Modifying Gaussian term structure models when interest rates are near the zero lower bound. Leo Krippner. March JEL classi cation: E43, G12, G13

Transcription

1 DP0/0 Moifying Gaussian term structure moels when interest rates are near the zero lower boun Leo Krippner March 0 JEL classi cation: E43, G, G3 Discussion Paper Series ISSN

2 DP0/0 Moifying Gaussian term structure moels when interest rates are near the zero lower boun Leo Krippner y Abstract With nominal interest rates near the zero lower boun (ZLB) in many major economies, it is theoretically untenable to apply Gaussian a ne term structure moels (GATSMs) while ignoring their inherent material probabilities of negative interest rates. I propose correcting that e ciency by ajusting the entire GATSM term structure with an explicit function of maturity that represents the optionality associate with the present an future availability of physical currency. The resulting ZLB-GATSM framework remains tractable, proucing a simple close-form analytic expression for forwar rates an requiring only elementary univariate numerical integration (over time to maturity) to obtain interest rates an bon prices. I emonstrate the salient features of the ZLB- GATSM framework using a two-factor moel. An illustrative estimation with U.S. term structure ata inicates that the ZLB-GATSM "shaow short rate" provies a useful gauge of the stance of monetary policy; in particular becoming negative when the U.S. policy rate reache the ZLB in late 008, an moving more negative with subsequent unconventional monetary policy easings. The Reserve Bank of New Zealan s iscussion paper series is externally referee. The views expresse in this paper are those of the author(s) an o not necessarily re ect the views of the Reserve Bank of New Zealan. The author thanks Katy Bergstrom, Iris Claus, Toby Daglish, Francis Diebol, Pero Gomis, Michelle Lewis, Anella Munro, Glenn Ruebusch, Peter Phillips, Christie Smith, Daniel Thornton, Christopher Waller, participants at the 0 Reserve Bank of New Zealan conference, a 0 Bunesbank seminar, an the 0 New Zealan Econometrics Stuy Group meeting for helpful comments. y Economics Department, Reserve Bank of New Zealan, The Terrace, PO Box 498, Wellington, New Zealan. aress: leo.krippner@rbnz.govt.nz. ISSN creserve Bank of New Zealan

3 Introuction In this article I propose a framework for imposing the zero lower boun (ZLB) for nominal interest rates on Gaussian a ne term structure moels (GATSMs). My primary motivation for eveloping the ZLB-GATSM framework is to aress the inherent theoretical e ciency of negative interest rates in GATSMs, an a practical, topical motivation is to provie an inicator of the stance of monetary policy in the presence of the ZLB. I expan on both motivations subsequently below. At the same time, I seek to preserve in the framework two key features that have mae GATSMs extremely popular; i.e. their exibility an tractability. Speci cally, GATSMs may be speci e arbitrarily (in terms of the number of factors an inter-factor relationships) while retaining both close-form analytic solutions for pricing stanar interest rate instruments (e.g. bons an options) an multivariate-normal transition ensities for the state variables. Those features make GATSMs straightforwar to estimate an apply relative to other term structure moels. As such, Hamilton an Wu (00) introuces GATSMs as the basic workhorse in macroeconomics an nance an notes seven recent examples of their application. Ruebusch (00) surveys GATSM applications in macro nance. However, it is well acknowlege that GATSMs cannot be theoretically consistent in the real worl where physical currency is available (e.g. see Piazzesi (00) p. 76). The inconsistency arises because any unconstraine Gaussian process for short rate ynamics technically implies non-zero probabilities of negative interest rates for all maturities on the term structure. On such a realization, one coul realize an arbitrage pro t by borrowing (therefore receiving the absolute interest rate) to buy an hol physical currency (with a known return of zero). Alternatively, one coul sell bon options base on the non-zero probabilities of negative interest rates in GATSMs, but with no probability of an out-of-the-money expiry in practice. Despite that known inconsistency, GATSMs are often applie with the assumption (as iscusse in Piazzesi (00) p. 76, but typically left implicit) that the inherent probabilities of negative interest rates in GATSMs are su ciently small to make the moel immaterially i erent to a real worl moel subject to the ZLB. While that may have been the case over history, when interest rate levels relative to their typical stanar eviation of unanticipate changes remaine well above zero, such an assumption is obviously untenable in several major evelope economies at the time of writing. For example, near-zero policy interest rates have been maintaine in the Unite States an the Unite Kingom since late-008/early-009, an in Japan since the 990s (an each of those countries have also engage in unconventional monetary policy easings ue to the ZLB constraint). Moreover, the levels of short- an meiummaturity interest rates in those countries also lie well within their typical stanar eviations of unanticipate changes. The material probability of negative interest rates in GATSMs in near-zero interest Hamilton an Wu (00) proposes a reliable metho for estimating GATSMs, an Joslin, Singleton, an Zhu (0) provies another approach. Arbitrage-free versions of the Nelson an Siegel (987) class are a sub-class of GATSMs that are also empirically reliable. (Their nesting within the GATSM class is clearly illustrate in Christensen, Diebol, an Ruebusch (009, 0), where arbitrage-free Nelson an Siegel (987) moels are erive via GATSM speci cations that reprouce Nelson an Siegel (987) factor loaings.)

4 rate environments in turn implies moel mis-speci cation relative to the ata being moele. In essence, if term structure ata are materially constraine by the ZLB but the GATSM applie to the ata assumes no constraints, then the estimate GATSM an its state variables cannot provie a vali representation of the term structure an its ynamics. The mis-speci cation will a ect even routine GATSM applications, such as monitoring the level an shape of the estimate term structure to provie a gauge of the stance of monetary policy. The mis-speci cation implications are compoune for any relationships establishe via GATSMs between term structure an macroeconomic ata (e.g. measures of in ation an real output growth), because macroeconomic ata are not constraine to be non-negative. A straightforwar moi cation to GATSMs that eliminates negative interest rates is the Black (995) framework, which is base on the observation that physical currency e ectively provies an option against negative interest rates at each point in time. Speci cally, the ZLB short rate may be e ne as r (t) max fr (t) ; 0g, where r(t) is the shaow short rate that is free to evolve with negative an positive values. Unfortunately, Black-GATSM implementations (i.e. using GATSMs to e ne the shaow short rate r(t) within the Black (995) framework) result in moels with limite tractability. For example, even the simplest Black-GATSM, the one-factor Black- Vasicek (977) moel by Gorovoi an Linetsky (004), oes not result in a close-form analytic solution. Speci cally, it requires the numerical evaluation of relatively complex functions (e.g. Weber-Hermite parabolic cyliner functions). In aition, the Gorovoi an Linetsky (004) approach oes not appear to generalize to multiple factors. 3 Bom m (003), Ueno, Baba, an Sakurai (006), an Ichiue an Ueno (007) respectively use the purely numerical methos of nite-i erence gris, Monte Carlo simulations, an interest rate lattices to illustrate an/or estimate two-factor Black- GATSMs. 4 However, the complexity of those numerical methos increases to the orer of the power of the number of factors. Therefore, while Black-GATSMs have arbitrary exibility in principle, in practice their rapi ecline in tractability woul preclue the full exibility o ere by larger multifactor GATSMs. The ZLB framework I propose in this article is conceptually similar to Black (995), in that it is base on the optionality provie by physical currency, but it is istinctly i erent in the following respect: I ensure the entire ZLB forwar rate curve remains Of course, a prevalent literature has evolve over several ecaes using non-gaussian ynamics esigne to avoi negative interest rates in term structure moels; e.g. James an Webber (000) pp iscusses a variety of positive interest rate moels. Given my focus is on Gaussian moels, I o not iscuss positive interest rate moels further beyon the following comments in this context of this article: () such moels lose the potential information provie by the shaow short rate an shaow term structure, as will be iscusse subsequently in the main text; an () such moels inevitably result in limite exibility an/or tractability relative to GATSMs. For example on the latter point, close-form analytic solutions an transition ensities are not available for arbitrary speci cations of multifactor Cox, Ingersoll, an Ross (985b)/square-root moels, an even the special case of inepenent factors requires the prouct of noncentral chi-square istributions (e.g. see Piazzesi (00) p. 77). 3 See Kim an Singleton (0) p.. Ichiue an Ueno (006) an Ueno, Baba, an Sakurai (006) apply the Gorovoi an Linetsky (004) moel to the Japanese term structure. 4 The Bom m (003) application is to Unite States ata, as I subsequently iscuss in section 4.3. The other applications are to the Japanese term structure. An analogous Japanese application is the two-factor Black-quaratic-Gaussian-moel from Kim an Singleton (0), which nests the two-factor Black-GATSM as a special case.

5 non-negative at each point in time by aing to the shaow forwar rate curve an explicit function of maturity that represents the option e ect from the present an future availability of physical currency. To highlight just the essence of the istinction at this stage, 5 the Black (995) framework e ectively allows for an option effect of max f r (t) ; 0g (which is positive if r(t) is negative, an zero otherwise) at each point in time. While that mechanism also imposes the ZLB constraint, i.e. r(t) + max f r (t) ; 0g max f0; r (t)g r (t), the function of maturity representing the future stream of physical currency optionality is left implicit. My propose ZLB framework in the Gaussian context uses the close-form expression for GATSM call options on bons to erive the option e ect as a function of time to maturity. Aing that to the shaow-gatsm forwar rate curve results in a ZLB-GATSM forwar rate curve with a simple close-form analytic expression, involving just exponential functions an the univariate cumulative normal istribution. ZLB-GATSM interest rates an ZLB-GATSM bon prices are obtaine using stanar term structure relationships; respectively, the mean of the integral of ZLB-GATSM forwar rates over time to maturity an the exponential of that (negate) integral. The integral is necessarily numerical in the Gaussian context, but its evaluation is elementary ue to the nature of the ZLB-GATSM forwar rate expression. I illustrate the salient features of the ZLB-GATSM framework using a calibrate two-factor GATSM for the shaow term structure. The rst exercise shows how the ZLB-GATSM framework transforms a selection of shaow-gatsm forwar an interest rate curves. The secon exercise is an application to U.S. term structure ata an illustrates that moel-implie shaow short rates move to increasingly negative values aroun the announcements of unconventional monetary policy easings unertaken after the U.S. policy rate reache the ZLB in late 008. Those movements inicate that the ZLB-GATSM framework provies a means for routinely istilling a quantitative inicator of the stance of monetary policy from term structure ata when interest rates are materially constraine by the ZLB. The outline of the remainer of the article is as follows. Section outlines the intuition of my approach to eveloping a ZLB term structure base on bon an bon option prices, an then iscusses the i erences between my framework an the framework of Black (995). Section 3 erives the generic ZLB-GATSM term structure. Section 4 contains the illustrative exercises alreay note earlier, incluing a brief iscussion of recent relate literature. Section 5 brie y iscusses potential extensions of the ZLB-GATSM framework introuce in this article, incluing aspects relating to a full empirical estimation an a thorough exploration of the resulting ZLB-GATSM shaow short rate an the shaow term structure as inicators of the stance of monetary policy. Section 6 conclues. A general ZLB framework In this section, I outline my approach to constructing ZLB term structures without referring to any particular moel of the term structure. Section. introuces the concept of the availability of physical currency e ectively proviing investors with a 5 Section.4 precisely e nes the istinction, in light of appropriate notation, between my propose ZLB framework an the Black (995) framework. 3

6 (European) call option on a bon, in aition to the bon investment itself. 6 Section. extens that optionality concept from the current point in time to future time horizons. Section.3 uses the results from section. to erive a general ZLB term structure in terms of bon an option prices. I iscuss in section.4 how my propose ZLB framework i ers from the Black (995) framework.. Physical currency over a single time-step Physical currency may be viewe as a bon with a price of an an associate interest rate of zero. To illustrate this, I introuce a nite-step shaow bon that settles for price P(t; ) at time t an pays at time t +, where > 0 is a (small) nite-step representing the time to maturity. 7 The annualize rate of return over the nite-step on a continuously-compouning basis is the nite-step shaow short rate R(t; ), i.e.: R (t; ) log () P (t; ) Clearly, if P(t; ), then R(t; ) 0, an so the return equals that o ere by physical currency. The availability of physical currency is therefore equivalent to nite-step shaow bons with a price P(t; ) being available as an investment, if esire. The if esire is the crucial turn of phrase that unerlies the optionality of physical currency; i.e. investors always have the right but not the obligation to hol it. Of course, if the prevailing market price of P(t; ) is less than, then R(t; ) > 0 an so investors will choose to hol P(t; ) rather than substituting physical currency at P(t; ). Alternatively, if the prevailing market price of P(t; ) is greater than, then R(t; ) < 0 an so investors will choose to maximize their return by holing physical currency at P(t; ). In summary, investors will choose P (t; ) min f; P (t; )g, where P (t; ) is my notation for the nite-step ZLB bon. P (t; ) may be expresse more conveniently as the sum of the nite-step shaow bon P(t; ) an the payo for a call option on P(t; ), i.e.: P (t; ) min f; P (t; )g P (t; ) + min f P (t; ) ; 0g P (t; ) max fp (t; ) ; 0g P (t; ) C (t; 0; ) () where C(t; 0; ) max fp (t; ) ; 0g is the payo for a call option, with immeiate expiry an a strike price of, on the nite-step bon P(t; ). The optionality arising from the availability of physical currency over the nite step establishes the lower boun of zero for the nite-step ZLB short rate (t; ). R 6 I typically take the perspective of investors, but all comments an calculations from the investors perspective can be reverse to give the borrowers perspective. 7 For example, overnight or on-call eposits (often colloquially referre to as cash or liquiity ) are e ectively nite-step bons with a single ay to maturity. 4

7 Explicitly: (t; ) R log (t; ) P log [P (t; ) max C (t; 0; )] log [min f; P (t; )g] log [] ; log [P (t; )] max f0; R (t; )g (3) Note that taking the in nitesimal limit of woul reprouce the Black (995) concept of physical currency proviing an option on the shaow short rate; heuristically, lim R (t; ) max [lim R (t; ) ; 0], so r (t) max [r (t) ; 0]. Section.3 contains a more formal exposition of that in nitesimal limit in the context of e ning the continuous-time term structure for the general ZLB framework.. Physical currency over multiple time-steps Beyon the rst nite-step, investors also know that the availability of physical currency at any point of time in the future will always o er them the choice between holing physical currency or investing in shaow bons at their prevailing market price. In notation, I introuce a positive quantity to represent a future horizon from time t, so future time may be represente as t +. 8 The multiple- nite-step ZLB bon is therefore e ne as: (t + ; ) min f; P (t + ; )g (4) P While the optionality inherent in P (t + ; ) is analogous to that for the rst- nite-step ZLB bon P (t; ), the i erence is that P (t; ) is a non-contingent quantity (because the choice between holing the bon or physical currency is alreay known at time t), while P (t + ; ) is a contingent quantity (because the choice between holing the bon or physical currency is unknown at time t, being epenent on the single- nitestep shaow bon price P(t + ; ) at the future time t + ). Nevertheless, P (t + ; ) provies a terminal bounary conition that may be use in conjunction with an appropriate pricing mechanism to obtain the multiple- nite-step ZLB bon price P (t; + ). An explicit mechanism is not require for the present iscussion, but section.4. introuces a generic expression for security pricing in the context of comparing my propose ZLB framework to the Black (995) framework. Analogous to the rst- nite-step ZLB bon P (t; ), the multiple- nite-step ZLB bon P (t; + ) may also be more conveniently expresse as the sum of the multiple- nite-step shaow bon P(t; + ) an a call option on P(t; + ). To establish that sum, I procee by rst separating the bounary conition for the ZLB bon into the 8 For the purpose of transparent exposition within a iscrete time formulation, I assume at this stage that t an are integer multiples of. However, the nite-step is the key component for the iscrete time formulation, while t an coul be regare as continuous variables. 5

8 bounary conition for a shaow bon an a put option, i.e.: P (t + ; ) min f; P (t + ; )g + min f0; P (t + ; ) g max f0; P (t + ; )g (5) where is the bounary conition for the multiple- nite-step shaow bon P(t; ) maturing at time t + (i.e. P(t + ; 0) ), an max f0; P (t + ; )g is the bounary conition for a put option Q(t; ; + ) expiring at time t + with a strike price of. 9 Note that the unerlying security for Q(t; ; + ) at expiry is the single- nite-step shaow bon P(t + ; ), i.e. Q(t + ; 0; ) max f0; P (t + ; )g, while the unerlying security at time t is the forwar price of that bon. The bounary conition P(t + ; 0) obtains the shaow bon price P(t; ), while the bounary conition max f0; P (t + ; )g obtains the negate put option price Q(t; ; + ). Therefore, the solution for the ZLB bon price is: P (t; + ) P (t; ) Q (t; ; + ) (6) The right-han sie of equation 6 may be re-expresse in terms of P(t; + ) an C(t; ; + ) using the put-call parity relationship for forwar bons an bon option prices, as note in Chen (995) p. 363, with a strike price of. That is, C(t; ; + ) Q(t; ; + ) P(t; + ) P(t; ), 0 an so: P (t; ) Q (t; ; + ) P (t; + ) C (t; ; + ) (7) The ZLB bon price therefore becomes: P (t; + ) P (t; + ) C (t; ; + ) (8) Finally, analogous to the rst- nite-step ZLB short rate (t; ), the optionality R arising from the future availability of physical currency establishes that each realize future single- nite-step ZLB short rate (t + ; ) will respect the ZLB. Explicitly: R (t + ; ) R log (t + ; ) P log [P (t + ; ) max C (t + ; 0; )] log [min f; P (t + ; )g] log [] ; log [P (t + ; )] max f0; R (t + ; )g (9) 9 Expressions for put option prices are often enote P(), but I have use the notation Q() to avoi any confusion with my notation P() for bon prices. 0 The stanar put-call parity relationship equates the forwar price to the call option price less the put option price, or F C Q (see, for example, Haug (007) p. 8 or the original reference therein to Nelson (904)). For forwar bons an options with a strike price of, F P(t; + ) P(t; ), so F P(t; + ) P(t; ). 6

9 .3 A general ZLB term structure While the nite-step securities in sections. an. coul be use irectly to establish a iscrete-time term structure moel, it is almost invariably more convenient to work with term structure moels in continuous time. For example, in the ZLB-GATSM framework that follows in section 3, the continuous-time expressions are much more parsimonious an computationally convenient than their iscrete-time counterparts woul be. To obtain the continuous-time term structure for the general ZLB framework, I begin with the stanar term structure relationship relating (continuously compouning) ZLB forwar rates to ZLB bon prices, i.e.: f (t; ) log [P (t; )] (0) an I obtain the require erivative via P(t; + ) in the in nitesimal limit of, i.e.: log (t; )] lim log (t; + )] () [P [ + ] [P Appenix A. contains the etails of this erivation. The resulting expression for the ZLB forwar rate is: f (t; ) f (t; ) + z (t; ) () where f(t; ) is the shaow forwar rate curve an z(t; ) is the option e ect: C (t; ; + ) z (t; ) lim (3) P (t; ) The ZLB forwar rate f (t; ) establishes the ZLB interest rate r (t; ) an the ZLB bon price (t; ) via stanar term structure relationships, i.e.: P r (t; ) Z 0 f (t; ) (4) where is a ummy integration variable for horizon/time to maturity, an: Z (t; ) exp f (t; ) P 0 exp [ (t; )] (5) r It is worthwhile explicitly establishing that the ZLB short rate r (t) from the general ZLB term structure will always be a well-e ne an non-negative quantity. Appenix A. contains the etails, which I summarize as follows: r (t) lim (t; ) f f (t; 0) + z (t; 0) r (t) + max f r (t) ; 0g max fr (t) ; 0g (6) See, for example, Filipović (009) p. 7 or James an Webber (000) chapter 3 for this relationship an other stanar term structure relationships that I subsequently refer to an use in this article. 7

10 .4 Comparing the Black (995) an ZLB frameworks Section.4. establishes by example that the Black (995) framework an my propose ZLB framework are not equivalent. In section.4., I iscuss the i erence between the two frameworks from a theoretical perspective..4. Numerical comparison It is straightforwar to establish by example that the Black (995) an ZLB frameworks are not equivalent. Speci cally, using an ientical moel for the shaow short rate, I compare the numerical results from my ZLB framework with those of the Black (995) framework. For the comparison, tables an respectively present what I call the BGL- GATSM() results from table 6. in Gorovoi an Linetsky (004), which are base on a one-factor GATSM/Vasicek (977) moel within the Black (995) framework. The precise GATSM() speci cation for the shaow short rate is the shaow-gatsm() moel subsequently speci e in section 4. of the present article with the state variable r(t) s (t) an the parameters 0:, 0:0, 0:0, an 0. 3 Gorovoi an Linetsky (004) uses initial values of r(t) 0 an r(t) 0:0 (i.e. percent) for the shaow short rate. I then use the same GATSM() speci cation within my ZLB-GATSM framework to evaluate the corresponing ZLB-GATSM() interest rates (the approach is subsequently outline for the ZLB-GATSM() in section 4.). The results are shown in tables an, along with the i erences between the Black (995) an ZLB framework results. Table : Interest rates (in percentage points) with r(t) 0 percent Moelntime to maturity (years) BGL-GATSM() ZLB-GATSM() BGL less ZLB GATSM() Note that, to facilitate comparison across the i erent times to maturity, I have converte the Gorovoi an Linetsky (004) bon price results to their interest rate equivalents using the stanar term structure relationship r(t; ) log [P (t; )]. The bon prices from Gorovoi an Linetsky (004) are accurate to ve ecimal places. That accuracy translates to the interest rates in tables an being accurate to at least three ecimal places (i.e percentage points or 0. basis points) in all cases. I have also reprouce the Gorovoi an Linetsky (004) results inepenently via Monte-Carlo simulation. 3 The state variable s (t) an its associate parameters in the GATSM() are, of course, all set to zero. 8

11 Table : Interest rates (percentage points) with r(t) percent Moelntime to maturity (years) BGL-GATSM() ZLB-GATSM() BGL less ZLB GATSM() While the respective results are very similar for the -year maturity, the i erences are su ciently large for the other maturities consiere (e.g. a maximum of 0. percentage points at the maturity of 30 years in table, relative to the signi cance of 0.00 percentage points) to con rm that r (t; ) 6 r Black (t; ), or equivalently P (t; ) 6 P Black (t; ). As an asie to the main point of this section, but relevant for practically illustrating the theoretical e ciency of GATSMs in low interest rate environments, note that the interest rates for the GATSM() are always substantially below their counterparts that use a ZLB mechanism. Inee, the relatively low mean reversion of 0: in these particular examples results in negative interest rates for the 30-year maturity (i.e. bon prices greater than ), which is a phenomenon I return to iscuss in the context of the GATSM() examples of section Theoretical comparison The natural follow-up question to section.4. is: how/where oes the i erence between the two frameworks arise? The rst point to note in response is that the two frameworks have ientical short rates uner the physical or P measure. That is straightforwar to establish by simply noting that implementations of the Black (995) framework are base on short rate i usion processes e ne as r Black (t) max fr (t) ; 0g, while equation 6 establishes that the short rate process associate with my propose ZLB framework is r (t) max fr (t) ; 0g. Therefore, realize short rates for a given shaow short rate process are ientical in the two frameworks; i.e. r (t) r Black (t). The result (t; ) 6 Black (t; ) from section.4. along with r (t) r Black (t) uner P P the P measure implies that the Black (995) an ZLB frameworks must have i erent short rate i usion processes uner the risk-neutral Q measure. To make that proposition more transparent, I introuce the stanar generic expression that e nes security prices via their expecte iscounte cash ows; i.e.: 4 X (t; ) E Q t exp Z 0 ¼r (t + ) A (t + ) where X(t; ) is the security to be price (base on a given set of state variables s (t) an parameters B), E Q t is the expectations operator uner the Q measure, A(t + ) 4 See, for example, the generic option price expression from Filipović (009) p. 09, which is given in appenix A. of the present article. Filipović (009) chapter 4 an James an Webber (000) chapter 4 are two examples of many texts that iscuss the stanar framework of risk-neutral security pricing. 9 (7)

12 is the terminal cash ow for the security X(t; ), ¼r(t + ) is the Q-measure R short rate i usion process, which is use to obtain the iscount factor exp ¼r (), an 0 is a ummy integration variable for future time relative to the current time t. Bon prices in the Black (995) framework, which I enote Black (t; ), are e ne P irectly via equation 7 using ¼r(t + ) max fr (t + ) ; 0g r Black (t + ) (i.e. the Black (995) e nition for the i usion process of the shaow short rate subject to the ZLB constraint) an A(t + ) (i.e. the terminal payo for a bon). The resulting expression for Black (t; ) is therefore: P Z Black (t; ) E P Q t exp Black r (t + ) (8) Forwar rates in the Black (995) framework are e ne with the following stanar term structure e nition, i.e.: 0 Black f (t; ) log [P Black (t; )] (9) although this expression is necessarily implicit given the absence of close-form analytic solutions for Black (t; ). P In the ZLB framework, I inirectly e ne the entire ZLB forwar rate curve via equation 7 using ¼r(t + ) r(t + ) (i.e. the shaow short rate itself) an A(t + ) min f; P (t + ; )g (i.e. the terminal payo for a multiple- nite-step ZLB bon). That set-up results in the multiple- nite-step ZLB bon price (t; + ) as an intermeiate P step, i.e.: P (t; + ) E Q t exp Z 0 r () min f; P (t + ; )g Following the iscussion in section.3, ZLB forwar rates f (t; ) associate with (t; + ) are then: P f (t; ) lim log (t; + )] () [ + ] [P an bon prices subject to the ZLB constraint are: Z (t; ) exp f (t; ) P The ZLB bon price (t; ) coul also be e ne, in principle, via equation 7 P using ¼r(t + ) r (t + ) an A(t + ), i.e.: Z (t; ) E P Q t exp r (t + ) (3) where r (t + ) is the implicit ZLB i usion process require to reprouce the prices (t; ) from equation. Written in this way, it is clear that the result (t; ) 6 P P Black (t; ) from section.4. implies that: P E Q t exp Z 0 r (t + ) 6 E Q t exp Z 0 Black r (t + ) (0) () (4)

13 Hence, the i erent short rate i usion processes uner the risk-neutral Q measure are i erent for the Black (995) an ZLB frameworks. That i erence, in turn, arises from the subtlely but funamentally i erent mechanism by which each framework converts shaow short rates into real worl short rates that respect the ZLB. The i erence also impacts on the nature of the term structure associate with each framework. In summary, the Black (995) framework begins with a simple an explicit process for the risk-neutral short rate subject to the ZLB, but results in a relatively complex an implicit term structure. Conversely, the ZLB framework e ectively e nes a complex an implicit process for the risk-neutral ZLB short rate, but results in a relatively simple an explicit term structure. To conclue this section, I introuce the natural follow-up question to the iscussion above, i.e.: which framework is best? To brie y respon, I believe the answer is an open issue at present. The ZLB framework certainly has an avantage over the Black (995) framework from the perspective of practical implementation, as sections 3 an 4 will illustrate. However, the comparison from a theoretical perspective is unresolve because both frameworks are essentially base on mechanical/statistical e nitions for the short rate subject to the ZLB rather than an explicit theoretical founation. To be clear on this point, GATSMs themselves have long ha explicit theoretical founations to their statistical speci cations, 5 but I am not aware of such founations for either the Black (995) framework or my propose ZLB framework (although, heuristically, both frameworks certainly appear sensible ). Developing theoretical justi cations for the Black (995) an ZLB frameworks an comparing them is well beyon the scope of this article, but as I iscuss in section 5, will be an important topic for future work. Similarly, as also iscusse in section 5, a irect empirical comparison may at least resolve the less ambitious question of which framework better represents the ata. 3 The ZLB-GATSM framework In this section, I evelop the ZLB-GATSM framework using the general ZLB framework outline in section.3. To establish notation, section 3. outlines the generic GATSM speci cation that I use to represent the shaow-gatsm term structure. While GATSM summaries are available in many articles an textbooks, I generally use Chen (995) as a convenient point of reference because it contains the explicit multifactor expressions for GATSM bon an option prices that I use respectively in sections 3. an 3.3. However, I also refer to Dai an Singleton (00) pp for the compactness of its matrix notation in some instances. Section 3. erives the shaow-gatsm forwar rate f(t; ) an section 3.3 erives the ZLB-GATSM option e ect z(t; ). 6 In section 3.4 I combine the results from 5 For example, Cox, Ingersoll, an Ross (985a) originally provie a general equilibrium basis for term structure moels (incluing the one-factor GATSM) base on a representative-agent economy, an that approach has since been extene by many authors to provie theoretical founations for multifactor GATSMs; see, for example, Berari an Esposito (999) an Berari (009). More recently, Wu (006) also shows how GATSMs may be given an explicit founation within ynamic stochastic general equilibrium moels. 6 Chen (995) actually provies the speci cation an results for a two-factor GATSM but notes the reay extension to N factors that I use. Chaplin (987) rst erive explicit bon an option prices for the two-factor GATSM, an Sharp (987) extene those expressions to N factors. The heritage

14 sections 3. an 3.3 to obtain the close-form analytic expression for ZLB-GATSM forwar rates f (t; ), which therefore e nes ZLB-GATSM interest rates r (t; ) an bon prices P (t; ) for the ZLB term structure. Section 3.5 makes a series of observations about the ZLB-GATSM framework an term structure. 3. Shaow-GATSM short rate process I use the stanar generic GATSM speci cation from Chen (995) to represent the shaow-gatsm short rate, i.e.: NX r (t) s n (t) (5) n where s n (t) are the N state variables that evolve as a correlate Ornstein-Uhlenbeck process uner the physical or P measure, i.e.: s n (t) n [ n s n (t)] t + n W n (t) (6) where n are constants representing the long-run levels of s n (t), n are positive constants representing the mean reversion rates of s n (t) to n, n are positive constants representing the volatilities (annualize stanar eviations) of s n (t), W n (t) are Wiener processes with W n (t) N (0; )t, an E [W m (t) ; W m (t)] mn t, where mn are correlations mn. Mainly for ease of notation, I aopt the Chen (995) speci cation of constant market prices of risk for each factor, which I enote n. 7 However, as in Vasicek (977), I e ne the market prices of risk n to be outright positive quantities. That e nition means that increases in n intuitively a risk premiums n n G ( n ; ) to forwar rates, as in the following section, therefore raising interest rates an lowering bon prices. 8 The i usion process uner the P measure in conjunction with the market price of risk speci cation of course allows one to erive a risk-neutral Q-measure i usion process for the shaow-gatsm short rate that may be use to price shaow-gatsm securities. However, I nee not explicitly unertake such an evaluation in this article because the shaow-gatsm bon price an option price expressions that I require for the ZLB-GATSM framework are alreay available in Chen (995). 3. Shaow-GATSM forwar rates I erive shaow-gatsm forwar rates f(t; ) using the close-form analytic expression for GATSM bon prices from Chen (995) within the stanar term structure e nition f(t; ) log [P (t; )]. Appenix B. contains the etails, an the resulting for multifactor GATSMs appears to begin with Langetieg (980). 7 However, the speci cation coul reaily be extene to the essentially a ne market prices of risk from Du ee (00); i.e. (t) 0 + s (t) in obvious matrix notation. 8 The literature, incluing Chen (995), often speci es term structure expressions where terms analogous an relate to n n G ( n ; ) are subtracte. However, in such speci cations the market prices of risk estimates are negative, so the e ect is equivalent to my speci cation. Dai an Singleton (00) table is an example that obtains negative market price of risk estimates, an Lun (00) p. notes that Vasicek (977) uses the opposite sign for the market price of risk (which he calls q instea of our )..

15 expression for shaow-gatsm forwar rates is: where: the matrix () is: f (t; ) NX n + [s n (t) n ] exp ( n ) n + NX n n G ( n ; ) n Tr [ () ] (7) G ( n ; ) n [ exp ( n )] (8) ij () ij i j i j G ( i ; ) G ( j ; ) (9) the matrix is ij i j, an Tr[] is the matrix trace operator. 3.3 ZLB-GATSM option e ect I erive the ZLB-GATSM option e ect z(t; ) using the close-form analytic expressions for GATSM bon prices an call option prices from Chen (995) within equation 3. Appenix B. contains the etails, an the resulting ZLB-GATSM option e ect is: z (t; ) f (t; ) N f (t; )! () +! () p exp f (t; )! () where N[] is the stanar univariate cumulative normal istribution function an! () is the annualize instantaneous option volatility, i.e.:! () lim (; + ) v ux t N NX NX n G ( n ; ) + mn m n G ( m + n ; ) (3) n 3.4 ZLB-GATSM forwar rates m nm+ Substituting the respective results for f(t; ) an z(t; ) from equations 7 an 30 into equation gives the generic ZLB-GATSM forwar rate expression: f (t; ) f (t; ) + z (t; ) f (t; ) f (t; ) f (t; ) N N f (t; ) +! ()! () f (t; ) +! ()! () p exp p exp! f (t; )! ()!! f (t; )! () ZLB-GATSM interest rates r (t; ) an bon prices P (t; ) are obtaine using f (t; ) within the respective expressions provie in equations 4 an 5. 3 (30) (3)

16 3.5 Observations on the ZLB-GATSM framework One avantage of eveloping the ZLB-GATSM framework base on the generic GATSM is that the associate observations will apply to any particular ZLB-GATSM speci - cation, regarless of the number of factors an factor inter-relationships. As a relate point, the rst observation is then that, because any GATSM may be use to represent the shaow term structure, the ZLB-GATSM framework obviously preserves the complete exibility of the GATSM class of moels. The ZLB-GATSM framework provies the explicit moi cation z(t; ) to ensure that the ZLB is respecte by the entire ZLB-GATSM forwar rate curve at each point in time, no matter how much shaow- GATSM forwar rates evolve below zero. Speci cally, as f(t; ) ecreases to larger negative values, N[f (t; )! ()] an exp [f (t; )! ()] both approach zero, so lim f(t;)!()! f (t; ) 0. The secon observation is that the ZLB-GATSM forwar rate curve will always be a simple close-form analytic expression. That is evient from the generic ZLB- GATSM being itself compose of simple close-form analytic expressions, i.e.: () f(t; ), which is compose of scalar exponential functions of time to maturity an the state variables; ()! (), which is compose of scalar exponential functions of an the state variable innovation variances an covariances; (3) the univariate cumulative normal istribution N[f (t; )! ()], which is a stanar function that is well tabulate or reaily approximate with close-form analytic expressions; an (4) the scalar exponential function exp [f (t; )! ()]. Thir, ZLB-GATSM interest rates r (t; ) for any given time to maturity must be obtaine via numerical integration over horizon/time to maturity, a property that arises from the Gaussian context. 9 However, the complexity of such integrals remain invariant to the speci cation of the ZLB-GATSM because, as alreay note earlier, the ZLB-GATSM forwar rate curve will always be a simple close-form analytic expression. ZLB-GATSM bon prices (t; ) for any given time to maturity are just the P scalar exponential of the same univariate numerical integral negate. Fourth, the relatively straightforwar evaluation of ZLB-GATSM interest rates an bon prices along with multivariate-normal transition ensities for the ZLB-GATSM state variables means that ZLB-GATSMs will retain a large egree of tractability for empirical applications an estimations. Fifth an nally, note that ZLB-GATSM forwar rates converge to GATSM forwar rates when the latter are su ciently positive relative to term structure volatility. Specifically, as the ratio f(t; )! () increases to larger positive values, N[f (t; )! ()] approaches one an exp [f (t; )! ()] approaches zero, so lim f(t;)!()! f (t; ) f(t; ). That convergence of forwar rates means that ZLB interest rates an bon prices will similarly converge to their shaow counterparts. 9 I have trie, obviously without success, to erive close-form analytic expressions for r (t; ), an hence P (t; ). The relatively simple form of f (t; ) tantalizingly suggests the possibility of an analytic integral, but integration by parts i not prove successful an neither i brute force analytic integration via Mathematica. 4

17 4 Illustrating the ZLB-GATSM framework In this section, I illustrate the salient features of the ZLB-GATSM framework using a two-factor ZLB-GATSM base on the two-factor GATSM from Krippner (0). What I will hereafter call the ZLB-GATSM(), to be outline in section 4., is ieal for the purposes of emonstration because: () it is parsimonious enough to facilitate simple an transparent examples; () it provies su cient exibility to prouce plausible term structures (an emphasize that the ZLB-GATSM framework reaily accommoates multifactor shaow-gatsm moels); (3) it contains unit-root ynamics that result in negative shaow forwar rates an interest rates at very long horizons, thereby proviing for an illustration of how that issue is resolve within the ZLB- GATSM framework; an (4) parameter estimates for the shaow-zlb-gatsm() may be taken irectly from Krippner (0). The latter avois the complexities of a complete empirical estimation that, while obviously a esirable extension, woul be beyon the scope of the present article for the reasons iscusse in section 5. Section 4. focuses on the cross-sectional perspective of the ZLB-GATSM(). The examples illustrate how, at a given point in time, the ZLB-GATSM() transforms the shaow-gatsm term structure into ZLB-GATSM term structure. Section 4.3 focuses on the time-series perspective of the ZLB-GATSM(), showing the evolution of the estimate state variables an the associate shaow short rate obtaine from Unite States (U.S.) term structure ata. 4. The ZLB-GATSM() The shaow-gatsm() that unerlies the ZLB-GATSM() is the two-factor GATSM from Krippner (0). The shaow-gatsm() is very parsimonious because it essentially speci es long-run levels of zero for the two state variables an a zero rate of mean reversion for the rst factor. Appenix C etails how the Krippner (0) moel may be replicate within the generic shaow-gatsm speci cation from section 3.. The resulting shaow-gatsm() forwar rate expression, with the non-zero rate of mean reversion for the secon factor set to for notational convenience, is: f (t; ) s (t) + s (t) exp ( ) + + G (; ) [G (; )] G (; ) (33) an the ZLB-GATSM() annualize instantaneous option volatility is: q! () + G (; ) + G (; ) (34) To brie y provie the intuition of the ZLB-GATSM(): s (t) is the Level component of the shaow forwar rate, an innovations to s (t) shift the shaow forwar rate curve equally at all horizons/times to maturity); an s (t) exp ( ) is the Slope component, an innovations to s (t) shift short-horizon/time to maturity shaow forwar rates by more than long-horizon/time to maturity shaow forwar 5

18 rates. 0 Following the iscussion in Krippner (0) pp. -4, the rst line of equation 33 represents the expecte path of the shaow short rate (as at time t an as a function of horizon ) which I hereafter enote concisely as E t [r (t + )], the secon line represents risk premiums ue to the combine e ect of the quantities an market prices of risk (i.e. innovation volatilities an the compensation require by the market to accept the unanticipate price e ects associate with those innovations), an the thir line represents the volatility e ect in the shaow forwar rate which captures the in uence of volatility on expecte returns ue to Jensen s inequality (i.e. the expecte compoune return from investing in a volatile short rate is less than the compoune return from investing in the expecte [or mean] short rate). The parameters for the shaow-gatsm() are the point estimates of the twofactor GATSM from Krippner (0), p., table 3, i.e.: 0:3884, 0:435, 0:895, 0:07, 0:050, an 0:4098. ZLB-GATSM() forwar rates are therefore e ne by equations 33 an 34, the parameter set A f; ; ; ; ; g, an the ZLB-GATSM expression in equation 3, which I repeat here for convenience:! f (t; ) f (t; ) f (t; ) f (t; ) N +! () p exp (35)! ()! () R ZLB-GATSM() interest rates for any given time to maturity are r (t; ) f (t; ). I calculate the latter by numerical integration using the rectangular/miorinate rule with constant increments, i.e.: 0 " r (t; ) IX f t; i # (36) i Equation 36 conveniently turns into a arithmetic mean, because I, so, I an then. Therefore: I I r (t; ) I IX i f t; i mean ff (t; 0:5) ; : : : ; f (t; 0:5)g (37) I ensure all integrals are accurate to within 5e-8, which is 5/0000th of a basis point. 0 Diebol an Li (006) contains further iscussion on the intuition of the Level an Slope components from the perspective of the Nelson an Siegel (987) moel, which is interprete in Diebol an Li (006) as a non-arbitrage-free latent three-factor (i.e. Level, Slope, an Curvature factors) ynamic term structure moel. Christensen, Diebol, an Ruebusch (0) is the arbitrage-free analogue of Diebol an Li (006). The GATSM() estimates from Krippner (0) are estimate from the same ataset to be outline in section 4.3 of the present article, but over the perio 988 to 00. One percentage point is 0.0 an one basis point is 0.0 of a percentage point, so 5e-8 5e-4 basis points. Note that 5e-8 is an arbitrary choice, but is certainly small enough to ensure the examples will not be unuly in uence by the metho of numerical integration. A value of 0:005 years prove suitable for obtaining my 5e-8 accuracy threshol in all of the examples presente in this article. 6

19 For comparison to the ZLB-GATSM() interest rates, I also calculate shaow- GATSM() interest rates using the following close-form analytic expression: r (t; ) s (t) + s (t) G (; ) G (; ) + G (; ) + G (; ) exp ( G (; ) ) (38) R which I erive via x(t; ) for each component of the shaow-gatsm() forwar 0 rate expression in equation Cross-sectional/time-to-maturity perspective Figure shows how the ZLB-GATSM framework accommoates a materially negative shaow short rate an its associate term structure. Speci cally, in this example I set the ZLB-GATSM() state variables to s (t) 0:05 an s (t) 0:0, which gives a shaow short rate value of r(t) s (t) + s (t) 0:05, or 5 percent. As inicate in the top-left subplot, the expecte path of the shaow short rate E Q t [r (t + )] rises from its initial value of 5 percent to ve percent (the value of s (t), which sets the long-horizon level for E Q t [r (t + )]). Regaring intermeiate horizons, E t [r (t + )] remains negative for aroun years in this example. The shae areas are respectively one an two stanar eviation regions for the Gaussian probability ensities that represent, as at time t, the expecte istribution of future values of the shaow short rate r(t + ) as a function of horizon. The mile-left subplot shows the shaow forwar rate f(t; ) an the option effect z(t; ). f(t; ) has a similar pro le to E t [r (t + )] but i ers by the shaow risk premium an volatility e ect components. Regaring z(t; ), there are two points of note: () z(t; ) takes on its highest value for 0, which in turn re ects the certainty that r(t) is negative (so the option to hol physical currency is in the money an exercise, thereby o setting the negative return that investors woul otherwise face on the shaow term structure in the absence of the option e ect from the availability of physical currency); an () z(t; ) eclines but remains materially positive for all horizons shown. That pattern in turn re ects the falling but still material probabilities of r(t + ) remaining negative over each horizon, as alreay inicate by the probability ensities for the shaow short rate. The bottom-left subplot shows R the shaow interest rate r(t; ) an the interest rate option e ect component z(t; ). These quantities show similar patterns to 0 f(t; ) an z(t; ), but being cumulative averages of their forwar rate counterparts, the interest rate components evolve more graually by horizon/time to maturity. The mile-right subplot shows the shaow forwar rate f(t; ) along with the associate ZLB forwar rate f (t; ). The important points to note are: () f (t; ) remains essentially at zero until aroun 0.5 years (an it never takes on negative values); an () f (t; ) remains istinctly above f(t; ) for all horizons, with the i erence being 7

20 the option e ect z(t; ) alreay iscusse. The bottom-right subplot shows shaow interest rates r(t; ) an ZLB interest rates r (t; ), which again have similar but more graual pro les relative to their forwar rate counterparts. Figure : Perspectives on the ZLB-GATSM() term structure with s (t) 0:05 an s (t) 0:0, so r(t) 5 percent. Figure shows that the ZLB-GATSM() term structure associate with a zero shaow short rate is similar to the shaow-gatsm() term structure. In this example, I set the ZLB-GATSM() state variables to s (t) 0:05 an s (t) 0:05. Those values result in a current shaow short rate of r(t) s (t)+s (t) 0, or 0 percent, an E t [r (t + )] again rises to ve percent. By comparison to the example from gure, forwar rates an interest rates are less elevate relative to their shaow counterparts. 8

21 That re ects the lower option e ect z(t; ), which in turn re ects the lower probabilities of r(t + ) becoming negative over short an moerate horizons compare to gure. Figure : Perspectives on the ZLB-GATSM() term structure with s (t) 0:05 an s (t) 0:05, so r(t) 0 percent. Figure 3 shows that the ZLB-GATSM() term structure associate with a positive shaow short rate is almost ientical to the shaow-gatsm() term structure for short horizons. In this example, I set the ZLB-GATSM() state variables to s (t) 0:05 an s (t) 0:00. Those value results in a current shaow short rate of r(t) s (t)+s (t) 0, or 0 percent, an E t [r (t + )] remains constant at ve percent. By comparison to the example from gure, forwar rates an interest rates i er much less from their shaow counterparts for horizons/times to maturity out to aroun one year. The latter re ects a very low option e ect z(t; ) for short horizons, which in turn re ects very 9

22 low probabilities of r(t + ) becoming negative over short horizons. However, gures an 3 become similar from horizons/times to maturity beyon aroun three years. That re ects the ientical Level component for the two examples an the ominance of that component (relative to the Slope component) for moerate horizons an beyon. Figure 3: Perspectives on the ZLB-GATSM() term structure with s (t) 0:05 an s (t) 0:00, so r(t) 5 percent. As somewhat of an asie to the main point of this section, an mainly out of curiosity, it is worth noting that the ZLB-GATSM framework also accommoates negative shaow forwar rates an interest rates that can arise for very long maturities when the shaow-gatsm has very persistent (i.e. slowly mean-reverting) state variables. The shaow-gatsm() is an example of such a persistent shaow-gatsm because it contains a unit root process for the Level state variable s (t) (which was obtaine 0