Backus, Foresi, Zin (1998) (Arbitrage Opportunities in Arbitrage-Free Models of Bond Pricing)

8/3/2019 Backus, Foresi, Zin (1998) (Arbitrage Opportunities in Arbitrage-Free Models of Bond Pricing)

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Arbitrage Opportunitiesn Arbitrage-Freeo d e l so f B o n d P r i c i n gDavid BACKUSStern School of Business, New YorkUniversity,New York,NY 10012-1126 ([email protected])Silverio FORESISalomon Brothers nc.,New York,NY 10048Stanley ZINGraduateSchool of IndustrialAdministration,arnegieMellonUniversity, ittsburgh,PA 15213-3890

Mathematicalmodels of bondpricingareusedby both academics nd WallStreetpractitioners,withpractitionersntroducingime-dependentarameterso fit "arbitrage-free"odels o selectedassetprices.Weshow, n a simpleone-factoretting, hat heabilityof suchmodels o reproducesubsetof securitypricesneed not extend o state-contingentlaimsmoregenerally.Weargue hatthe additional arametersf arbitrage-free odelsshouldbe complemented y close attentionofundamentals, hichmight ncludemeanreversion,multiple actors, tochasticvolatility,and/ornonnormalnterest-rate istributions.KEY WORDS: Fixed-income erivatives; ptions;Pricingkernel;State-contingentlaims.

Since Ho and Lee (1986) initiated work on "arbitrage-free" models of bond pricing, academics and practitionershave followed increasingly divergent paths. Both groupshave the same objective-to extrapolate from prices of alimited range of assets the prices of a broader class of state-contingent claims. Academics study relatively parsimoniousmodels, whose parametersare chosen to approximate"aver-age" or "typical"behavior of interest rates and bond prices.Practitioners, on the other hand, use models with moreextensive sets of time-dependent parameters, which theyuse to match current bond yields, and possibly other assetprices, exactly.To practitioners, the logic of this choice is clear: Theparsimonious models used by academics are inadequateforpractical use. The four parameters of the one-factor Va-sicek (1977) and Cox-Ingersoll-Ross (1985) models, forexample, can be chosen to match five points on the yieldcurve (the four parameters plus the short rate) but do notreproduce the complete yield curve to the degree of ac-curacy required by market participants. Even more gen-eral multifactor models cannot generally approximatebondyields with sufficient accuracy. Instead, practitioners relyalmost universally on models in the Ho and Lee (1986) tra-dition, including those developed by Black, Derman, andToy (1990), Black and Karasinski (1991), Cooley, LeRoy,and Parke (1992), Heath, Jarrow,and Morton (1992), Hulland White (1990, 1993), and many others. Although analyt-ical approaches vary across firms and even within them, theBlack-Derman-Toy model is currentlyclose to an industrystandard.

Conversely, academics have sometimes expressed worrythat the large-parametersets of arbitrage-freemodels maymask problems with their structure.A prominent exampleis Dybvig (1997), who noted that the changes in parametervalues required by repeated use of this procedure contra-

dicted the presumption of the theory that the parametersare deterministic functions of time. Black and Karasinski(1991, p. 57) put it more colorfully: "When we value theoption, we are assuming that its volatility is known andconstant. But a minute later,we startusing a new volatility.Similarly,we can value fixed income securities by assumingwe know the one-factor short-rateprocess. A minute later,we startusing a new process that is not consistent with theold one." Dybvig argued that these changes in parametervalues through time implied that the framework itself wasinappropriate.

We examine the practitioners' procedure in a relativelysimple theoreticalsetting, a variantof Vasicek's (1977) one-factor Gaussian interest-ratemodel that we refer to as thebenchmarktheory.Ourthoughtexperimentis to apply mod-els with time-dependent drift and volatility parameters toasset prices generated by this theory. We judge the modelsto be useful, in this setting, if they are able to reproduceprices of a broadrange of state-contingent claims. This ex-periment cannot tell us how well the models do in practice,but it allows us to study the role of time-dependentparam-eters in an environment that can be characterizedprecisely.We find, in this environment, that, if the world exhibitsmean reversion, then the use of time-dependent parametersin a model without meanreversioncan reproduceprices of alimited set of assets but cannot reproducethe prices of gen-eral state-contingent claims. In this sense, these arbitrage-free models allow arbitrage opportunities: A traderbasingprices on, say, the Black-Derman-Toy model can be ex-ploited by a trader who knows the true structure of theeconomy.

9 1998 American Statistical AssociationJournal of Business &Economic StatisticsJanuary 1998, Vol.16, No. 113


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14 Journalof Business &EconomicStatistics,January1998A strikingexampleof mispricingn thissetting nvolvesoptionson long bonds.Optionsof this type vary acrosstwo dimensionsof time,the expirationdate of the optionandthematurity f thebondonwhichtheoption s written.TheBlack-Derman-Toymodelhasa one-dimensionalrrayof volatility parametershat can be chosen to reproduceeither hepricesof optionswithanyexpiration ateon one-periodbondsor of optionswith commonexpirationdateson bonds of any maturity.But, if the worldexhibitsmeanreversion,his vectorcannot eproducehe two-dimensionalarrayof pricesof bondoptions.If thevolatilityparametersarechosento matchpricesof optionson one-periodbonds,thenthe modeloverprices ptionson long bonds.Althoughwe think that meanreversions a useful fea-ture in a model, it illustratesa moregeneralpoint-thatmisrepresentationsf fundamentals,whatever heir form,cannotgenerallybe overcomeby addingarraysof time-dependentparameters.Fromthis perspective, he role ofacademicresearch s to identifyappropriateundamentals,whichmightincludemultiple actors,stochasticvolatility,and/ornonnormalnterest-ratemovements.We conjecturethatarbitrage-freemodelswith inaccuracies long any ofthesedimensionswill mispricesome assets as a result.

1. A THEORETICALFRAMEWORKWe use a one-factorGaussian nterest-ratemodel as a

laboratoryn whichto examine hepractitioners' rocedureof choosing ime-dependentarameterso fitabond-pricingmodelto observed ssetprices.The model s a closerelativeof one describedby Vasicek(1977).Although n some re-spects t is simpler han hose usedby practitioners,ts log-linearstructures extremelyuseful in clarifying he rolesplayedby its variousparameters.To fix the notation, et bVbe the price at date t of azero-coupon ond of maturityn, the claim to one dollaratdate t + n. By conventionb - 1 (one dollartodaycostsone dollar). Bond yields are y7 = -n-1 log bZandforwardrates are

ft" = log(bt"/bn+). (1)We label the short rate rt = yl f,.We characterizessetprices n our benchmarkheory,orlaboratory,with a pricingkernel,a stochasticprocess gov-erningpricesof state-contingentlaims.Existenceof sucha process is guaranteedin any arbitrage-freeenvironment.We describe the kernel for our theoretical environment intwo steps. The first step involves an abstract state variablez, whose dynamics follow:

Zt+i = pzt + (1 - p)6 +et+l= zt + (1 - cp)(6- zt) +et+1, (2)with {et} distributed normally and independently withmean 0 and variance a2. The parameter controls meanreversion: With = 1, the state follows a random walk,but with values between 0 and 1, the conditional mean offuture values of z converges to the unconditional mean 6.Step 2 is the pricing kernel m, which satisfies

-log mt+l = zt +/ACt+l. (3)The parameterA,whichwe refer to as theprice of risk,determineshecovariance etween nnovationso thekerneland the state and thus the riskcharacteristics f bonds andrelatedassets.Given a pricingkernel,we derivepricesof assetsfromthepricingrelation

1 = Et(mt+xRt+x), (4)which holds for the grossreturnRt+l on anytradedasset.Because the one-period eturnon an n + 1-periodbondisbi+ /b"+1, hepricingrelationgives us

bn+1 = Et(mt+lbtn,+), (5)whichallowsus tocomputebondpricesrecursively, tartingwith the initial condition bt = 1.Considera one-periodbond. FromEquation 5) andtheinitial conditionb? = 1, we see that the price is the con-ditionalmean of thepricingkernel:b1= Etmt+l. Becausethe kernel s conditionallyognormal,weneed hefollowingproperty f lognormal andom ariables: f log x is normalwith meanp andvariancea2, thenlog E(x) = p + a2/2.FromEquation 3) we see that log mt+l has conditionalmean -zt andconditionalvariance Aa)2.The one-periodbond price satisfies log b1 = -zt + (Aa)2/2 and the shortrateis

rt = - log b1= zt - (Aa)2/2. (6)Thus,the shortrate r is the state z with a shift of origin.The meanshortrate is 6 - (Aa)2/2,whichwe denoteby pin the restof the article.The stochastic rocess orz, Equation2),impliessimilarbehavior or the shortrate:

rt+ = rt + (1 - P)(p - rt) + Et+l, (7)a discrete-time ersionof Vasicek's 1977)short-rate iffu-sion. Futurevalues of the shortrateare

for n _ 1, which yields conditional first and second mo-ments ofEt(rt+n) = rt + (1 - (P)(p - rt) (8)

andvart(rt+n) 2 2(nj- 2 1 2 . (9)j=1

We return to these formulas later.Prices of long bonds follow from (5); details are pro-vided in Appendix A.1. Their properties are convenientlysummarizedby forwardrates, which are linear functions of


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Backus,Foresi,andZin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 15the shortrate:

fth= rt + (1- pn)(p - rt) + [A2 (A? ] 2/2(10)

for all n > 0. Given forwardrates,we can computebondprices andyields from their definitions.The rightside ofEquation 10) has a relativelysimpleinterpretation.f wecompare t to (8), we see thatthe first two terms aretheexpectedshortrate n periods n the future.We refer to thelast term as a riskpremiumand note that it dependsonthreeparameters-themagnitudeof risk (a), the priceofrisk (A),and meanreversion(po).Both the Ho and Lee (1986) and Black, Derman,andToy (1990)modelsarecapableof reproducing narbitraryforward ate curve(equivalently,ieldsorbondprices), n-cludingone generatedby a theoreticalmodellike this one.An issue we address ater s whether hiscapability xtendsto morecomplexassets. With this in mind,considera Eu-ropeancall optionat datet, withexpirationdate t + 7 andstrikeprice k, on a zero-couponbond with maturityn atexpiration.Giventhe lognormalityof bond prices in thissetting, he call priceis givenby theBlack-Scholes(1973)formula,cn" = br+nN(dx) - kb[N(d2), (11)

whereN is the cumulativenormaldistributionunction,log[bl+n/(bVk)]+ v2 /2V1-,n

d2 dl - vr,n,and the option volatility is

v2,n= vartlogb ) = 1-2 1- o 2. (12)See Appendix A.4. Jamshidian (1989) reported a similarformula for a continuous-timeversion of the Vasicek model.The primary difference from conventional applications ofthe Black-Scholes formulais the role of the mean-reversionparametern in (12).

2. PARAMETERVALUESThe parametervalues used in the following numerical ex-amples come from an informal moment-matching exercisebased on propertiesof monthly yields for U.S. governmentsecurities computed by McCulloch and Kwon (1993). Someof the properties of these yields are reportedin Table 1 forthe sample period 1982-1991.We choose the parameters to approximate some of thesalient features of bond yields using a time interval of one

Table1. Propertiesof U.S. Government ondYieldsMaturity Mean Standarddeviation Autocorrelation1 month 7.483 1.828 .9063 months 7.915 1.797 .9206 months 8.190 1.894 .9269 months 8.372 1.918 .92812 months 8.563 1.958 .93224 months 9.012 1.986 .940

36 months 9.253 1.990 .94348 months 9.405 1.983 .94660 months 9.524 1.979 .94884 months 9.716 1.956 .952120 months 9.802 1.864 .950NOTE: Dataaremonthlystimatesofannualized,ontinuouslyompounded,ero-couponU.S.governmentondyieldscomputed yMcCullochnd Kwon1993).ThesampleperiodsJanuary1982to February992.

month.FromEquation7)we see thatp is theunconditionalmeanof theshortrate,so we set it equal o thesamplemeanof the one-month ieldin Table1, 7.483/1,200. (The1,200convertsan annualpercentage ateto a monthlyyield.)Themean-reversionarameter o s the firstautocorrelationfthe shortrate.InTable1 the autocorrelations .906, so weset ?pequalto this value. This indicatesa high degreeofpersistence n the short ratebut less thanwith a randomwalk. The volatilityparameter is the standard eviationof innovations o the shortrate,which we estimatewiththe standardrrorof the linearregression 7).Theresult sa = .6164/1,200. Thus the values of (p,ua, o) are chosento match he mean,standard eviation,andautocorrelationof the shortrate.We choose the finalparameter,he priceof riskA,to approximatehe slopeof the yieldcurve.Notefrom(10)that mean forward ates, n the theory,areE(f) = p + A2 A + a2/2.

This tells us that to producean increasingmeanforward-rate curve (implyingan increasingmeanyield curve)weneed A to be negative.Thepriceof riskparameter,n otherwords,governsthe averageslope of the yield curve.Oneway of fixingA, then,is to select a value thatmakesthetheoreticalmean yield curve similarto the samplemeanyield curve,given our chosen values for the other threeparameters.An example s pictured n Figure1, in whichwe see that A= -750 produces theoretical mean yields (theline in the figure) close to their sample means (the stars)for maturitiesbetween one month and 10 years. With morenegative values, the mean yield curve is steeper, and withless negative (or positive) values, the yield curve is flatter(or downwardsloping).Thus we see that all four parameters are required forthe theory to imitate the dynamics of interest rates and theaverage slope of the yield curve. We use these benchmarkvalues later to illustrate differences in prices across bond-pricing models.

3. HO AND LEE REVISITEDWe turn now to the use of time-dependentparameterstofit theoretical models to observed asset prices. We apply,


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16 Journalof Business & EconomicStatistics,January1998

9.5

9-

8-

0 20 40 60 80 100 120MaturitynMonthsFigure1. Mean Yields n Theoryand Data. The line is theory, hestarsare datapoints.

in turn, analogs of the models of Ho and Lee (1986) andBlack, Derman, and Toy (1990) to a world governed by thebenchmarktheory of Section 1.Our first example is a Gaussian analog of Ho and Lee's(1986) binomial interest-ratemodel. The analog starts witha state equation,zt+l = zt + at+l + rlt+, (13)

with time-dependent parameters {at} and normally and in-dependently distributedinnovations {t } with mean 0 andconstant variance 32. The pricing kernel is- log mt+ = zt + 7r7t+1. (14)

We use different letters for the parameters than in thebenchmarktheory to indicate that they may (but need not)takeon differentvalues. OurHo andLee analogdiffers fromthe benchmarktheory in two respects. First, the short-rateprocess does not exhibit mean reversion, which we mightthink of as setting ? = 1 in Equation (2). Second, thestate equation (13) includes time-dependent "drift"param-eters {at}.Given Equations (13) and (14), the pricing relation (5)implies a short rate rt = zt - (y3)2/2 and forward rates

fZZ= rt + at+j + [.2 - (y + n)2]/32/2 (15)j= 1for n > 1. See Appendix A.2. Note that (10) differs from(15) of the benchmark in two ways. One is the impact ofthe short rate on long forwardrates. A unit increase in r isassociated with increases in fn of 1 in Ho andLee, but (1-p") < 1 in the benchmark. The other is the risk premium,the final term in Equation (15).Despite these differences, time-dependent drift parame-ters allow the Ho and Lee model to reproduce some ofthe features of the benchmarktheory. One such feature isthe conditional mean of future short rates. The future short

rates implied by this model aren

rt+n rt + Y (at+j + rlt+j), (16)j=1which implies conditional means of

nEt(rt+n) = Tt+ Zat+j (17)j=1

for n > 0. If we compare this to the analogous expressionfor the benchmarktheory,Equation(8), we see that the twoare equivalent if we setn

Z at+, = (1 - yn)( - Tt). (18)j=1Thus, the time-dependent drift parametersof the Ho andLee model can be chosen to imitate this consequence ofmean reversion in the benchmarktheory.

In practice it is more common to use the drift parametersto fit the model to the current yield curve. To fit forwardratesgenerated by the benchmarktheory,we need [compare(10) and (15)]n

Zat+j - (1- ,9n)4I - rt)j=1+ [A2 (A+ (1 - )/(1 - (p))2]a2/2- [Y2 ( + n)2132/2. (19)

The drift parameters implied by (18) and (19) are, in gen-eral, different. Because1 - (pnlim =, no--1 1 -

we can equate the two expressions when ? = 1 by setting3 = a and y = A. But when 0 < p < 1 the two expressionscannot be reconciled. This is evident in Figure 2, in which

4-3 ,' MatchExpectedShortRates

-3 MatchCurrent ondYields4

-50 5 1 15 20 25 30 35 40 45 50Time Horizon in MonthsFigure 2. TwoChoices of Ho and Lee DriftParameters. One matchesthe currentbond yields. The other matches expected future short rates.


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Backus,Foresi,and Zin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 17we graph he two choices of cumulativedriftparameters,n=, at+j, using the parameters stimated n Section 2,with / = a, -y A,and r = 3.0/1,200. The drift parametersthatreproducehe conditionalmeanconverge apidlyas theeffectsof meanreversionwearoff. Butthe driftparametersthatfit the currentyield curveget steadilysmalleras theyoffset the impactof maturityon the risk premium n thismodel.Thisresults, or therangeof maturitiesn thefigure,in a declining erm structure f expected utureshortrates,whereas hebenchmarkmpliesthe reverse.Thediscrepancyn thefigurebetween he twochoicesofdriftparameterss a concreteexampleof the following:

Remark1. The parameters f the Ho and Lee modelcanbe chosen o match he current ieldcurveorthecondi-tionalmeansof future hortrates,buttheycannotgenerallydo both.Thisproperty f theHo andLeemodel s a hintthat ime-dependentdriftparameters o not adequately apture heeffectsof meanreversionn thebenchmarkheory.Dybvig(1997)madea similarobservation.A closerlook indicatesthat hedifficultyies in thenonlinearnteractionn theriskpremiumbetweenmeanreversionand the priceof risk.Inthe benchmarkheory,the risk premiumon the n-periodforward ate[seeEq. (10)]is

A2A- A+ - 2/2.In the Ho andLee model,Equation15),the analogous x-pression is [y2 - (y + n)2]032/2. If we choose - = A and0 = a, the two expressions are equal for n = 1, but theymove apartas n grows.The discrepancy otedin Remark1 is a directconsequence.A similar omparison f conditional ariances lso showssigns of strain.The conditionalvariancesof futureshortratesimplied by the Ho andLee model are vartrt+n) =np2. If we compare histo the analogous xpressionn thebenchmarkheory,Equation 9),we see thattheygenerallydiffer when Wp 1. With p < 1 and 0 = a, the condi-tionalvariances re the samefor n = 1,but for longertimehorizonsthey are greater n the Ho and Lee model. Wesummarize his discrepancyn the following:

Remark2. The parameters f the Ho and Lee modelcannot be chosen to reproduce the conditional variances offuture short rates.Thus, we see that additional drift parameters allow theHo and Lee model to imitate some of the effects of meanreversion on bond yields. They cannot, however, reproducethe conditional variances of the benchmarktheory. Dybvig(1997) summarized this feature of the Ho and Lee modelmore aggressively: "[T]he Ho and Lee model starts withan unreasonable implicit assumption about innovations ininterest rates,but can obtain a sensible initial yield curve bymaking an unreasonableassumptionaboutexpected interestrates. Unfortunately,while this ... give[s] correct pricing ofdiscount bonds ..., there is every reason to believe that itwill give incorrectpricing of interest rateoptions."Dybvig's

120.100

20

o

0 5 10 15 20 25Time o ExpirationnMonthsFigure3. The Premium f Ho and Lee CallPrices Over he Bench-markPrice.

intuition boutoptions s easilyverified.TheBlack-Scholesformula,Equation11),applies o theHo and Lee modelifwe use optionvolatility

v,n vart(1ogb,) = T(n3)2. (20)This formulacannotbe reconciledwith that of thebench-marktheory,Equation 12), for all combinations f T andn unless p = 1. If we choose the volatility parameter0 toreproducehe optionvolatilityv1,l of a shortoptionon ashortbond, henwe overstate hevolatilitiesof longoptionson long bonds. As a result,the model overvaluesoptionswith moredistantexpirationdatesand/orlongerunderly-ing bonds.

We see in Figure3 thatcall pricesc,'1 impliedby theHo and Lee model can be substantially igherthanthosegeneratedby the benchmarkheory.The figureexpressesthe mispricingas a premium f the Ho and Lee priceoverthe pricegeneratedby the benchmarkheory.Benchmarkpricesare based on the parameter alues of Section 2. HoandLeepricesarebasedondriftparametershatmatchcur-rent bondprices,Equation 19),and a volatilityparameter0 = a thatmatches the option volatility v1,1 of a one-periodoptionon a one-periodbond.Both areevaluatedat strikeprice k = b/b+n. For T = 1 the two models generatethe same call price,but for optionswith expirationdates12 months in the future the Ho and Lee price is more than50% higher.

4. BLACK,DERMAN,AND TOYREVISITEDBlack, Derman, and Toy (1990) extended the time-dependentparametersof Ho and Lee to a second dimension.They based bond pricing on a binomial process for the log-arithm of the short rate in which both drift and volatilityare time dependent.We build an analog of their model thatretains the linear, Gaussian structure of previous sectionsbut includes these two sets of time-dependent parameters.Given a two-parameterdistribution like the normal, time-dependentdrift and volatility can be used to match the con-ditional distributionof future short rates exactly and thus


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18 Journalof Business &EconomicStatistics,January1998to mitigate he tendencyof theHo andLee modelto over-price ong options.Thequestion s whether heyalsoallowus to reproducehe pricesof otherinterest-rate-derivativesecurities.Ouranalogof theBlack-Derman-Toymodeladdstime-dependent olatility o thestructure f Section3: A randomvariable follows

Zt+1= Zt+ at+1+ 7?t+l, (21)andthepricingkernel s

- log mt+ = Zt+ yTt+1. (22)Thenewingredients thateachrqthastime-dependentari-ance 3t2.This structure iffers romthe benchmarkheoryin its absenceof meanreversion thecoefficientof 1 on ztin the stateequation)and in its time-dependent rift (thea's) andvolatility the /'s).We approachbondpricingas before.We show in Ap-pendix A.2 that the short rate is rt = -log(Etmt+1) =Zt- (y3t+1)2/2 and forward atesare

nft = rt + at+j + [72 - ( + n)2]t2+1/2j=1

n+ + n_ j)2( 2 o2+ + n - )2 ;+j - t+j+l)/2 (23)j=1for n > 1.Equation23)reduces o the HoandLee expres-sion, Equation 15), when 3t = /3 for all t. As in the HoandLee model,forward atesdiffer romthebenchmarknboththe impactof short-ratemovementson long forwardratesandthe form of the riskpremium.We can use both sets of time-dependent arametersoapproximatessetprices n thebenchmarkheory.Considerthe conditionaldistributionf futureshortrates.The shortratefollows rr+1 = rt + 2 21+ -t+2)/2 + at+1so futureshortratesare

n2 (2 _o2rt+n = rt2+t+ 1-- t+n+l )/2+ (at+j +,7t+j). (24)j=1Theirconditionalmeanand varianceare

Et(rt+n) = rt +72-(/+1i t+n+ 1)/2 +l- at+j (25)j.=1and

vart(rt+n) = 3t+j (26)j=1This model,in contrast o that of Ho andLee, is able tomatchthe conditionalvariancesof the benchmarkheory,which we do by choosingvolatilityparametershatdeclinegeometrically:

=t+j= pj-lo. (27)

This impliesconditional ariances fvart(rt+n) -o=2EW2 (X-a j)j=1

the same as Equation 9) of the theory.Similarpatternsof declining ime-dependentolatilitiesarecommonwhenthese methodsareused in practice,Black et al.'snumeri-cal example ncluded see their tableI). To match he con-ditional mean of the benchmarkheory,Equation 8), wechoose driftparametershatsatisfy

Zat+j- (1 - p)(p - rt) - 2(t+l - t+n+l)/2. (28)j=1Thuswe see, as Blacket al. (1990, p. 33) suggested, hatwe canfit the firsttwo momentsof the shortrate with two"arrays"f parameters:

Remark3. The parameters f the Black-Derman-Toymodel can be chosento reproduce he conditionalmeansand variances f futureshortrates.Given the criticalrole played by volatilityin pricingderivativeassets, this representsan essentialadvancebe-yondHo andLee.Despite time-dependent olatilityparameters,he modelcannotsimultaneouslyeproducehe conditionalmomentsof the shortrate and the forwardratesof the benchmarktheory.Thedriftparametershatreproduceheforward atecurve,Equation 10),are

- [T2- (y+ n)2]t2+1/2nS +n -) (It+j - t+j+l)/2. (29)j=1

Comparing 28) and (29), we see that the two are notequivalent,n general.We note the difference n the fol-lowing:Remark . Givenvolatilityparameters27) thatrepro-ducethe conditional ariances f futurerates,the driftpa-rametersof the Black-Derman-Toymodelcan be chosen

to matchthe currentyield curve or the conditionalmeansof futureshortrates,buttheycannotgenerallydo both.Figure4 plotsthe differencebetween(29)and(28).Theparametervalues are those of Section 2, with y = A, 3t+j =qj-l , and r = 3.0/1,200. The differences are smaller thanthose of Figure2 for the Ho andLee modelbutare nonzerononetheless.Remark4 is reminiscentof Remark1 for the Ho andLee model.Moresurprising,n ourview,is thatthe Black-Derman-Toymodel'sabilityto reproduce he conditionalvarianceof future short rates does not extendto optionprices. ConsiderEuropeancall options on zero-couponbonds. Once more the lognormalstructureof the model


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Backus,Foresi,andZin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 191.4

1.200

. > 8

C0.6-0CD0.4

0.2

0 5 10 15 20 25 30 35 40 45 50TimeHorizonnMonthsFigure4. CumulativeDifferenceBetweenBlack-Derman-ToyDriftParametersChosen to Match the CurrentBond Yieldsand ExpectedFutureShort Rates.

means that the Black-Scholesformula,Equation 11), ap-plies. If we choose driftparameterso fit the currentyieldcurve,thenanydifferencebetweencallprices n the modelandthe benchmarkheorymust lie in theiroptionvolatili-ties. The optionvolatilityfor the Black-Derman-Toy na-log is2 =n2 y 2

j=1See AppendixA.4. If we restrictourselvesto optionsonone-periodbonds (so that n = 1), we can reproduce hevolatilitiesof ourtheoretical nvironment y choosingf'sthat declinegeometricallywithtime,the same choice thatreplicates onditional ariances f futureshortrates,Equa-tion (27).The Black-Derman-Toy nalogcannot,however, imul-taneouslyreproducepricesof optionson bonds of longermaturities.f we use thegeometrically ecliningparametersof Equation 27),the optionvolatility s

-= n2 r2V-rn 1--2 ) "n0From(12)we see thatthe ratioof optionvolatilities s

BDT volatility n2benchmarkvolatility (1 + p + ... + p"n-1)2

This ratio is greater than 1 when 0 < p < 1 and n > 1 andimplies that, when the Black-Derman-Toy analog pricesoptions on short bonds correctly, it will overprice optionson long bonds. We see in Figure 5 that this mispricing getsworse the longer the maturityof the bond andis greaterthan150% for bonds with maturities of two years or more. As inFigure 3, benchmarkprices are based on the parametersofSection 2, the drift parametersof the Black-Derman-Toyanalog are chosen to reproduce the currentyield curve, thestrike price is k = bT/b+,t, and the expiration period is7 = 6. Thus we have the following:

Remark5. The parameters f the Black-Derman-Toymodelcannotreproducehepricesof call optionsonbondsfor all maturities ndexpirationdates.The difference n optionvolatilities,and hence in callprices,between the two models stems from two distinctroles played by mean reversion n determiningprices oflong bonds.Meanreversionappears, irst, n the impactof

short-rate nnovationson future short rates. We see from(9) thatthe impactof innovations n futureshortrates, nthe theory,decaysgeometricallywith the time horizonT.This feature s easilymimicked n the Black-Derman-Toymodel,as we haveseen,by usingvolatilityparametersIOtthatdecay at the same rate.The secondrole of mean re-version concerns the impactof short-ratemovementsonlong-bondprices.In the Black-Derman-Toymodel,a unitdecrease n the shortrateresults n an n unit ncrease n thelogarithmof the priceof an n-periodbond;see AppendixA.2. In the benchmarkheory,the logarithmof the bondpricerisesby only (1 +p+.. - + 9-l) = (1 -_")/(1- p);see AppendixA.1. Thisattenuationf the impactof short-rate innovationson long-bondprices is a direct conse-quenceof meanreversion. t is not, however,reproducedby choosinggeometricallydecliningvolatilityparametersand is thereforemissing from call prices generatedbythe Black-Derman-Toyanalog. Stated somewhat differ-ently, the Black-Derman-Toyanalogdoes not reproducethe hedgeratiosof the benchmarkheory.As a practicalmatter, hen,we might expectthe Black-Derman-Toyprocedureo work well in pricingoptionsonshort-termnstruments,ncludingnterest-rateaps.Forop-tions on longbonds,however, he modeloverstates he op-tionvolatilityandhence thecallprice.A commonexampleof such an nstruments a callablebond.Thisprocedurewillgenerallyoverstate he valueof thecallprovision o the is-suer, n the theoretical nvironment,ndthusunderstatehevalue of a callable ong-termbond.The discrepancy etweenoptionprices generatedby thebenchmarkheoryand heBlack-Derman-Toynalog llus-

150

CDC100

o-U)

O 50

0o 5 10 15 20 25Bond Maturity n MonthsFigure5. The Premium f Black-Derman-ToyCallPrices OvertheBenchmarkPrice.Expiration eriod7 is 6.


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20 Journalof Business &EconomicStatistics,January1998trates one difficultyof using models with time-dependentparameters-a one-dimensional ector of time-dependentvolatilityparametersannotreproducehe conditional ari-ances of bondpricesacross the two dimensionsof matu-rity andtime-to-expiration.We turnnow to a second ex-ampleof mispricing, class of "exotic"derivativeswhosereturnsdisplaydifferent ensitivity han bonds to interest-rate movements.

Consideran assetthat delivers he power0 of the priceof ann-periodbondoneperiod n thefuture.Thisasset hassomeof the flavorof derivativeswithmagnified ensitivityto interest-ratemovementsmadepopularby BankersTrustyetretains he convenientog-linearity f bondprices nourframework.As withoptions,we compareheprices mpliedby the benchmarkheory,log dZ'= -(rt + (Ao)2/2) + 0 log b'

(A+ 01 )2c 2/2 - 0(1 -p1)(/ - rt), (30)withthosegeneratedby theBlack-Derman-Toy nalog,logd =-n(rt + (7y3t+1)/2)+0logT t)2 2+ (,y+ On)2); +1/2 - 0(nat+l)n

- 0>1 [(n -- j)-(t+j+l --Ot+j)j=1

- (-y+ n - j)2(3t2j+1 3t+j)2/2].(31)

Bothfollow frompricingrelation 4) withthe appropriatepricingkernel.Suppose we choose the parametersof the Black-Derman-Toyanalogto match the conditionalvarianceoffutureshort rates [Eq. (26)], the currentyield curve [Eq.(29)], and the price of risk [y = A].Can this model re-produce he pricesof our exoticasset for all valuesof thesensitivityparameter ? The answer s generallyno. Whenn - 1, the two modelsgenerate he samepriced' for allvalues of 0, just as we saw thatthe two sets of cumula-tivedriftparameters ereinitially he same(seeFig.4 andthe discussion ollowingRemark4). Forlongermaturities,however, the prices are generally different. Figure 6 is anexample with n = 60 and r = 3.00/1,200. We see that, forvalues of 8 outside the unit interval, the Black-Derman-Toy analog overprices the exotic, although the differencebetween models is smaller than for options on long bonds.

Remark 6. The parameters of the Black-Derman-Toymodel can be chosen to match both the currentyield curveand the termstructureof volatility for options on one-periodbonds, but they cannotgenerallyreplicatethe prices of moreexotic derivatives.These examples of mispricing illustrate a more generalresult-the Black-Derman-Toy analog cannot replicate thestate prices of the benchmark theory. This result is statedmost clearly using stochastic discountfactors,

0.05

0.04

0.03

- 0.02

0 0.010 0

-0.01,-10 -8 -6 -4 -2 0 2 4 6 8 10Sensitivity arameter hetaFigure6. The Premium f Black-Derman-ToyPrices of the ExoticOverthe BenchmarkPrice.Maturity is 60.

nMt,t+n - mt+j,j=1

which definen-period-aheadtateprices.We showin theAppendix hat the benchmarkheory mpliesdiscount ac-tors- log Mt,t+, = n6 + 1 (rt - IL)

+ 1 :)A+ ?0 Et+jj=11andtheBlack-Derman-Toy nalog mplies- log Mt,t+n = n[rt + (7l3t+1)2/2]n n

+ :(n - j)at+j + (7 + n - j)rnt+j.j=1 j=1As long as (o , 1 anda , 0, the secondexpression annotbe madeequivalento the first:

Proposition1. The time-dependent rift andvolatilityparametersof the Black-Derman-Toymodel cannot bechosen to reproducehe stochasticdiscount actors of thebenchmarktheory.A proof is given in Appendix A.3. The difficulty lies inthe innovations

+ 1 -(pl t+jand (7 + n - j)rlt+j. There is no choice of volatility pa-rameters {f3t} in the Black-Derman-Toy analog that canreplicate the nonlinear interaction of mean reversion ()and the price of risk () in the benchmarktheory.In short, the time-dependent drift and volatility param-eters of a model like the Black-Derman-Toy analog can-not replicate the prices of derivative assets generated by amodel with mean reversion.


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Backus,Foresi,and Zin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 215. ARBITRAGE NDPROFITOPPORTUNITIES

In the laboratory f the benchmarkheory,the Black-Derman-Toyanalog systematicallymispricessome assets.In this section we considerstrategies hatmightbe usedto profitfromtradersusingpricesindicatedby the Black-Derman-Toymodel.One way to exploitsomeonetradingat Black-Derman-Toy pricesis to arbitragewith someonetradingat bench-markprices. If, as we have seen, a Black-Derman-Toytraderoverpricesoptionson long bondsrelative to thoseon short bonds anda benchmarkraderdoes not, then wecanbuyfrom the latterandsell to the former, husmakinga risk-freeprofit.This is as clearanexampleof arbitragesthere s. Butbecausesuchpricedifferences re so obvious,they areunlikelyto be verycommon.It is less easy to exploit a Black-Derman-Toy raderwhen trades akeplace onlyatpricesdictatedby themodel.Because Black-Derman-Toyprices are consistent with apricingkernel, heypreclude isklessarbitrage.Wecan of-ten devise dynamicstrategies,however,whosereturnsarelargerelative o theirrisk. One suchstrategy nvolves calloptionson long bonds,which (again) he Black-Derman-Toymodelgenerallyovervalues.f we sell theoption o thetraderat datet, andliquidateat t + 1, we mightexpecttomakea profit.Thisseemsparticularlyikelyin theoption'sfinalperiodbecause he option s overvalued utthebondson which theoption s writtenarenot.Before we evaluate his strategy,we needto explainhowtheBlack-Derman-Toyrader ricesassetsthroughime.AtraderusingtheBlack-Derman-Toymodelgenerally inds,both n practiceand n our theoretical etting, hat he time-dependentparametersmust be reset each period.In oursetting, supposethat the traderchooses volatilityparam-eters at date t to fit impliedvolatilitiesfrom optionsonshort bonds. As we have seen, this leadsthe trader o set{ft+1,3t+2, 3t+3,... } equalto {a, cpr,W2 ,...}. If the pa-rameterswereliterallytimedependent,henin the follow-ing period ogic dictatesthat we set {ft+2, /t+3, /t+4, ..

0.550.s55- \ RevisedVolatilityurve0.5

0.45 ',0.4

~ .35 Initial olatility urve

0.250.2

Date nMonthsFigure 7. Initial and Revised VolatilityParameters for theBlack-Derman-ToyModel.

350

300 Ho-Leew250

a 200150 -

d 100

2 50

0 Benchmark0 5 10 15 20BondMaturitynMonths

Figure8. AverageRisk-Adjusted xcess Returns.equalto {scp, p2a,V3a,... }. But if we calibrate nce moreto options on one-periodbonds, we would instead use{a, r, pV2a, ...}. This leads to a predictable pward umpin the volatilityparametersrom one periodto the next,as in Figure7. Similarly,he driftparametersmust be ad-justedeachperiod o retain hemodel'sability o reproducethe current ieldcurve.Thesechanges n parameteraluesthroughime areanindicationhat hemodel s imperfectlyimitating heprocessgenerating tateprices,buttheynev-ertheless mprovetsperformance.n thissense the internalinconsistencynotedby Dybvig(1997)is a symptomof un-derlyingproblemsbut not a problemn its ownright.Thus,our traderpricescall optionsusingdrift andvolatilitypa-rameters hatmatch,each period,the yield curve and theterm structureof volatilitiesimplied by optionson one-periodbonds[Eqs.(29)and(27),respectively].Now considera strategyagainst uch a trader f sellinga7--periodptionon ann-periodbondandbuying t back oneperiod ater.Thegrossone-period eturn romthisstrategyis

Rt+1 = -c -1,n/cT ,for T > 1, withthe convention hat an optionat expirationhas value c?'" = (b" - k)+. The profit from this strategyis not risk free, but it can be largerelative o the risk in-volved. In the world of the benchmark theory, the appro-priate adjustment for risk is given by the pricing relation(4). We measure the mean excess risk-adjusted return bya = E(mt+lRt+l) - 1, using the pricing kernel m for thebenchmarkconomy.This measure s analogouso Jensen'salpha or thecapitalassetpricingmodeland s theappropri-ate riskadjustmentn this setting.For the trading trategyjust describedand the parameter alues of Section2, thereturnsarein theneighborhoodf severalhundred ercentperyear;see Figure8. Thereturns n the figurewere com-puted by Monte Carlo and concern two-period (7 = 2) op-tions onbonds or maturitiesn) between1 and24 months.The returnsaregreaterwith 7 = 1 but declinesignificantlyas 7 increases. For 7 > 6, the overpricing is difficult to


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22 Journalof Business &EconomicStatistics,January1998exploitbecausethe overpricing f a five-period ptionthefollowingperiod s almostas great.

6. MEANREVERSION NDOTHERFUNDAMENTALSWehaveseen thatthetime-dependentrift andvolatilityparametersf analogsof the Ho andLee (1986)andBlack-

Derman-Toy1990)models cannotreproducehepricesofstate-contingentlaims generatedby a model with meanreversion.This thoughtexperimentdoes not tell us howwell thesemodelsperformn practice,butit indicates hatextraparameters re not a solution to all problems:Theyneed to be used in the contextof a structurewith soundfundamentals.We focusedon meanreversionbecausewe find it an ap-pealingfeature n a bond-pricingmodel,yet it is missingfrom the most popularbinomial nterest-ratemodels. Theestimatedautocorrelationf .906reportedn Table1 is notsubstantially ifferent rom 1, but a valueof 1 has, in thebenchmarkheory, wo apparently ounterfactualmplica-tionsforbondyields.Thefirst s thatthe meanyieldcurveeventuallydeclines tominus nfinity)withmaturity.nfact,yield curvesare,on'average,upward loping,Figure1 be-ing a typicalexample.A secondimplicationwas pointedoutby GibbonsandRamaswamy1993)andmaybe moretelling:With o close to 1,averageyieldcurvesexhibitsub-stantiallyess curvaturehanwe see in the data.Anexamplewith? = .99 is picturedn Figure9, in which we have setA= -200 to keepthe theoretical10-yearyield close to itssamplemean. These two implicationsllustrate he addedpowerof combining ime-seriesand cross-sectionnforma-tion,andsuggestto us thatrandom-walkmodels overstatethepersistenceof the short-termate of interest.For thesereasons,we feel that mean reversion s suggestedby theproperties f bondprices.Forsimilarreasons,Blackand Karasinski1991),Heath(1992),and Hull and White(1990, 1993)developedwaysof introducingmean reversion nto arbitrage-freemodels.Despite hesedevelopments,meanreversions generally g-

109.5

8.5

0 o20 40 60 80 o100 120Maturity n MonthsFigure9. TheoreticalMean YieldCurveWhencp= .99. The line istheory, he starsare datapoints.

noredby all but themostsophisticatedmarketparticipants.Financialprofessionalsdo not generallyrevealtheirmeth-ods,buttheycanbe inferred romothersources.Onesourceis bookson fixed-incomemodeling.Tuckman's1995)bookwas aimed at quantitativelyrientedM.B.A. studentsandwas used in SalomonBrothers'sales andtrading rainingcourse,yethe devotedonlyfourpagesto modelswith meanreversion.Sundaresan1996) targeteda similaraudienceanddisregardedmeanreversion ltogether. erhaps bettersource s Bloomberg 1996),theleadingpurveyor f fixed-income information ervices. Calculationson Bloombergterminals f the "fairvalue"of interest-rate erivatives rebasedon a lognormal nalogof the Ho andLeemodel,andcalculations f "option-adjustedpreads"or callablebondsuse the same model as the default Bloomberg1996),withBlacket al. (1990)andHullandWhite(1990)availableasalternatives.We think, however, hat the centralissue is not meanreversionbut whetherpricing models-arbitrage-freeornot-provide a good approximationo the fundamentalsdrivingbondprices.Meanreversion,n thiscontext, s sim-ply anexampleof a fundamentalhatcannotbe mimickedby time-dependentriftandvolatilityparameters. secondexample s multiple actors.Workby, amongothers,Bren-nanandSchwartz1979),ChenandScott(1993),DuffieandSingleton inpress),Dybvig(1997),Garbade1986),Litter-manandScheinkman1991),andStambaugh1988)clearlyindicates he benefitsof additional actors n accountingorthe evolutionof bondprices through ime. LongstaffandSchwartz1992)arerepresentativef a growingnumber fstudiesthatidentifya second factor with volatility,whosevariationhrough ime is apparentn the pricesof optionson fixed-incomenstruments. et another xamples the fattails in weekly interest-rate hangesdocumentedby Das(1994),who introducedumpsinto the standard iffusion-basedmodels.Wewouldguessthatmodelsthat gnoreanyor all of these fundamentalswill, as a directconsequence,produce naccurate tateprices.We argue,in short,that fundamentalsmatterand thatpoorchoice of fundamentalsannotgenerallybe rectifiedby judicioususe of time-dependent arameters.Whether urexamplesof mispricing an be usedto directtrading trategiesnrealistic ettingsdepends nthe relativemagnitudes f themispricing nd he inevitable pproxima-tionerrorsof moreparsimoniousmodels.We wouldnot bewilling, at present, o bet our salaries on our benchmarktheory.Nevertheless,we thinkthe exercise ndicates hat tis importanto get the fundamentalsight.In ourthoughtexperiment,undamentalswererepresented y the degreeof meanreversion n the shortrate,and we saw that amis-take in this dimension ould lead to largepricingerrorsonsome securities.The extratime-dependent arameters,notherwords,are not a panacea:Theyallowus to reproducea subsetof assetpricesbutdo notguarantee ccurate ricesfor the full rangeof interest-rate-derivativeecurities.Inmoregeneralsettings,we expecta model with n arraysofparameterso be able to reproduce he term structureofprices of n classes of assets,but if the fundamentals rewrong,therewill be some assets that aremispriced.


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Backus,Foresi,and Zin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 23Thedifficultynpractices thateven the bestfundamentalmodelsprovideonly a roughapproximationo the marketpricesof fixed-income erivatives.That eaves us in theun-certain erritorydescribedby Black and Karasinski1991,p. 57): "[One]approach s to search for an interestrateprocess generalenoughthatwe can assume t is trueandunchanging. ... While we may reach this goal, we don'tknowenough o usethisapproachoday."Bestpractice,wethink, is to combinecurrentknowledgeof fundamentalswith enoughextraparameterso makethe approximationadequateor tradingpurposes.

7. FINALTHOUGHTSWe have examined the practitioners'methodologyofchoosing time-dependentparameters o fit an arbitrarybond-pricingmodel to currentassetprices.We showed, na relativelysimpletheoretical etting, hat this methodcansystematicallymispricesome assets. Like Ptolemy's geo-centric model of the solarsystem,these models can gen-erally be "tuned" o provide good approximationso (in

ourcase)pricesof a limitedsetof assets,butthey mayalsoprovideextremelypoor approximationsor otherassets.AsLochoff 1993,p. 92)put t, "Evenbadmodelscan be tunedto give good results" or simpleassets.ACKNOWLEDGMENTS

We thank Fischer Black and the associate editor forextensivecomments;JenniferCarpenter,VladimirFinkel-stein, and BruceTuckman or guidanceon industryprac-tice;AnthonyLynchfor pointingout an error n an earlierdraft;andNed Elton orinadvertentlynitiatinghisproject.Backus thanks he NationalScience Foundation or finan-cial support.BackusandZin are ResearchAssociateandFacultyResearchFellow,respectively, f the NationalBu-reau of EconomicResearch.

APPENDIX: MATHEMATICALUMMARYWe derivemanyof the formulasused in the text. Forboth the benchmarkheoryand our analogof the Black-Derman-Toymodel,we derivethe stochasticdiscount ac-tors impliedby the pricingkernelslisted in the text andthe impliedbondprices,forwardrates,andpricesof calloptionson discountbonds.The result s effectivelya math-ematical summary of the article.

A.1 Benchmark TheoryWe characterize bond-pricing theories with stochasticdiscount factors

Mt,t+n - n mt+jj=1

or - log Mt,t+7 - - Z=l- log mt+j. Given the pricing ker-nel m of our benchmark theory [Eqs. (2) and (3)], thestochastic discount factors are

S-n-j+j=1 1for n > 1. Given the discountfactors,we computebondprices from bn = EtMt,t+n [a consequence of (5)]:- log b= n + - (zt - 6)

n 2S 1 -A+ n--(P) a2/2.Forward ates[see (1)] are

f n = log bn - logbn+1= 6+n(6-z)- 1

-(A+ -2o/2,which includes a short rate of rt = fo = zt - (AU)2/2.It is conventionalo expressdiscountactors,bondprices,and forwardratesin terms of the observable hort raterratherhan he abstract tatevariable , which s easilydonegiven the linear relationbetweenthem. Becausep = 6 -(AU)2/2, we get discount actors- logMt,t+n= n6(+ (rt- P)"-

+ +1+ Et+j, (A.1)bondprices

fa s tt + (1E- na)t( -t() +-E- A+ _1-( a2/21as stated n Equation 10).A.2 Black, Derman, and Toy

We approachour Gaussian analog of the Black-Derman-Toy model the same way, using Equations (21) and (22) todefine the pricing kernel. Our analog of the Ho and Leemodel is a special case with it = P for all t. The stochasticdiscount factors are- log Mt,t+n = nzt + E (n - j)at+j

+ E ( + n- j)nt+j.=l


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24 Journalof Business & EconomicStatistics,January1998They implybondpricesof

n n- logb - nzt + (n - j)at+j - (y + n- j)232 /2j=1 j=1and,for n > 1, forward ates offt= Zt+ at+3 -(y + n)2 2+1/2j=1

- + )2 2+j - t+j+1)/2.j=1Because he shortrate s rt = fto = zt- (,yt+1)2/2, we canrewrite hese relationsas- log Mt,t+n = n[rt+ (7yt+1)2/2]

"n n+ (n - j)at+ + E ( + n - j)r/t+j, (A.2)j=1 j=1-log bZ= n[rt+ (70t+1)2/2]n n+ (n- j)at+j -E (7+ n - j)2+j/2,j=1 j=1and

nfZ = rt + E at+ + [y2 _ (y + n)2]3+1/2j=1 n

(7 + n - j)2 - +j+l)/2,j=1

as stated in Equation (23). With /t+j = 0, this reduces ton

ft= = rt + E at+3+ [72 - (7 + n)2]02/2,j=1

equation 15)of the Ho and Lee model.A.3 Nonequivalenceof DiscountFactors

Ourexamplesof assetsthat aremispricedby the Black-Derman-Toy model indicate that Black-Derman-Toy can-not generally reproduce the stochastic discount factors ofour benchmarktheory:Proposition 1. The parameters{7, at, 13t} f the Black-Derman-Toy model cannot be chosen to reproduce thestochastic discount factors of the benchmarktheory.The proof consists of comparingthe two discount factors,(A.1) for the benchmark theory and (A.2) for the Black-Derman-Toy model. The deterministic terms of (A.1) and(A.2) are relatively simple. For, say, the first n discountfactors, we can equate the conditional means of (A.1)and (A.2) by judicious choice of the n drift parameters

{at+l, ... , at+n}. The stochastic terms, however, cannotgenerally be equated.We can representthe initial stochastic

terms for the benchmarkheory n an array ike this:j=1 j=2 j=3n = 1 At+ln = 2 (A+ 1)Et+1 AEt+2n=3 (A+ 1 + Po)t+l (A+1)Et+2. At+3.

For theBlack-Derman-Toymodel, heanalogousermsarej=1 j=2 j=3n = 1 yt+1n = 2 ( + 1)rt+i r]t+2n = 3 (7+2)rt+1 (7 +1)rt+2 77t+3.

If a = 0, the model is riskless andreplicablewith driftpa-rametersalone.Alternatively,f p= 1 we canreplicate hebenchmarktheory by setting y = A and rt+j = Et+j, whichimplies 3t+j = a for all j. But with a $ 0 and ?p 1, itis impossibleto choose the price of risk 7 and volatilityparameters { 3t+1, t+2, /it+3} to reproduce the benchmarktheory.Supposewe tryto match he termssequentially.Tomatch the term (n,j) = (1, 1) we need y77t+l= AEt+l,which requires7y3t+l = A. Similarly,equivalenceofthe (2, 1) terms,(7 + 1)r7t+1= (A+ 1)Et+l, tells us (fornonzero A)that the parametersmust be 7 = Aand 3t+l = a.The(3, 1)termnowrequires + 2 = A+ 1+ p, which s in-consistentwithourearlierparameterhoiceswhen p 1.[WhenA= 0 the same conclusionholds,butthe argumentstartswith the (2, 1) term.]Thus we see that our attemptto reproducehe discount actorsof the benchmarkheorywith those of the Black-Derman-Toymodel has failed.It is easyto imaginesimilarproblemswith extensionsofthe model.We could, for example, et y dependon time.But for similarreasons, his one-dimensional rraycannotreproducehe discount actorsof thebenchmarkmodel.A.4 BondOptions

Any streamof cashflows{ht} canbe valuedbyn

Et 1 Mt,t+j ht+jj=1usingthe stochasticdiscount actorsM. We use this relationto price European ptionson zero-coupon onds.Considera European alloptionat date t withexpirationdate t + 7. The option gives its owner the right to buy ann-period bond at date t + 7 for the exercise price k, thusgenerating the cash flow ht+, = (b+, - k)+, where X= =max{O, x} is the nonnegative part of x. The call price is

c" = Et[Mt,t+,(bt+, - k)+]. (A.3)Computingthis price involves evaluating (A.3) with the ap-propriatediscount factor and bond price.Both of our theories have lognormal discount factorsand bond prices, so to evaluate (A.3) we need two prop-erties of lognormal expectations. Let us say that log x =(logx1, log x2) is bivariate normal with mean vector puandvariance matrix E. Formula 1 is

E[xil(x2- k)] = exp(Cti+ a1/2)N(d)


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Backus, Foresi,and Zin:ArbitrageOpportunitiesnArbitrage-FreeModelsof BondPricing 25with

d-2 - logk+ 012d'2whereN is the standard ormaldistribution unctionandI is an indicator unction thatequals 1 if its argumentspositive,0 otherwise.A similar esultwas statedandprovedby Rubinstein 1976, appendix).Except for the N term,this is the usualexpression or the mean of a lognormalrandomvariable.Formula2 follows from 1 with a changeof variables(X212for xl):E[(xl22)I(x2 - k)]= exp(Apl+ 2 + a1/2 + a2/2 + a12)N(d),with

d #2 - log k + "12 02-'2Ourapplication f these formulas o (A.3)uses Mt,t+, asx1 and bn+, as X2.For the benchmarkheory,we use discount actor(A.1)andevaluate A.3)usingthe expectationormulas.Tokeepthe notationmanageable,et

An -- A+ 1-- 2/2.j=1Now we look at the discount actorandfuturebondprice.The discount actor s, from(A.1),- log Mt,t+, = + 1- (rt - I)

+ jcj l"It has conditionalmeangiven by the first two termsandconditional ariance A,. Thefuturebondpricecan berep-resented

-logb = n6- An+ (r++ n j ) + -(p \(rt+j

j=1=- og n + log b + (An -A -A/

1 -f /n) 1

di =d2 = dl - v,,n

andoptionvolatility2 0.2 (1--(fln)~2V = 012 2(-r--j)1 j== 20(1 pn)2 (1-2r)-"1 cp 1 -_2 '

the conditional ariance f thelogarithm f the futurebondprice.Thecall pricefor theBlack-Derman-Toymodelfollowsa similar route with discountfactor (A.2). If we choosethe model'sparameterso matchbondprices, hen theonlydifference n the call formula s the option volatility,2 2E 2

j=1Whether his is the sameas thebenchmarkheorydependson the choice of volatility parameters {tM}.The difficultywith the Black-Derman-Toymodelwithrespectto pricingoptionsin our benchmark conomyissimilar o thatwith stochasticdiscount actors Proposition1). Volatilitiesare definedover the two-dimensional rrayindexedby thelengthr of theoptionand thematurityn ofthe bond on whichthe option s written.This arraycannotbe replicatedby the one-dimensional ector of volatilityparameters {3tI}.

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