AppliedMathematical Finance 8,27–48 (2001)

Calibrating the Black–Derman– Toy model: some theoretical results

PHELIM P.BOYLE, KEN SENG TAN andWEIDONG TIAN Universityof Waterloo,Ontario, Canada, N2L 3G1

ReceivedJune 2000

TheBlack– Derman– Toy (BDT) modelis a popularone-factor interest rate model that is widely used by practitioners. Oneof itsadvantages is that the model can be calibratedto both the current market term structure of interestrate and thecurrent term structure of volatilities.The input term structure of volatilitycan be eitherthe short term volatility or theyield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditionalshort rate volatility is used.In this paper conditions are investigated under which calibration to the yield volatilityis feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicatedthan when the short rate volatilities are used since the calibration at each time step now involves the solutionof two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situationsthe calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibrationfrom period n to period n +1,thecorresponding yield volatility has to lie within certain bounds. Under certaincircumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed shortrates for the period ( n, n +1)either become negative or else explode and this feature corresponds to the economicintuition behind the breakdown.

Keywords: interestrate models, Black– Derman– T oymodel, volatility, short term, yield

1.Introduction

Themodern approach to the modelling of stochastic interest rates started with the classic paper by Vasicek(1977). The early models postulated a stochasticdifferential equation for theevolution of theshort rate of interest and, by invoking no-arbitrage arguments, developed expressions for theprices ofpure discount bond and other securities of interest such as options. Other examples of such modelsinclude Brennan and Schwartz (1979) and Cox, Ingersoll and Ross (1985). One of theproblems withthese models was thatthey did not have enough degrees of freedom to match the model prices ofpure discount bonds with the corresponding market prices. This was anuncomfortable situation

Theauthors are gratefulto Baoyan Ding, Fred Guan, Houben Huang, George Lai, David Li, Chonghui Liu, and Ken Vetzal foruseful discussion.Phelim P .Boylethanks the Social Science andHumanities Research Councilof Canada andthe Natural Sciences and EngineeringResearch Councilof Canada,for research support.Ken SengTan acknowledges the research supportfrom the Natural Sciences andEngineering Research Councilof Canada.Weidong Tian thanks CITO (Communicationsand Information Technology Ontario)for research support.

AppliedMathematical Finance ISSN1350-486X print/ ISSN1466-4313 online # 2001Taylor & FrancisLtd http://www.tandf.co.uk/journals DOI: 10.1080/13504860110062049 28 Boyle,T anand Tian sincethe model could not be described as arbitrage free whenthe model prices differed from thecurrent marketprices. Oneway to resolve this problem is to makethe parameters of theshort term rate time dependent. In the Vasicekcase, the resulting model is known as the extended V asicekmodel. This approach has been popularizedby Hull and White (1993). Another approach is to model the uncertainty by assuming a stochasticprocess for theevolution of the forward rate.Since this approach starts from thecurrent observedforward rate,the market prices of today’ s zero-couponbonds are built directly into the foundationsof themodel. The Ž rstpaper to applythis approach was Hoand Lee (1986) in a discrete-time framework.Subsequently Heath, Jarrow andMorton (1990) (HJM) provideda muchmore extensive and rigorousapproach in continuous-timeframework. The HJM modelis quite general; it canbe calibratedto currentbond prices and option prices. In practice, the model parameters are selected by Ž ttingmodel pricesto the current market prices of the most liquid instruments. From atradingperspective, this approachis useful since the model reproduces market prices over for themost liquid traded securities. 1 For manypricing applications, it is convenient to have a simplebinomial type model that Ž tsthe currentterm structure of bondprices and the current term structure of volatilities.In this connection, the Black,Derman and Toy (1990) (BDT) modelis awidelyused model that can be calibratedto matchthe termstructure of zero-couponbond prices and the term structure of volatilities.In the original BDT paper, theauthors used the yield volatilities as the input term structure of volatility. The yield volatility correspondsto the volatility of the yields on long term bonds. In practical applications, it is often more convenientto use the term structure of shortrate volatilities since they can be directly inferred from the marketprices of interestrate caps. Thus,there are two ways to calibrate the BDT model.The Ž rstone is the short rate volatility method which uses thecurrent term structure of zerocoupon bond prices; theterm structure of futureshort rate volatilities. Thesecond one is the yield rate volatility method which uses thecurrent term structure of zerocoupon bond prices; theterm structure of yieldson zero coupon bonds. For eachof theseapproaches it is of interestto investigate the conditions under which Ž ttingthe BDT model results in a reasonablecalibration .Our conditionsfor areasonablecalibration are quite weak. W ewillrequire thatall the short rates and all the output volatilities 2 inthe calibrated BDT modelare positive. Sandmann and Sondermann(1993) have analysed the case when the BDT calibrationis based on theshort rate volatilities. Theyprovide necessary and sufŽ cient conditions for thisto happen. Their result is: if the current implied forward ratesare all positive (i.e. the pure discount bond price declines as the time to maturity increases) and theshort rate volatilities are all positive, then it is possible to calibrate the BDT model.Their conditions are simplefrom amathematicalperspective and have an intuitive economic interpretation.

1 Thisapproach has disadvantagesfrom an econometric perspective since, by Ž ttingnew parameters ona dailybasis, we are effectivelyassuming a new modelevery day. 2 Theoutput volatility under the Ž rst approachis theyield volatility as computedfrom the calibrated BDT modelsince theinput volatilityis theshort rate volatility.The output volatility under the second approach is theshort rate volatilityas computedfrom the calibratedBDT modelsince theinput volatility is theyield rate volatility. Calibratingthe Black– Derman– Toy model 29

Thispaper investigates the conditions under which we cancalibrate the BDT modelwhen the yield volatilityis used. W ederivethe precise mathematical conditions which the input data must satisfy so that the BDT modelcan be calibrated.These conditions are less elegant than in the case when the short rate volatility isused. However ,we Žndthatit isnot possible to calibrate the BDT modelfor seeminglyplausible input termstructures. T oobtainour conditions we useresults from thetheory of polynomialequations. Theoutline of the rest of the paper is as follows. In the next section, we reviewthe details of the procedurethat is used to calibrate a BDTmodel.W eexamineboth the original and modiŽ ed BDT models whichcalibrate to the term structure of yieldvolatilities and short rate volatilities respectively. W eshow thatthe calibration equations can be reducedto a systemof polynomialequations so that we candraw on resultsfrom theso-called QuantiŽ er Elimination;an algebraic approach that provides conditions for polynomialsto have real roots. In Section 3, we providea detailedanalysis of thecalibration of athree- periodBDT model.W eprovideboth necessary and sufŽ cient conditions for thecalibration to be feasible. Theconditions are quite complicated even in the three period case and it appears difŽ cult to extend this typeof analysis to the n periodscase. Hence in Section4, we providea sufŽcient condition for calibration tobe feasible from the nthstep to the ( n +1)thstep given that the calibration was successfulfor the preceding n periods.In Section 5 we provideseveral examples which illustrate the conditions developed inSection 4. The Ž nalsection concludes the paper.

2.Calibration ofthe BDT model

Inthis section, we reviewthe procedure used to calibrate a BDT interestrate model. In this model, a recombiningbinomial lattice is constructed so that it matches the current yield curve and the current yield volatilitycurve. W eassumethe calibrated binomial lattice has N periodsand each period is of size t years.Hence the total time horizon of thebinomial lattice is T = N t years.The recombining nature of thebinomial lattice ensures that at time period n, there are n +1states.W elabelthese states as i = 0, 1, . . ., n. Let r(n, i)bethe(annualized) one-period short rate at period n and state i.Theshort rate r(n, i) evolves either to r(n + 1, i)(i.e.down-state) or to r(n + 1, i +1)(i.e. up-state) one period hence with equal risk-neutralprobability. Let Y^(0, n) Y^(n)bethecurrent (market) yield on a n-periodzero-coupon bond (i.e. with maturity n t) and ^Y (n)bethe corresponding current yield volatility. Then the price of an n-periodzero-coupon bond, P^(0, n) P^(n), isgiven as n P^(n) 1 Y^(n) t ¡ (1) ˆ ‰ ‡ Š Similarly,let P (n) and Y (n)denotethe model price and model yield of an n-periodzero-coupon bond and Y (n)denotethe volatility corresponding to the n-periodyield implied from themodel. In other words, the set {P(n), Y(n), Y(n)}issimilarto { P^ (n), Y^(n), ^Y (n)} exceptthat the Ž rstset of values is computed from themodel while the second set is the market inputs. W ecalibratethe model to the market by ensuring that P(n) = P^(n) and Y (n) = ^Y (n) for all n = 1, 2, . . ., N. Letus denote the two possible yield realizations at period 1 (i.e.nodes (1, 0) and (1, 1)) ona zero- couponbond which matures at the end of period n by Yd (n) and Yu (n).Inthe BDT model,these two yieldsare related by

Y (n) Y (n) exp 2 (n) t (2) u ˆ d ‰ Y Š p 30 Boyle,T anand Tian

Ina similarmanner, let Pd (n) and Pu (n)denotethe prices of zero-coupon bonds corresponding to the yields Y (n) and Y (n), respectively.Therefore, P (n) and P (n)representbonds with ( n 1) t years to d u d u ¡ maturityand are related to Yd(n) and Yu(n) as follows 1 Pd(n) n 1 ˆ 1 Yd(n) t ¡ ‰ ‡ Š 1 Pu(n) n 1 ˆ 1 Yu(n) t ¡ ‰ ‡ Š Wealsohave the following relationship: 1 P(n) Pd(n) Pu(n) (3) ˆ 2 1 r(0, 0) t ‰ ‡ Š ‰ ‡ Š As pointedout by Jamshidian (1991), the calibration procedure is facilitated by the use of a forward inductiontechnique. This involves using the Arrow– Debreu securities, which are deŽ ned as follows. Assumewe havea securitywhich pays one unit at time n in state i andzero elsewhere and let A(n, i) denotethe price at node (0, 0) ofthisArrow– Debreu security. The Arrow– Debreu security is sometimes referred toas theGreen’ s functionbecause of its continuous-time analogue. TheArrow– Debreu prices satisfy the following recursive relation: A(n 1,i 1) ¡ ¡ , i n 2 1 r(n 1,i 1) t ˆ 8 ‰ ‡ ¡ ¡ Š > > > A(n 1,i 1) A(n 1, i) > A(n,i) > ¡ ¡ ¡ , i 1, 2, . . . , n 1 ˆ > 2 1 r(n 1, i 1) t ‡ 2 1 r(n 1, i) t ˆ ¡ <> ‰ ‡ ¡ ¡ Š ‰ ‡ ¡ Š > > A(n 1, i) > ¡ , i 0 > 2 1 r(n 1, i) t ˆ > ‰ ‡ ¡ Š > Themodel price :>of an n-periodzero-coupon bond can be written in terms of Arrow– Debreu prices as n P(n) A(n, i (4) ˆ i 0 † Xˆ withouthaving to work backwards through the lattice one period at a timeto obtain the required value. 3 Let Ad(n, i)denotethe Arrow– Debreu price at node(1, 0) ofa contingentclaim that pays $1 if state i is realizedin period n andzero otherwise. Similarly, let Au(n, i)denotethe corresponding price of the Arrow–Debreu security at node (1, 1). Then, Pd(n) and Pu(n)canbe computedfrom Ad(n, i) and Au(n, i) as n Pu(n) Au(n, i) ˆ i 0 Xˆ

3 Moregenerally, if X(n, i)denotesthe payoff of a Europeancontingent claim at node( n, i),thenthe price ofthecontingent claim at n node(0, 0) isconveniently computed as i 0 A(n, i) X(n, i). ˆ Calibratingthe Black– Derman– Toy model 31

n Pd(n) Ad(n, i) ˆ i 0 Xˆ Note that Ad(n, n) and Au(n,0)arezero for all n. Incalibrating a ( N +1)-periodbinomial lattice, the task reduces to Ž ndingthe values of r(n, i), for n = 0, 1, . . ., N, i = 0, 1, . . ., n,for whichthe model values are consistent with the input market values. Normallythis is carried out one time step at a time.For instance,in the ( n +1)thperiod calibration, the taskis to Ž nd r(n, i), i = 0, 1, . . ., n sothat the resulting lattice matches the input Y^ (n + 1) and ^Y (n + 1), assumingthat all the earlier short rates, r(m, i), m = 0, 1, . . ., n, i = 0, . . ., m havealready been calibrated to theinput term structures { Y^(m), ^Y (m), m = 0, 1, . . ., n}.Inother words, the ( n +1)thperiod calibration involvessolving the following non-linear equations: n A n i ^ ( , ) a n 1 P(n 1) P(n 1) i (5) ‡ ˆ ‡ ˆ ‡ ˆ i 0 1 r(n, 0)(b n) t Xˆ ‡ 1 1 u¡ 1 ( ¡ 1) n 1 (6) ¡ ˆ ¡ ‡ where n n Au(n, i) u i (7) ˆ i 0 1 r(n, 0)(b n) t Xˆ ‡ n n Ad(n, i) i (8) ˆ i 0 1 r(n, 0)(b n) t Xˆ ‡

2^Y (n 1)p t n 1 e ‡ (9) ‡ ˆ  n n a n 1 u 2 ‡ (10) ‡ ˆ a 1

Notethat the non-linear Equations 5 and6 containonly two unknowns, r (n, 0) and n.Theseequations haveto be solved numerically for r(n, 0) and n andmethods such as the Newton– Raphson iteration approachare often used. Once these unknowns are determined, the short rates in other states are computed viathe following recursive relationship among the short rates in each period: r(n, i) r(n, 0)(b )i ˆ n for i = 1, 2, . . ., n. For n =1or2,the calibration is simple. The required short rates are determined as r(0, 0) Y^(1) (11) ˆ

a 2 (1 2)(1 2 ) t pD r(1, 0) ¡ ¡ a 1 ‡ (12) ˆ 4 a 2 ( t)2 a 1 2 

r(1, 1) r(1, 0) (13) ˆ 2 32 Boyle,T anand Tian where 2 a 2 a 2 2 a 2 D (1 2) 1 2 t 16 2( t) 1 2 ˆ ¡ ¡ a 1 ‡ a 1 ¡ a 1 h i For theseshort rates to be positive and not exceeding the maximum rate !,thefollowing conditions on the marketinput must be satisŽ ed: Y^(1) ! (14)

! t(1 ) 2 a ‡ 2 ‡ 2 2 1 (15) 2! t(1 ! t ) 2 a ‡ ‡ 2 ‡ 2 1 Toverify that most reasonable term structures satisfy the second inequality, we Žrst notethat the left-hand mostexpression is an increasing function of andconverges to (2 + ! t)/2(1 + ! t) as . 2 2 ! 1 Suppose f1 isthe one-period forward ratefrom period1 toperiod2; i.e. 2(1 + f1 t) = 1. Then any f1 that satisŽes thethe following inequality ! 0 f 1 2 ! t ‡ alsosatisŽ es (15). In practical situations, setting ! =1(or 100%)would be a veryconservative upper bound. Also t isusually less than 1 year.If we assume ! = 1 and t =1,theupper bound on f1 is 1/3. Thisimplies that as long as the forward rate f1 ispositive and does not exceed 33%, inequality (15) is satisŽed andthe calibration is feasiblefor anypositive input ^Y (2).This provides a justiŽcation that most reasonableinput term structures satisfy inequalities (14) and (15). Itis nontrivial to extend the calibration to one more period; i.e. from period2 toperiod 3. 4 In the next sectionwe willanalyse the calibration issues for thethree-period BDT model. Wenowexplain a generalmethod to handle the non-linear Equations 5 and6 usingelimination theory.The original system of non-linear equations over the two unknown variables r(n, 0) and n canbe expressedas a systemof non-linearequations over the four unknowns r (n, 0), n, u and . More precisely,Equations 5, 6,7and8 canbe expressedas four polynomial equations. This allows us toinvoke animportantset of resultsfrom classicalalgebra known as QuantiŽ er Elimination(QE). TheQE provides amethodologyto solve polynomial equations. The underlying principle can be summarized as follows: supposewe aregiven several polynomial equations. To see if a givenpolynomial equation has real solutions,it sufŽ ces to check if thecoefŽ cients of the polynomial equations satisfy certain conditions. A moreprecise statement of this is given in Appendix A. As afamiliarillustration, let us consider the quadraticequation ax2 + bx + c =0.Inthis case, we onlyneed to check whether the coefŽ cients ( a, b, c) satisfythe relation b2 4ac 0for realsolutions to exist. ¡ Wenowconsider a simpleexample using QE. Supposewe areinterested in Ž ndingcriteria for the existenceof real solutions r(n,0)inthe range ( a1, a2), where a1 < a2.Letus introducetwo new variables x and y, so that r(n, 0) = a + x2 and r(n, 0) = a y2.Theelimination theory provides conditions among the 1 2 ¡ coefŽcients of the polynomial equations such that rn (a , a )ifand only if thecoefŽ cients satisfy , 0 2 1 2 certainrelations. Althoughelimination theory is constructive in the sense that there are algorithms for Žndingthe relationsamong the coefŽ cients, all algorithms are either impractical or verycomplicated to implement. 4 Ourcolleague, Ken Vetzal pointedout that extensions of resultsfrom n = 2 to n =3are sometimes notvery easy andcited Fermat’ s Last Theoremas anillustration. Calibratingthe Black– Derman– Toy model 33

Evenin the three-period BDT model,which we willdiscuss in the following section, the most efŽ cient algorithmfor QEisalready difŽ cult to handle. In subsection 3.2, we givean efŽ cient algorithm for our problemin the three-period BDT modelusing ideas from QE.

2.1 Calibrationof the modiŽed BDT model

Inthis subsection, we briey discussa modiŽed version of theBDT modelwhich takes the term structure ofshort rate volatilities as input instead of the term structure of yield volatilities. In this case, the calibrationis alotsimpler .First,the term structure of shortrate volatilities can be inferredfrom themarket pricesof interestrate caps. Second, the calibration reduces to solving one non-linear equation since under theassumption of lognormalityof the short rates, we have

2 r n p t r(n, i) r(n, i 1)e … † , for i 1, 2, . . . , n ˆ ¡ ˆ where ^r (n)isthe input short rate volatility  for the nthperiod. In other words, the parameter n in (5) becomesknown and is equivalent to e2^r(n)p t.Hencethere is onlyone equation with one unknown for thiscalibration.  For themodiŽ ed BDT model,Theorem 2.1 of Sandmannand Sondermann (1993) provides a necessary andsufŽ cient condition under which it is possible to calibrate a BDT modelas long as the short rate volatilitiesare positive and Ž nite.They show that there exists a BDTmodel(with positive short rates) if andonly if the forward ratesare positive. This result can easily be shown as follows: Supposethe modiŽ ed BDT modelhas been calibrated up toperiod n.Toproceedone more period, we needto solve the following equation (from (5)) n A(n, i) a n 1 (16) ‡ i ˆ i 0 1 r(n, 0)b n t Xˆ ‡ 2^r n p t where n = e … † . Let fn bethe one-period forward ratefrom period n to n +1.Then we have n 1 a n 1 ai ‡ f t a i 1 n ˆ n ˆ i 0 1 r(n, 0)b n t ‡ Xˆ ‡ where A(n, i) ai 0 ˆ a n and i ai = 1 since i A(n, i) = n. i If r(n, 0) n arepositive for all0 i n, then 1 (0, 1) 1 r(n, 0)b i t 2 ‡ n Thisimplies that the convex combination

n a i (0, 1) i 0 i ˆ 1 r(n, 0)b n t 2 X ‡ Consequently,we musthave fn > 0. 34 Boyle,T anand Tian

Conversely,suppose fn >0.W eneedto show that there exists a uniquepositive number r(n, 0) such that(16) holds. Let n a 1 h(x) i i f t ˆ i 0 1 xb n t ¡ 1 n Xˆ ‡ ‡ Firstnote that h(x)isstrictlydecreasing. Second, by assumption we have 1 h(0) 1 > 0 ˆ ¡ 1 fn t ‡ Third, 1 h( ) < 0 ‡ 1 ˆ ¡ 1 fn t ‡ Thisimplies that there exists a uniquepositive root for h(x)=0,as required.

3.Three-period BDT model

3.1 AsufŽcient condition

Inthis section, we considerthe calibration issue for theoriginal three-period BDT model.W eassume thatthe Ž rst twoperiods have already been calibrated successfully so that we onlyneed to Ž nd r(2, 0), r(2, 1), and r(2,2) such that the resulting interest rate lattice matches to the input term structures Y^(3) and ^Y (3).Although this is only the third period calibration, we showthat the BDT latticemay not existfor certainterm structures. T oexaminethese conditions, we Žrst notethat eliminating v in (6) and (10) with n =2,the parameter u becomesthe root of a polynomialequation of degree 4; i.e. g1 (u) = 0 where 2 4 2 3 3 1 2a 3 2 4a 3 2a 3 g1(x) x x ‡ 2 x x 2 (17) ˆ ‡ 3 1 ‡ ( 3 1) ¡ a 1 ¡ a 1( 3 1) ¡ a 1( 3 1) ¡ h ¡ i ¡ ¡ TheŽ rst resultcan be stated as follows:

Theorem 1 If thereexist positive short rates in the interval (0, !), then

g1( (!)) > 0, where

1 2a 1 (x) min , 3 : ˆ 1 r(1, 1) t a ¡ 1 r(1, 0) t (1 x t) (s‡s1‰‡Š‡)

Inother words, if g1 ( (!)) 0,then there exists no positive short rates in the interval (0, !).

Proof:SeeAppendix B. Calibratingthe Black– Derman– Toy model 35

Aconsequenceof theabove theorem is that calibration of theBDT modelis not feasible when the yield curveis increasing sharply while the yield volatility curve is decreasing dramatically. This observation can beveriŽ ed as follows: Substituting (!)into(17), we obtain

4 2a 3 2 1 3 4a 3 1 2 2 2a 3 g1( (!)) (!) (!) 2 (!) (!) 2 ( 3 1) (!) ˆ ¡ a 1 ‡ 3 1 ¡ a 1 ‡ ( 1) ‡ ¡ a 1 ¡ 3 ¡ Assumingthat the yield volatility is positive, we have 3 >1.Noticethat the terms in the Ž rst twosquare bracketsare always negative. This implies that g1( (!)) 0ifthe term in the third bracket is also negative;i.e. 2a ( 2 1) (!)2 < 3 ‡ a 1 Hencewe havethe following corollary:

Corollary 2 (a) Suppose

2a 1 (!) 3 ˆ a ¡ 1 r(1, 0) t (1 ! t) s1‰‡Š‡ Thereexists no short rates { r(2, 0), r(2, 1), r(2,2)} inthe range (0, !) if

1 2a 3 Y (3) log 1 r(1, 0) t (1 ! t) 1 (18) ¡ 4p t a 1 ‰ ‡ Š ‡ ¡ and  a < 2a 1 r(1, 0) t (1 ! t) 1 3‰ ‡ Š ‡ (b) Suppose

1 (!) ˆ 1 r(1, 1) t s‡ Thereexists no short rates { r(2, 0), r(2, 1), r(2,2)} inthe range (0, !) if

1 2a 3 Y (3) log 1 r(1, 1) t 1 (19) 4p t a 1 ‰ ‡ Š ¡ and  a < 2a 1 r(1, 1) t 1 3‰ ‡ Š

3.2 Necessary andsufŽ cient conditions

Theorem1 establisheda sufŽcient condition for thethird period calibration to be feasible. In this subsection,we providea necessaryand sufŽ cient condition for theshort rates to lie in the interval (0, !), where ! > 0. 36 Boyle,T anand Tian

Weassumethe largest short rate lies on thetop branch of theBDT latticeand denote it as y*, then the 2 othertwo short rates in the third period are y*/ 3, y*/ 3, where 3 1. Since y*isthe largest attainable rate,it is sufŽ cient to consider the conditions on y*for whichit lies in (0, !). From (7) and(8), u and v mustsatisfy the following equations: 1 1 2(1 c)u2 (20) 1 y ‡ 1 y=b ˆ ‡ ‡ ‡ 1 1 2(1 b) 2 (21) 1 y=b ‡ 1 y=b 2 ˆ ‡ ‡ ‡ where = 3, y = y* t, b = r(1, 0) t and c = r(1, 1) t.Theassumption that 1impliesthat the bound on is

B1 B2 (22) where

2a 3 1 c 2a 3 1 B1 ‡ and B2 ˆ a 2 b c ˆ a ¡ (1 c)(1 !) r1‡‡ s1‡‡ DeŽ ne m and n as 1 1 m (23) ˆ 2(1 c)u2 ˆ 2(1 c)(2 a 3 2) ‡ ‡ a 1 ¡ 1 n (24) ˆ 2(1 b) 2 ‡ Note that m n since 1.Substituting the above expressions m and n into(20) and (21), we obtain y2 (b 1)(1 m)y (1 2m)b 0 (25) ‡ ‡ ¡ ‡ ¡ ˆ y2 b (b 1)(1 n)y (1 2n)b 3 0 (26) ‡ ‡ ¡ ‡ ¡ ˆ Eliminatingthe y2 term gives b 2m 1 (2n 1)b 2 y ‰ ¡ ¡ ¡ Š ˆ (b 1) (n 1)b (m 1) ‡ ‰ ¡ ¡ ¡ Š or b 2m 1 (2n 1)b 2 y* ‰ ¡ ¡ ¡ Š (27) ˆ t(b 1) (n 1)b (m 1) ‡ ‰ ¡ ¡ ¡ Š Toensure that the condition 0 y* ! issatisŽ ed, we needto consider the cases where the denominator in(27) is either positive or negative.

Case 1: (n 1) (m 1) > 0 ¡ ¡ ¡ Inthis situation, the only admissible case is n >1.Here iswhy this is the only case. If n = 1 then (m 1)> 0whichimplies that m <1contradictingthe assumption that m n. ¡ ¡ Calibratingthe Black– Derman– Toy model 37

If n < 1, then < (1 m)/(1 n)<1because( m n)andthis contradicts the assumption that 1. ¡ ¡ For n >1,the conditions on are m 1 b > ¡ (28) n 1 ¡ 2m 1 b ¡ (29) 2n 1 r¡ and ! t(b 1) (n 1)b (m 1) b 2m 1 (2n 1)b 2 > 0 (30) ‡ ‰ ¡ ¡ ¡ Š ¡ ‰ ¡ ¡ ¡ Š Let 1 betheexpression on theleft-hand side of theabove inequality with m and n replacedby (23) and (24). 1 becomesa functionin terms of theunknown variables v and .Anequivalent condition to (30) becomes

1(b , ) > 0 (31) where v isaroot(satisfying the boundary condition (22)) ofthe function g2 deŽ ned as 2 4 2 3 3 3 1 2a 3 2 4a 3 3 2a 3 3 g2(x) x x ‡ 2 x x 2 (32) ˆ ‡ 3 1 ‡ ( 3 1) ¡ a 1 ‡ a 1( 3 1) ¡ a 1( 3 1) ¡ h ¡ i ¡ ¡ Theabove function is derived from (6) and(10) by eliminating u.5 Inequality(31) provides one condition for which and v mustbe jointly satisŽ ed. Ina similarmanner, we deŽne m 1 b ¡ 2 ˆ ¡ n 1 ¡ and 2m 1 b 2 ¡ 3 ˆ ¡ 2n 1 ¡ andsubstituting (23) and (24) into the above two expressions. Inequalities (28) and (29) are respectively, equivalentto

2(b , ) > 0 (33)

3(b , ) < 0 (34) For agivenroot v,theremay exist many possible values of for whichconditions (31), (33), and (34) arefulŽ lled. Hence we needanother condition on sothat the uniqueness of isensured. This is achievedby substituting (27) into (25). If we denotethe resulting expression by 4( , v), is then computedfrom thefollowing equation (b , ) 0 (35) 4 ˆ Theabove series of steps provides necessary conditions for whichthe short rates lie in the required range (0, !). Itremains to consider the other situation where the denominator in (27) is negative. In

5 Alternatively, g2 (x)can beobtainedfrom g1(x)byreplacing 3 in (17) by 1/ 3. 38 Boyle,T anand Tian thiscase, there are two admissible possibilities depending on the value n.Wesummarizethe results as follows:

Case 2a: (n 1) (m 1) < 0 and n > 1 ¡ ¡ ¡ Usingthe above notation, the conditions on v and are

1(b , ) < 0

2(b , ) < 0

3(b , ) 0 (b , ) 0 4 ˆ Case 2b: (n 1) (m 1) < 0 and n 1 ¡ ¡ ¡ Inthis case, the conditions on v and are

1(b , ) < 0

2(b , ) > 0

3(b , ) 0 (b , ) 0 4 ˆ From theabove discussion, we alsoknow that the process can be reversed.The reason is as follows: theinequality B1 v B2 impliesthat m n.Thenin case1, bothrelations 2( , v) > 0 and 3( , v) < 0 imply that y > 0, and 1( , v) > 0 yields y* < !.Thenboth relations 4( , v) = 0 and g2 (v) = 0 areequivalent to the original non-linear relations (5) and(6). The results for Cases2a and 2b are similar. Toconclude this section, we providean algorithm which checks the existence of the third period BDT shortrates in (0, !).

Step1. First check whether the equation g2(v)=0hasa solutionin the range ( B1, B2). This can beaccomplished using Sturm’ s algorithm(see AppendixC for abriefdescription). Alternatively, some mathematicalsoftware packages such as Maple have a built-inversion of Sturm’s algorithm.If nosuch root v exists,stop.

Step2. If thereexists such roots, there are at mostfour roots. Since the degree of g2 is4, theseroots can befound relatively easily. For each v,we obtainthe corresponding by solving 4( , v)=0.Sturm’ s algorithmcan again be usedas aŽrststep to checkthe existence of theroot. If theroot exists,then go to step3; otherwise stop.

Step3. For eachroot (onlyŽ nitelymany) and v, compute m and n using(23) and (24) and check if any ofthe following conditions holds:

6 1 3:1 : 1(b , ) > 0, 2(b , ) > 0, 3(b , ) < 0 and : ‰ Š s2(1b) ‡ 6 Thelast conditionon v is equivalentto the condition n 1. Calibratingthe Black– Derman– Toy model 39

1 3:2a : (b , ) < 0, (b , ) < 0, (b , ) 0 and : ‰ Š 1 2 3 2(1 b) s‡ 1 3:2b : 1(b , ) < 0, 2(b , ) > 0, 3(b , ) 0 and : ‰ Š s2(1b) ‡ If oneof the above conditions holds, then there exists short rates, say y*, y*/ , y*/ 2,inthe range (0, !), where y*canbe determined from (27).Otherwise, positive short rates in the range (0, !) do not exist.

4.General case

Inthe previous section, we provideda necessaryand sufŽ cient condition for theexistence of the positiveshort rates for thethree-period BDT model.W esaw thateven with three periods, the task was alreadyvery challenging. Hence it seems virtually impossible to extend the methodology to the general n-periodcase. Inthis section we assumefor agiveninput term structures, the model has been calibrated (successfully) up to n periods.W ethenprovide a sufŽcient condition that jointly characterizes Y^ (n + 1) and ^Y (n + 1) for whichthe calibration would fail in the ( n +1)thperiod. More precisely, we summarizethe result in the followingtheorem.

1 1 Theorem 3 Let a = Pd(n) n , b Pu(n) n, and ‰ Š ˆ ‰ Š 1 n x a n 1 n ¡ (x) 2 ‡ x 1 (36) ˆ 1 x a ¡ ¡ ¡ " 1 # for x >0.Assume for anyarbitrary term structure, the BDT modelhas been calibrated up to n periods. In thecalibration to the ( n +1)thperiod, there exists no positive short rates if the inputs Y^(n + 1) and ^Y (n +1)satisfyany of thefollowing conditions: 1 (i) ^y(n 1) > log( (a)) (37) ‡ 2p t

1 1 (ii) ^y(n 1) < log (38) ‡ 2p t (b) Proof: SeeAppendix D. 

Remark 1. Theproof to the above theorem only requires the monotone-property of the two-variable function f (x, y)andthe basic relation among the Arrow– Debreu securities. The same conclusion applies to anyinterest rate tree as long as the short rates satisfy the BDT relations;i.e. r(n, i) = i(n, 0) i , n, i. n 8 Remark 2. If (a) (b) 1,thenthere exists no positive short rates in the ( n +1)thperiod regardless of the value ^y(n + 1). 40 Boyle,T anand Tian

Table 1. Examplesof input term structures.

Example (a) (b) (c) (d ) (e) ( f )

N 4 12 52 52 10 10 t (yrs) 1/4 1/12 1/52 1/12 1/4 1/12 1 1 5 T = N t (yrs) 1 1 1 4 3 2 2 6 Y^ (1) 8% 8% 8% 8% 10% 10% Y^ (N) 4% 4% 4% 4% 5% 5% ^Y (1) 20% 20% 20% 20% 20% 20% ^Y (N ) 20% 20% 20% 20% 17% 17%

5.Numerical examples

Inthis section, we presentnumerical examples which illustrate the calibration issues based on theresults establishedin the earlier sections. Table1 givesdetails of thecases which we usedto construct N-periodBDT models.In particular, the Žrst threeexamples, ( a), (b), and (c),haveidentical term structures in the sense that all three of themhave aatyield volatility of 20% while the yield curve is decreasing linearly from 8%to 4% over one year horizon.The only difference is the number of timesteps Ž ttedto these term structures. Example ( a) uses 4 periods, (b)uses12 periods,while ( c)uses52 periods.This accounts for thedifference in magnitude of t reportedin the third row ofthe table. In each of these examples, we didnot encounter any problem in constructingthe BDT models(for varying N). Supposewe makea minorchange to the input yield volatilities in these examples ( a), (b), and (c) and attemptto re-calibrate the BDT lattice.The minor change is by perturbing the N-periodyield volatility from 20%to 19% (i.e. ^Y (N)=19%).In all three situations, we foundthat we canonly calibrate up to N 1periodsand fail in the Nthperiod. It might be argued that this phenomenon results from tothe ¡ numericalmethods used to solve the two non-linear equations (5) and(6), for instance,if a Newton– Raphsonprocedure were usedto solve these equations. Failure to Ž ndconvergence does not necessary implythat there exists no solution. This could merely be due to apoorset of badinitial values in carrying outthe iteration process. It is well-known that the convergence of the Newton– Raphson algorithm cruciallyrelies on the initial values. Wenowshow that Theorem 3 canbe usedto explainthe failure to calibrate the nthperiod short rates. In N 1periods,the quantities P (N 1) and P (N 1)arereadily available. From theassumed value of ¡ d ¡ u ¡ the N-periodyield (i.e. Y^(N)), we candetermine which condition(s) in Theorem 3 is(are) satisŽed and henceŽ ndtheappropriate bound of ^Y (N)for whichpositive short rates in the Nthperiod are not possible. Itturns out that in all three cases, either condition ( i)orcondition( ii)issatisŽ ed. The ranges on theyield volatilitiesare reported in Table 2. For instancefor case( a), theredoes not exist positive short rates in the fourthperiod when the yield volatility is either greater than 20.568% or less than 19.238%. In our modiŽed examples,we have ^Y (4) =19%,which falls in the required range. This explains the failure in solvingthe fourth period short rates that calibrate to this yield volatility. It should be emphasizedthat the aboveresult is based on our theory and is independent of thenumerical techniques. Calibratingthe Black– Derman– Toy model 41

Table 2. Theimplied ranges on ^Y (N )fromTheorem 3 forexamples in T able1.

Example (a) (b) (c) (d ) (e) ( f )

N 4 12 52 52 10 10

(i): ^Y (N) > 20.568% 20.057% 20.005% 20.011% 17.481% 17.425% (ii): ^Y (N) < n/a n/a n/a n/a n/a n/a (ii): ^Y (N) < 19.238% 19.930% 19.994% 19.999% 17.135% 17.224% (i): ^Y (N) > n/a n/a n/a n/a n/a n/a

Theresults in Table 2 alsoindicate that as the size of the period becomes smaller, the convergence of theBDT modelbecomes more sensitive. For example,a smallperturbation of the n-periodyield volatility, 1 sayto 19.9%, would lead to no positive short rates for Example( c) with t = 52 whilethe result is inconclusivefor both( b) and (c)withlarger time steps. Wenowconsider the rest of theexamples (( d), (e) and ( f ))inTable 1. Example ( d)issimilar to ( c) 1 1 exceptthat each period is oflength 12 yearsso that the total time horizon in 52 periods are 4 3 years. In 1 example (e), theyield curve is downward sloping, decreasing linearly from 10%to 5% over 2 2 years. The yieldvolatility is also downward sloping which decreases linearly from 20%to 17% over the same time horizon.The input term structures for example( f )alsoexhibits the same shape as ( e)exceptthat the 5 curvesare spread over a muchshorter horizon; i.e. 6 years.W ewilluse a 10-periodBDT latticeto Žtthe termstructures given in examples ( e) and ( f ). Itis found that even with a muchlonger time horizon (compared to ( c)), thereis no problem in calibratingthe term structure in ( d).Onthe other hand, the calibration fails in the tenth period for the decreasingterm structures in ( e) and ( f ).Thisissue can again be addressed using Theorem 3. The last threecolumns in Table 2 summarizethe results. Theorem 3 indicatesthat if the 10-period yield volatility in cases (e) and ( f )isless than 17.14% and 17.22%, respectively, then there exists no positiveshort rates inthis period. The input term structure, which is ^Y(10)= 17%,clearly falls within the range and hence accountsfor thebreak-down of the BDT inthe tenth period. Analternative characterization of Theorem3 isto examinethe ‘ regionsof no solution’. Theconditions inTheorem 3 jointlyprovide the bounds on Y^(n), and ^Y (n)for whichthe BDT modelcannot be calibrated.Figure 1 depictsthe regions of interest for Example( e). Theshaded region is derived from condition (i)whilethe striped region from condition( ii). Henceany input data ( Y^(n), ^Y (n))whichlie in thesezones makes the calibration infeasible. This also explains the phenomenon observed in Example ( e) wherethe input data lies within the failure zone (marked with an X onthe Ž gure). Itshould be pointed out that in all the examples we haveconsidered, the forward ratesare positive. If the n-periodyield volatility satisŽ es the bounds in Table 2, then positive short rates do not exist. Thisindicates that the implied short rate volatility is either negative or inŽ nite, a consequenceof Theorem2.1 of Sandmann and Sondermann (1993) (or seeSection 2). For instance,in Example ( b) with Y^(12) = 4% and ^Y (12)= 20%,the implied short rate volatility turns out to be 36.65%. Theorem2.1 of Sandmann and Sondermann ensures that the BDT modelcan be calibrated since the forward ratesare positive and the short rate volatility is positive and Ž nite.Now ifone changes themagnitude of ^Y (12)so that it approaches the bounds in Table 2, the implied short rate volatility mustnecessarily be approaching zero or inŽ nity. This is illustrated in T able3 whichgives the implied shortrate volatility in the last period by Ž ttingto ^Y (12)= 19.994%,19.995%, 20.005% and 20.006% 42 Boyle,T anand Tian

Fig. 1. Regionsof nosolution for Example ( e).

Table 3. Theimplied short rate volatility in Example ( b)fordifferent input ^Y (n).

n-periodimplied short

^Y (12) (%) ratevolatility (%)

19.994 3.554 19.995 9.050 20.005 64.712 20.006 70.414

whilemaintaining Y^ (12)= 4%.In these cases, even with a smallperturbation of the yield volatility, the impliedshort rate volatility changes dramatically. More importantly, when the yield volatility is approachingthe bound given by condition ( i),theimplied short rate volatility gets larger .Onthe other hand,when the yield volatility is approaching the bound by condition( ii),theimplied short rate volatility getssmaller .

6.Conclusion

Wealreadyknow that all one-factor models such as the BDT modelhave limitations in modelling yieldcurve behaviour .Littermanand Scheinkman (1991) Ž ndthat three factors are required to provide anadequaterepresentation of yieldcurve dynamics. Hull and White (1995) have noted that if we overŽt aone-factorMarkovian model, we endup with unrealistic dynamics for thefuture evolution of volatility.Indeed they suggest that one should only Ž ttheinitial term structure of bond prices and Calibratingthe Black– Derman– Toy model 43 notoverparametrize the model. Radhakrishnan (1998) points out that the BDT modelgenerates pricingerrors irrespectiveof whether the model is Ž ttedto the yield volatilities or the short rate volatilities.By using the HJM modelas a benchmark,he Ž ndsthat if the yield volatility is used to calibratethe corresponding BDT model,the BDT modelunderprices options with long maturities. Conversely,he Ž ndsthat if the short rate volatility is used, the BDT modeloverprices options on long term bonds. Thispaper has pointed out some additional technical problems that may arise when Ž ttinga BDT latticeto an input set of yields and term structure of yield volatilities. The situation is morecomplicated when we Žttheyield volatility rather than the short rate volatility. W eŽndexplicit mathematicalconditions which indicate when it is feasible to Ž taBDT model.In this case they involvejoint restrictions on both the input bond prices and the yield volatilities. When we usethe yield volatilitiesas input, the resulting mathematical conditions to ensure a feasiblecalibration are quite complicated.If we usethe short rate volatilities as the input, then both the economic intuition and the mathematicalrestrictions are much simpler .AnalysisconŽ rms thatthe conventional practice of Žttingthe modelto the short rate volatilities rather than the yield volatilities has theoretical as well as practical advantages.

References

Arnon,D. (1981)Algorithms for the Geometry of Semi-Algebraic Sets. PhD thesis,T echnicalReport no. 436, ComputerScience Dept. University of Wisconsin-Madison. Black,F .,Derman,E. andT oy,W .(1990)A one-factormodel of interest rates and its application to treasury bond Options, FinancialAnalysts Journal , 46(1), 33–9. Brennan,M.J. and Schwartz, E. (1979)A continuous-timeapproach to pricing of bonds, Journalof Banking and Finance, 3, 135–55. Cox,J.C., Ingersoll, J.E. and Ross, S.A. (1985)A theoryof the term structure of interest rates, Econometrica , 53, 385–407. Collins,G.E. (1975)Quantifier elimination for real closed fields by cylindricalalgebraic decomposition. In G. Goos andJ. Hartmanis,editors, AutomataTheory and Formal Languages 2nd Gl Conference,Springer Lecture Notes inComputer Science ,Vol.33, pp. 134– 83. Heath,D., Jarrow,R. andMorton, A. (1990)Bond pricing and the term structure of interest rates: a discretetime approximation, Journalof Financial and Quantitative Analysis , 25(4),419– 40. Ho,T .S.Y.andLee, S. (1986)T ermstructure movements and pricing interest rate contingent claims, Journal of Finance, 41(5),1011– 29. Hull,J. andWhite, A. (1995)A noteof the models of Hull and White for pricing options on the term structure: a response, Journalof Fixed Income , 5(2),97– 102. Hull,J. andWhite, A. (1993)One-factor interest rate models and the valuation of interestrate derivative securities, Journalof Financial and Quantitative Analysis , 28, 225–254. Jacobson,N. (1985) BasicAlgebra ,VolumeI (secondedition). W .H.Freemanand Company, New York. Jamshidian,F .(1991)Forward induction and construction of yield curve diffusion models, Journalof FixedIncome , June, 62–74. Litterman,R. andScheinkman, J. (1991)Common factors affecting bond returns, Journalof Fixed Income , 1, 54–61. Pan,V .Y.(1997)Solving a polynomialequation: some history and recent progress, SIAM Review, 30(2), 187–220. 44 Boyle,T anand Tian

Radhakrishnan,A.R. (1998)Mispricing of discount bond options in the Black– Derman– Toy model calibrated to term structureand cap volatilities: an empirical study. W orkingpaper, New YorkUniversity, August. Sandmann,K. andSondermann, D. (1993)A termstructure model and the pricing of interest rate derivatives, The Reviewof Futures Markets , 12(2),391– 423. Tarski,A. (1951) ADecisionMethod for Elementary Algebra and Geometry ,secondedition. University of California Press,Berkeley. Vasicek,O.A. (1977)An equilibrium characterization of the term structure, Journalof Financial Economics , 5, 177–88.

Appendix A:QuantiŽer elimination

Webriey reviewthe main theorem of QuantiŽer Elimination(QE) inthis appendix.

Theorem 4 (QE Theorem)Let A = [t , . . ., t ], B = A[x , . . ., x ] where the t’s and x’sareindeterminates < 1 r 1 n and isthe real number Ž eld.Let < G F , . . . , Fm, G B ˆ f 1 g Thenwe candetermine in a Žnitenumber of steps (and in a constructiveway) aŽnitecollection {G1, . . ., Gs} where

Gj fj, 1, . . . , Fj, mj, gj A ˆ f g suchthat for any( c , . . ., c ) (r) thesystem of equationsand equation G(c , . . ., c ): 1 r 2 < 1 r F (c , . . . , cr; x , . . . , xn) 0 1 1 1 ˆ . .

Fm(c , . . . , cr; x , . . . , xn) 0 1 1 ˆ G(c , . . . , c ; x , . . . , x ) 0 1 r 1 n 6ˆ issolvable for the xs in R ifandonly if the ci satisfyone of the systems Gj(c1, . . ., cr):

fj, 1(c1, . . . , cr) 0 ˆ . .

fj, mj (c1, , cr) 0 ˆ gj(c , . . . , cr) 0 1 6ˆ for every 1 j s.Thereare no variables x1, . . ., xn inthefunctions fj, k, gj, K 1 j s, K 1 k mj. Thiselimination process was Žrstgiven by T arski(1951), and the techniques traced back to the work ofSturm, Euler and Bezout in the 18th century. T arski’s originalmethod was notpractical and was substantiallymodiŽ ed and improved by Collins (1975). Collins’ s algorithmhas been implemented in a computeralgebra system by Arnon (1981). Even in the three-step case, this efŽ cient algorithm is so complicatedthat it is impossible to give the criterion explicitly by the analysis of the computing time givenin Collins’ s originalpaper. Nevertheless, we canstill provide criteria to checkwhether there exists reasonableinterest rates in the BDT modelas discussed in Section 4. Calibratingthe Black– Derman– Toy model 45 Appendix B:Proofof Theorem 1

Firstwe needthe following lemma:

Lemma 5 If r(2, 0), r(2, 1), r(2, 2) (0, !) (39) 2 then u2 (max L , L , min M , M ) (40) 2 ‰ 1 2Š ‰ 1 2Š where 1 L 1 ˆ (1 ! t) 1 r(1, 1) t ‡ ‰ ‡ Š 2a 1 L 3 2 ˆ a ¡ 1 r(1, 0) t 1 ‡ 1 M 1 ˆ 1 r(1, 1) t ‡ 2a 1 M 3 2 ˆ a ¡ 1 r(1, 0) t (1 ! t) 1 ‰ ‡ Š ‡ 1 1 1 Proof. DeŽ ne r0 = (1 + r(2, 0) t)¡ , r1 = (1 + r(2, 1) t)¡ , r2 = (1 + r(2, 2) t)¡ .Itfollows from (39) that 1 r , r , r , 1 0 1 2 2 1 ! t ‡ BydeŽ nition, the variable u2 and v2 canbe expressed as 1 u2 (r r ) (41) ˆ 2 1 r(1, 1) t 1 ‡ 2 ‰ ‡ Š 1 v2 (r r ) (42) ˆ 2 1 r(1, 0) t 0 ‡ 1 ‰ ‡ Š Substituting(42) into (10) and rearranging, we obtain 1 4a r r u2 3 0 ‡ 1 (43) ˆ 2 a ¡ 1 r(1, 0) t 1 ‡ Since 2 r r , r r , 2 0 ‡ 1 1 ‡ 2 2 1 ! t ‡ 2 Equations41 and43 implythat the bounds on u are (L1, M1) and (L2, M2),respectively.This completes theproof to the lemma. 46 Boyle,T anand Tian

Toprove Theorem 1. Firstnote that the root of g1 solves(6) and(10). Furthermore, the positivity of the shortrates implies that we areonly interested in the positive root of g1.Itfollows from theabove lemma thatcondition (39) implies that 0 < u < (!). j Suppose g1( (!)) 0,we needto prove g1 (x)hasno rootin (0, (!)). Let ej denotethe coefŽ cient of x in(17). W ehave g (0) = e < 0 and g0 (0) = e <0.This implies that we onlyneed to examine the 1 ¡ 0 1 ¡ 1 behaviourof g1 (x) in (0, (!)). Wearguethat the function g1 iseither decreasing from x = 0 and then increasinguntil x = (!),oris decreasing in theentire range. This is equivalentto saying that g10 (x) = 0 has atmost one root in (0, (!)) 7 Suppose isthe root of g10 (x)=0inthe interval (0, (!)), and let , beother roots of this equation. Then we have 3e a b 3 ‡ ‡ ˆ ¡ 4 e a b a b 2 ‡ ‡ ˆ 2 a b e ˆ 1 If one of , isnot a realnumber, so is the other .Inthis case the proof is complete. Suppose both , arereal numbers. From thelast equation we have >0.Therefore,either > 0, > 0 or < 0, < 0. However, + = 3e /4 <0,whichleads to < 0, < 0. Hence, g0 (x)hasat most one root in the ¡ 1 ¡ 1 interval(0, (!)). Thiscompletes the proof of Theorem1.

Appendix C:Sturm’s Theorem

Inthis appendix, we summarizethe key result of Sturm’ s Theoremwhich is taken from Jacobson(1985). If c = {c , c , . . ., c }isa Žnitesequence of non-zero elements of ,thenwe deŽne the number of 1 2 m < variationsin sign of c tobe thenumber of i, 1 i m 1, such that c c <0.Onthe other hand, if c is ¡ i i + 1 anarbitrary sequence of elements in ,thenthe number of variations in sign of c isdeŽ ned to be the < numberof variations in the sign of thesubsequence c0 obtainedby dropping the 0s in c. Now let f (x)beany polynomial in [x]ofpositivedegree. W edeŽne the standardsequence for f (x) as < f (x) f (x) 0 ˆ f (x) f 0(x) (formal derivativeof f (x)) 1 ˆ f (x) q (x) f (x) f (x), deg f < deg f 0 ˆ 1 1 ¡ 2 2 1 . . . . (44)

fi 1(x) qi(x) fi(x) fi 1(x), deg fi 1 < deg fi ¡ ˆ ¡ ‡ ‡ . . . .

fs 1(x) qs(x)fs(x) i:e: fs 1(x) 0 ¡ ˆ ‡ ˆ 7 Note that , may becomplexnumbers. Calibratingthe Black– Derman– Toy model 47

Thus, the fi(x)areobtained by modifying the Euclid algorithm for Žndingthe g.c.d. of f (x) and f 0(x) in sucha waythat the last polynomial obtained at each stage is thenegative of theremainder in the division process. Wenowstate Sturm’ s Theorem.

Theorem 6 (Sturm’s Theorem)Let f (x)bea polynomialof positive degree with coefŽ cients in the realnumber Ž eld and let { f0(x) = f (x), f1(x) = f 0(x), . . ., fs(x)} bethe standard sequence (44) for f (x). Assume [a,< b]isan interval such that f (a) 0, f (b) 0.Then the number of distinct roots 6ˆ 6ˆ of f (x) in (a, b) is V V where V denotesthe number of variations in sign of { f (c), f (c), . . ., a ¡ b c 0 1 fs(c)}.

Remark: TheNewton– Raphson algorithm is a commonlyused technique for Žndingthe zeros of the polynomials.There are many other efŽ cient algorithms for approximatingpolynomial zeros as well. See therecent survey paper by Pan (1997).

Appendix D:Proofof Theorem 3

Proof. Substituting(6) into(10) and eliminating u, we obtain n n n 1 ¡ a n 1 1 n 1 ‡ 2 ‡ 0 (45) ‡ ¡ ‡ ‡ ¡ a 1 ˆ orequivalently,

1 a n 1 n ¡n 2 ‡ 1 n 1 (46) 1 a ¡ ¡ ˆ ‡ ¡ 1 Replacing v bythevariable x and n + 1 bythevariable y, and let f (x, y) and (x)tobe theexpression on theleft-hand side of (45)and (46) respectively; i.e. n n y ¡ a n 1 f (x, y) x 1 y 2 ‡ (47) ˆ ‡ ¡ ‡ x ¡ a 1

1 x a n 1 n ¡n (x) 2 ‡ x 1 (48) ˆ 1 x a ¡ ¡ ¡ 1 Byconstruction, we have f (x, (x)) = 0, for x > 0. 1=n 1=n Let ( n 1= 1) and a Pd(n) . Then ( ) = 1 and (x) > 1 for x ( , 1), (x) < 1 for ˆ ‡ ˆ ‰ Š 2 x (0, ). 2 In (i), note that f (a, (a)) = 0 and a (0,1). Moreover f isstrictly increasing as a functionof x for any 2 Ž xed y > 0. Hence f (x, (a)) < 0, x (0, a). Onthe other hand, f (x, y)isstrictly decreasing as a function of y for anyŽ xed0 < x <1.This 8 implies 2 that f (x, ) < 0, > (a), x (0, a). 8 2 BydeŽ nition, v canbe written as n n Ad(n, i) 1 Pd(n) i ˆ i 0 Pd(n) 1 r(n, 0)b n t Xˆ ‡ 48 Boyle,T anand Tian

Sinceall the short rates up totimeperiod n areassumed to be positive, it follows from theconvex property that v (0, a). Hencethere exist no positiveinterest rates in ( n +1)thperiod when the yield volatility ^ 2 y (n +1)satisŽes that = exp[2^y(n + 1) p t)] > (a). Thus (i) is proved. Toprove part ( ii), we considerthe equation in terms of u andassume that <1(bysymmetry). Let b =  [P (n)]1/n (0,1). Then the proof is similar for u 2 exp 2^ (n 1)p t > (b) ˆ ‰¡ y ‡ Š whichis equivalent to  1 1 ^y(n 1) < log ‡ 2 t (b)