, Figure 2 shows he probabiliy obained from he AFNS(3) model ha he shor rae hree monhs ou will be negaive. Over much of he sample, he probabiliies of fuure negaive ineres raes are negligible. However, near he ZLB from 1999 o 2005 and from 2009 hrough he end of our sample he model is ypically predicing subsanial likelihoods of impossible realizaions. Anoher serious limiaion of he sandard Gaussian model is he assumpion of consan yield volailiy, which is paricularly unrealisic when periods of normal volailiy are combined wih periods in which yields are grealy consrained in heir movemens near he ZLB. Again, a shadow-rae model approach can miigae his failing significanly. Figure 3 shows he implied hree-monh condiional yield volailiy of he wo-year yield from he AFNS(3) and B-AFNS(3) models 24 along wih a comparison o he hree-monh realized volailiy of he wo-year yield calculaed from our sample using daily frequency. 25 While he condiional yield volailiy from he AFNS(3) model is consan, he condiional yield volailiy from he B-AFNS(3) model closely maches he realized volailiy series wih a 24 For he AFNS(3) model condiional yield volailiies can be calculaed using he formulas provided in Fisher and Gilles (1996), while condiional yield volailiies in he B-AFNS(3) model mus be generaed via Mone Carlo simulaion. 25 As in Kim and Singleon (2012), hese are he rolling sandard deviaion of daily yield changes over 60 rading-day windows.

20 20 Journal of Financial Economerics Probabiliy AFNS(3) model Figure 2 Probabiliy of negaive shor raes. Illusraion of he condiional probabiliy of negaive shor raes hree monhs ahead from he AFNS(3) model. correlaion of 72 percen. Paricularly noeworhy is he B-AFNS(3) model s abiliy o produce nearzero yield volailiy when yields are a heir lowes (during and ). 4.3 How Good is he Opion-Based Approximaion? As noed above, he opion-based approach does no consiue a formal derivaion of arbirage-free pricing relaionships, bu merely represens an approximaion of such relaionships. Therefore, in his subsecion, we analyze how closely he opionbased bond pricing from he esimaed B-AFNS(3) model maches an arbirage-free bond pricing ha is obained from he same model using Black s (1995) approach based on Mone Carlo simulaions. As a moivaing comparison, Figure 4 shows analyical and simulaion-based yield curves and opion-based and simulaion-based shadow yield curves from he esimaed B-AFNS(3) model as of January 9, 2004 which is during a Japanese ZLB period. The simulaion-based shadow yield curve is obained from 25, year long facor pahs generaed using he esimaed Q-dynamics of he sae variables in he B-AFNS(3) model, which, ignoring he nonnegaiviy Equaion (1), are used o consruc 25,000 pahs for he shadow shor rae. These are convered ino a corresponding number of shadow discoun bond pahs and averaged for each mauriy before he resuling shadow discoun bond prices are convered ino

21 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 21 Rae in basis poins AFNS(3) model B AFNS(3) model Three monh realized volailiy of wo year yield Correlaion = 72.4% Figure 3 Three-monh condiional volailiy of wo-year yield. Illusraion of he hree-monh condiional volailiy of he wo-year yield implied by he esimaed AFNS(3) and B-AFNS(3) models. Also shown is he subsequen hree-monh realized volailiy of he wo-year yield based on daily daa. yields. The simulaion-based yield curve is obained from he same underlying 25,000 Mone Carlo facor pahs, bu a each poin in ime in he simulaion, he resuling shor rae is consrained by he nonnegaiviy Equaion (1) as in Black (1995). The shadow-rae curve from he B-AFNS(3) model can also be calculaed analyically via he usual affine pricing relaionships, which ignore he ZLB. Noe ha he simulaed shadow yield curve is almos idenical o his analyical shadow yield curve. Any difference beween hese wo curves is simply numerical error ha reflecs he finie number of simulaions. More ineresingly, he differences beween he simulaion-based and opion-based yield curves are also hard o discern. The minuscule discrepancies beween hese wo yield curves show ha he approximaion error associaed wih he opion-based approach o calculaing bond yields near he ZLB is also very small in his insance. To documen ha he close mach beween he opion-based and he simulaion-based yield curves is no limied o one specific dae, we repeaed he simulaion exercise for he firs observaion in each year of our sample. Table 3 repors he resuling shadow yield curve differences and yield curve differences for various mauriies on hese 19 daes. Again, he errors for he shadow yield curves solely reflec simulaion error as he model-implied shadow yield curve is idenical o he analyical arbirage-free curve ha would prevail wihou currency

22 22 Journal of Financial Economerics Table 3 Approximaion errors in yields for hree-facor model Mauriy in monhs Daes Shadow yields /2/95 Yields Shadow yields /5/96 Yields Shadow yields /10/97 Yields Shadow yields /9/98 Yields Shadow yields /8/99 Yields Shadow yields /7/00 Yields Shadow yields /5/01 Yields Shadow yields /4/02 Yields Shadow yields /10/03 Yields Shadow yields /9/04 Yields Shadow yields /7/05 Yields Shadow yields /6/06 Yields Shadow yields /6/07 Yields Shadow yields /6/08 Yields Shadow yields /2/09 Yields Shadow yields /1/10 Yields Shadow yields /7/11 Yields Shadow yields /6/12 Yields Shadow yields /4/13 Yields Average Shadow yields absolue difference Yields A each dae, he able repors differences beween he analyical shadow yield curve obained from he opion-based esimaes of he B-AFNS(3) model and he shadow yield curve obained from 25,000 simulaions of he esimaed facor dynamics under he Q-measure in ha model. The able also repors for each dae he corresponding differences beween he fied yield curve obained from he B-AFNS(3) model and he yield curve obained via simulaion of he esimaed B-AFNS(3) model wih imposiion of he ZLB. The boom wo rows give averages of he absolue differences across he 19 daes. All numbers are measured in basis poins.

23 CHRISTENSEN &RUDEBUSCH Shadow-Rae Term Srucure Models 23 Rae in percen Yield curve, B AFNS(3) model Shadow curve, B AFNS(3) model Yield curve, Black (1995) simulaion Shadow curve, Black (1995) simulaion Observed yields Time o mauriy in years Figure 4 Fied yield curves in hree-facor shadow-rae models. Fied and shadow yield curves from an opion-based esimaed B-AFNS(3) model are shown as of January 9, In addiion, he corresponding curves are shown based on a simulaion using Black s (1995) approach and N = 25,000 pahs of he sae variables drawn using he opion-based esimaed B-AFNS(3) model facor dynamics under he Q-measure. in circulaion. These simulaion errors in Table 3 are ypically very small in absolue value, and hey increase only slowly wih mauriy. Their average absolue value shown in he boom row is less han one basis poin even a a 10-year mauriy. This implies ha using simulaions wih a large number of draws (N = 25,000) arguably delivers enough accuracy for he ype of inference we wan o make here. Given his calibraion of he size of he numerical errors involved in he simulaion, we can now assess he more ineresing size of he approximaion error in he opion-based approach o valuing yields in he presence of he ZLB. In Table 3, he errors of he fied B-AFNS(3) model yield curve relaive o he simulaed resuls are only slighly larger han hose repored for he shadow yield curve. In paricular, for mauriies up o seven years, he errors end o be less han 1 basis poin, so he opion-based approximaion error adds very lile if anyhing o he numerical simulaion error. A he 10-year mauriy, he approximaion errors are undersandably larger, bu even he larges errors a he 10-year mauriy do no exceed 4 basis poins in absolue value and he average absolue value is around 2 basis poins. Overall, he opion-based approximaion errors in our hree-facor