Numerical Procedures2

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NUMERICALROC EDU RES FORIMPLEMENTINGERMSTRUCTUREODELS11:TWO-FACTORODELSJohn Hullis a professor in the Faculty o f Management at the University o f Toronto inOntario, Canada.Alan Whiteis also a professor in the Faculty o f Management at the University o f Toronto.

In the last issue ofthislournal we explained a newprocedure fO r building trinomial trees fo r one-fctor no-arbi-trage models o f the term structure. The procedure is appro-priate for models where there is some function, x , o f theshort rate, r, that follows a mean-reverting arithmetic process.It is relatively simple an d computationally more eflcientthan previously proposed procedures.

In this article we extend the new tree-building procedurein two ways. First, we show how it can be usedto model theyield curves in two dgu en t countries simultaneously Second, weshow how to develop a variety ofMarkov two-factor models.

In order to model the yield curves in ,two dgerentcountries simultaneously, wefirst build a treefor each oftheyield curves separately. The trees are then adjusted so that

they are both constructed >om the viewpoint of a risk-neu-tral investor in the country in which the cashJows are real-ized. We then combine the two trees on the assumption thatthere is no correlation. In th e fi na l step we induce therequired amount ofcorrelation by adjusting the probabilitieson the branches emanatingfrom each node.

The two-jactor Markov models o f the term structurethat we consider incorporate a stochastic reversion level.They have the property that they can accommodate a muchwider range o f volatility structures than the one-jactor modelsconsidered in our earlier article. In particular, they can giverise to a volatility hump similar to that observed in thecap market. We show that one version o f the two-factormodels is somewhat analytically tractable.

he most general approach to constructingmodels of the term structure is the one sug-gested by Heat h, Jarrow, and Mor ton[1992]. This involves specifying rhe volatili-

ties of all forward rates at all times. The expected

drifts of forward rates in a risk-neutral world are cal-culated from their volatilities, and the initial values ofthe forward rates are chosen to be consistent with theinitial term structure.

Unfortunately, the models that result from theT

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Heath, Jarrow, and Morton approach are usually non-Markov, meaning that the parameters of the process forthe evolution of interest rates at a hture date dependon the history of interest rates as well as on the termstructure at that date. To develop Markov one-factormodels, an alternative approach has become popular.This involves specifying a Markov process for theshort-term interest rate, r, in terms of a function oftime, e(t).The function of time is chosen so that themodel exactly fits the current term structure.

Hull and White [1994] consider models of theshort rate, r, of the form

dx =[e(t)- axldt +odz (1)where x =f(r) for some function f; a and are con-stants; and e(t) s a function of time chosen so that themodel provides an exact fi t to the initial term struc-ture. This general family contains many of the com-mon one-factor term structure models as special cases.When f(r) =r and a =0, the model reduces to

dr =e(t)dt +odzThis is the continuous time limit of the Ho and Lee[1986] model. When f(r) = r and a # 0, the modelbecomes

dr =[e(t)- arldt +odzand is a version of the Hull and White [1990] extend-ed-Vasicek model. When f(r) =log(r), the model is

dlog(r) = [e(t)- log(r)]dt+odzwhich is a version of Black and Karasinski [1991].

In this article we extend the ideas in Hull andWhite [1994] to show how the yield curves in twodifferent countries can be modeled simultaneously.We also show how the Hull-White approach can beused to develop a variety of dfferent Markov two-fac-tor models of the term structure.I. THE HULLWHITE ONE-FACTORTREE-BUILDING PROCEDURE

The first step for building a tree for the model

in Equation (1) is to set e(t) = 0 so that the modelbecomes

A time step, At, is chosen, and a trinomial tree is con-structed for the model in Equation (2) on the assumptionthat the initial value of x is zero. The spacing between thex-values on the tree, Ax , is set equal to o G . heprobabilities on the tree are chosen to match the meanand the standard deviation of changes in x in time At.

The standard branching process on a trinomialtree is one up, no change, or one down. This meansthat a value x =jA x at time t leads to a value of xequal to one of (j +l)Ax, Ax , or (j- 1)Ax at time t +At. Due to mean reversion, for large enough positiveand negative values of j, when this branching processis used, it is not possible to match the mean and stan-dard deviation of changes in x with positive probabili-ties on all three branches. To overcome this problem,a non-standard branching process is used in which thevalues for x at time t + At are either (j + 2)Ax,(j +l)Ax, and Ax, or Ax , (j - )Ax, and (j- ) A x .

Once the tree has been constructed so that itcorresponds to Equation (2), it is modified by increas-ing the values of x for all nodes a t time iAt by anamount a,. he a i s are chosen inductively so that thetree is consistent with the initial term structure. Thisproduces a tree corresponding to Equation (1).Whenthe Ho-Lee or Hull-White model is used so that x =r, the values of ai can be calculated analytically. Inother cases an iterative procedure is necessary.

This procedure provides a relatively simple andnumerically efficient way of constructing a trinomialtree for x and implicitly determining e(t).The treehas the property that the central node at each time isthe expected value of x at that time.

Exhibit 1 shows the tree that is produced when x=r, a =0.1, o =0.01, At =one year, and the function0.08- .05e4.8t defines the t-year zero rate.l Note thatthe branching is non-standard at nodes E (where j ispositive and large) and I (where is large and negative).II. MODELING TWO INTEREST RATESSIMULTANEOUSLY

Certain types of interest rate derivatives require

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EXHIBIT 1TREE SSUMED OR BOTHrl AND r2- ORRESPONDSOTH E PROCESSr =[e@)- rldt +odz WHERE a =0.1,IS 0.08- .05e-0.'8t0 =0.01, At =ONE YEAR, AND THE f-YEAR ZER O RATE

N o d e A B C D E F G H Ir 3.82% 6.93% 5.20% 3.47% 9.71% 7.98% 6.25% 4.52% 2.79%p, 0.167 0.122 0.167 0.222 0.887 0.122 0.167 0.222 0.087p, 0.666 0.656 0.666 0.656 0.026 0.656 0.666 0.656 0.026pd 0.167 0.222 0.167 0.122 0.087 0.222 0.167 0.122 0.887

the yield curves in two different countries to be mod-eled simultaneously. Examples are diff swaps andoptions on diff swaps. Here we explain how the pro-cedure in Hull and White [1994] can be extended toaccommodate two correlated interest rates.

For ease of exposition, we suppose the twocountries are the United States and Germany, and thatcash flows from the derivative under consideration areto be realized in U.S. dollars (USD). We 'first build atree for the USD short rate and a tree for thedeutschemark (DM) short rate using the procedureoutlined above and described in more detail in Hulland White [1994].As a result of the construction pro-cedure, the USD tree describes the evolu~on f USDinterest rates from the viewpoint of a risk-neutralUSD investor, and the DM tree describes the evolu-tion of DM rates from the viewpoint of a risk-neutralDM investor.

Adjusting the Deutschemark TreeSince cash flows are realized in USD, the DM

tree must be adjusted so that it reflects the evolutionof rates from the viewpoint of a risk-neutral U.S.,investor rather than a risk-neutral DM investor.

r, :r2:r l :xox:Px:P:

and

DefineUSD short rate from the viewpoint of a risk-neutral U.S. investor;DM short rate from the viewpoint of a risk-neu-tral DM investor;DM short rate from the viewpoint of a risk-neu-tral U.S. investor;the exchange rate (number of USD per DM);volatility of exchange rate, X;instantaneous coefficient of correlation betweenthe exchange rate, X, and the DM interest rate, r2;instantaneous coefficient of correlation betweenthe interest rates, rl and r2.

Suppose that the processes for rl and r2 ared x , = [el(t)- lxl]dt +o,dzl

dx , =[e2(t) 2x2]dt+02dz2where x1 =fl(rl) and x2 =f2(r2) or some functions f,and f2, and dz, and dz2 are Wiener processes with cor-relation, p. The reversion rate parameters, al and a2 ,and the standard deviations, (3, and 02 , re constant.The drift parameters, 8, and e,, are functions of time.

We show in Appendix A that the process forr l isdx; = [e,(t) - pxo20x - a2x;]dt + 02dz2

where x l = f2 (r l) . The effect of moving from aDM risk-neutral world to a USD risk-neutral world isto reduce the drift of x2 by pX(3,(3,. The expectedvalue of x2 at time t is reduced by

To adjust the DM tree so that it reflects the viewpointof a risk-neutral U.S. investor, the value of x2 a t nodesa t time iAt (i.e., after i time steps) should therefore bereduced by2



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BD BC BB0.0278 0.1111 0.0278

Note that this is true for all functions f2, not j u s tf2(r)=r.To give an example, we suppose that fl(rl )=rl ;

f2(r2)=r2;al =a2 =0.1; and 0, =o2=0.01. We alsosuppose that the t-year zero rate in both USD andDM is 0.08 - 0.05e4.18f initially. (This resembles theU.S. yield curve at the beginning of 1994.)

The tree initially constructed for r l and r2when At =1 is shown in Exhibit 1. Suppose the cor-relation between r2 and the exchange rate px =0.5,and the exchange rate volatility ox =0.15. To createthe tree for ri from the r2 tree, the interest rates rep-resented by nodes on the tree a t the one-year pointare reduced by 0.5 x 0.01 x 0.15 (1 - e4.')/0.1 =0.00071 (0.071%). Similarly, interest rates at the two-and three-year points are reduced by 0.00136 and0.00194, respectively.

Constructing the TreeAssuming Zero CorrelationWe next combine the trees for USD and

DM interest rates on the assumption of zero corre-lation. The result is a three-dimensional tree wherenine branches emanate from each node. The prob-ability associated with any one of the nine branch-es is the product of the unconditional probabilitiesassociated with corresponding movements in thetwo short rates.

In the three-dimensional tree with the assumedparameter values, there is one node at time 0, ninenodes a t time At, twenty-five nodes a t time 2At,twenty-five nodes a t time 3At, and so on. (Note thatthe effect of mean reversion is to curtail the rate a twhich the number of nodes on the tree increases.)Each node at time iAt on the combined tree corre-sponds to one node at time iAt on the first tree andone node at time iAt on the second tree. We will use anotation where node XY on the combined tree cor-responds to node X on the first tree and node Y onthe second tree.

The nine nodes at time At and the probabilityof reaching each of the nodes from the initial nodeAA are:

CD cc CB0.1111 0.4444 0.1111DD DC DB0.0278 0.1111 0.0278For example, the probability of reaching node

BC from node AA is 0.1667 x 0.6666 =0.1111. Toillustrate how probabilities are calculated over the sec-ond time step, consider node BD, where rl =6.93%,r2 =3.47%, and rl =3.40%. From B we branch to E,F, G, and from D to G, H, I . So from BD we branchto the nine combinations of E, F, G and G, H, I. Thenine nodes that can be reached from node BD togeth-er with their probabilities are:

E1 EH EG0.0149 0.0800 0.0271FI FH FG0.0800 0.4303 0.1456

GI GH GG0.0271 0.1456 0.0493For example, the probability of reaching node EG h mnode BD is 0.122 X 0.222 =0.0271, the product of theprobability of branching h m B to E and from D to G.

To express the calculations more formally, wedefine node (i, j) as the node on the rl tree at timeperiod iAt at which the number of prior up-moves inthe interest rate minus the number of prior down-moves is j . Similarly, node (i, k) is the node on the r;tree at time period iAt at which the number of priorup-moves in the interest rate minus the number ofprior down-moves is k.Let rl(i, j) be the value of rlat node (i , j), and rl (i, k) be the value of rl at node(i, k) on the r i tree.

The three-dimensional tree is a combination ofthe two two-dimensional trees. At time iAt the nodesare denoted (i, j , k) where rl = rl(i, j) and rl = rl(i, k) for all j and k.Assume the probabilities associat-



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ed with the upper, middle, and lower branches ema-nating from node (i, j) in the rl tree are p p,, andpd. Similarly, assume that the upper, middle, and lowerbranches emanating from node (i, k) n the r i tree are

In the three-dimensional tree there are ninebranches emanating from the (i,j, k) node. The prob-abilities associated with the nine branches are:

qm, and qd'

rl -MoveLower Midd le , Upper

Upper Pdq, Pmqu PU9,Middle Pdq, Pmqm PuqmLower Pdqd P m q d puqd

For correlations between 0 and 1, the correla-tion matrix is the result of interpolating between thematrix for a correlation of 0 and the matrix for a cor-relation of 1 3

Suppose p = 0.2 so that E 10.00556. Theprobabilities on the branches emanating from nodeAA become:BD BC BB0.0222 0.0889 0.0556CD cc CB0.0889 0.4889 0.0889DD DC DB0.0556 0.0889 0.0222Building in Correlation

We now move on to consider the situationwhere the correlation between rl and r l , p, is non-zero. Suppose first that the correlation is positive. Thegeometry of the tree is exactly the same, but the prob- E1 EH EGabilities are adjusted to be: 0.0093 0.0578 0.0549

while the probabilities on branches emanating fromnode BD become

Note that the sum of the adjustments in eachrow and column is zero. As a result, the.adjustmentsdo not change the mean and standard deviations ofthe unconditional movements in rl and rl. Theadjustments have the effect of inducifig a correlationbetween rl and ri of 36s. The appropriate value of Eis therefore p/36.

The choice of the probability adjustments ismotivated by the fact that in the limit as' At tends tozero the probabilities tend to p, =q, = lY6, p, =q,= 2/3, and pd = qd = 1/6. When the correlation is1O, the adjusted probability matrix is in the limit:

FI FH FG0.0578 0.4747 0.1234

GI GH GG0.0549 0.1234 0.0437For example, the probability of moving from

node AA to node BC is 0.1111 - 4 x 0.00556 =0.0889; the probability of moving from node BD tonode EG is 0.0271 +5 x 0.00556 =0.0549.

For negative correlations, the procedure is thesame, except that the probabihties are:

rl-MoveLower Middle Ume r

r, -Move In this case, & = -p/36. For a correlation ofLower Middle , Upper -0.2, the probability on the branch from AA to BC is

Upper 0 0 1 6 as before: 0.1111- 4 x 0.00556 =0.0889; the proba-;-Move Middle 0 2/3 0 bility of moving from node BD to node EG is 0.0271

Lower 1/6 0 0 -0.00556 =0.0215.

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The only &fficulty with this procedure is thatsome probabilities are liable to become negative atsome nodes. We deal with this as follows. At any nodewhere E =p/36 leads to negative probabilities, we usethe maximum value of E for which probabilities arenon-negative.

To illustrate this rule, suppose that p = 0.8instead of 0.2 in our earlier example, so that the cal-culated value of E is 0.02222. Using this value of E atnode BD would cause the probabilities on the branch-es to EH, EI, and FI to become negative. The maxi-mum values of E for which the three probabilities arenon-negative are 0.0200, 0.0149, and 0.0200, respec-tively. We would therefore use a value of E of 0.0149,which corresponds to p = 0.5364, for all branchesemanating from BD.As At approaches zero, pu, p,, and pd approach1/6, 2/3, and 1/6. Also qu , q,, and qd approach 1/6,2/3, and 1/6. The proportion of the probability spaceover which it is necessary to adjust E reduces to zero.Although the procedure we have outlined does inducea small bias in the correlation, this bias disappears inthe limit as At approaches zero. As a result the proce-dure converges.

An ExampleWe illustrate the procedure by using it to pricea security whose payoff is calculated by observing the

three-month DM rate in three years' time and apply-ing it to a USD principal of 100. This is one compo-nent of a diff swap in which the swap payment in dol-lars is determined by the difference between theGerman and U.S. interest rates.As before, we suppose that the process for bothshort rates is

dr =[e(t)- arldt +odzwith a =0.1and 0=0.01.

The t-year zero-coupon bond rate in eachcountry is assumed to be 0.08 - 0.05e-0.'8f, and thevoladity of the exchange rate is assumed to be 15%per year. This example enables us to test the conver-gence of the tree-building procedure since the securi-ty can be valued analytically using the formulas in Wei[1994].

Recall that px is defined as the correlation

EXHIBIT 2VALUE OF A SECURITY WHO SE PAYOFF IS CALCULATED BYOBSERVINGHE THREE-MONTHM RATE NTHREEEARS'TIMEND APPLYINGT TO A US DPRINCIPAL OF 100Number ofTime Steps

51020406080

100Analytic

p =0.5px =0.51.39711.39761.39781.39801.39811.39811.39811.3981

p =0.51.48341.48391.48421.48441.48451.48451.48451.4845

px =-0.5 p ~ - 0 . 5 p =-0.5px =0.5 px =-0.51.4045 1.49091.4055 1.49181.4059 1.49231.4062 1.49261.4063 1.49271.4063 1.49271.4064 1.49281.4064 1.4928

Note: Both USD an d D M rates are assumed to follow the processdr = [e(t) - arldt +o dz where a =0.1, o =0.01, At = one year,and the t-year zero rate is 0.08 - .05e-0.'8t.

between the exchange rate and the DM rate, and p isthe correlation between the two interest rates. Exhibit2 shows results for a number of combinations of p andpx. It illustrates that prices calculated by the tree doconverge rapidly to the analytic price in a variety ofdifferent correlation environments.III. TWO-FACTOR MODELS OF ASINGLE TERM STRUCTURE

Now we consider two-factor models of the formdf(r) = [0(t) +u - f(r)]dt +oldzl (3)

where the drift parameter u has an initial value ofzero, is stochastic, and follows the process

As in the one-factor models considered in Hulland White [1994], the parameter 0(t) is chosen tomake the model consistent with the initial term struc-ture. The stochastic variable u is a component of thereversion level of r and itself reverts to a level of zeroat rate b.4 The parameters a, b, o,, and '3, are con-stants, and dz, and dz, are Wiener processes with

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instantaneous correlation p.This model provides a richer pattern of term

structure movements and a richer pattern of volatil-ity structures than the one-factor models considereda t the outset. Exhibit 3 gives an example of forwardrate standard deviations and spot rate standard devi-ations that are produced using the model when f(r)= r, a =3, b =0.1, 0, =0.01, B, =0.0145, and p=0.6.5 he volatilities that are observed in the capmarket often have the humped shape shown inthis plot.

When f(r) = r, the model is analyticallytractable. As shown in Appendix B, the bond price inthat case has the form

EXHIBIT 3FORWARDATE OLATILITIES (F-VOLS) ND SPOT R A T EVOL ATI LIT IES (Y -V OL S) IN TWO-FACTOR MODELWHEN f(r) =r, a =3, b =0.1, (zl=0.01,oz=0.0145,AN D p =0.6

StandardDeviations (%)T

F-VOISY-Vols

0.20

The price, c, at time t of a European call option on adiscount bond is given by

c =P(t, s)N(h)- XP(t, T)N(h-Op)0.00 : : : : : : : : ; : I ; : : ; : : : : I

O - N O P ~ Q Cwhere T is the maturity of the option, s is ithe maturi-ty of the bond, X is the strike price, Years

OPOp P(t, T)X 2+ -P(t, s)h = -log Uy = x + - b - a

and Op is as given in Appendix B. Since this is a two-factor model, the decomposition approach inJamshidian [1989] cannot be used to price options oncoupon- bearing bonds.

So thatdy =[0(t)- yldt +6,dz3du =-budt +02h2Constructing the Tree

To construct a tree for the model in whereEquation (3), we simplifjr the notation by definirlgx =f(r) so that 2P01020; = 0; + 4 +-- a ) , b - adx =[0(t)+u -~ ] d t B,dz,with and dz3 is a Wiener process. The correlation betweendx, and dz, isdu =-budt +O,dz,Assuming a f b, we can eliminate the dependence ofthe first stochastic variable on the second by defining

PO, + O,/@ - a)0 3



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The approach described in Section I1 can beused to construct a tree for y and u on the assumptionthat e(t) =0 and the initial values of y and u are zero.Using a simdar approach to that described in H d ndWhite [1994], we can then construct a new tree byincreasing the values of y at time i h by a,. he aisare calculated using a forward induction techniquesimilar to that described in Hull and White [1994] sothat the initial term structure is matched. The detailsare given in Appendix C.

An ExampleExhibit 4 shows the results of using the tree to

price three-month options on a ten-year discountbond. The parameters used are those that give rise tothe humped volatility curve in Exhibit 3 . The initialt-year zero rate is assumed to be 0.08 - 0.05e4.18'Five different strike prices are considered.

Exhibit 4 illustrates that prices calculated fromthe tree converge rapidly to the analytic price. Thenumbers here and in other similar tests we have car-ried out provide support for the correctness of thesomewhat complicated formulas for A(t, T) and op nAppendx B.

be used to model two correlated interest rates wheneach follows a process chosen from the family of one-factor models considered in our earlier article. It canalso be used to implement a range of different two-factor models.An interesting by-product of the researchdescribed here is a method for combining trinomialtrees for two correlated variables into a single three-dimensional tree describing the jo in t evolution ofthe variables. This method can be extended so that itaccommodates a range of binomial as well as trino-mial trees.APPENDIX A

In this appendix we show how to calculate the pro-cess for the DM interest rate from the viewpoint of a USDinvestor. Our approach is an alternative to that in Wei[19941.

Define Z as the value of a variable seen from theperspective of a risk-neutral DM investor and Z* as thevalue of the same variable seen fiom the perspective of arisk-neutral USD investor. Suppose that Z depends only onthe DM risk-free rate so that

dZ =p(Z)Zdt +o(Z)Zdz,IV. CONCLUSIONS

This article shows that the approach in Hulland White [1994] can be extended in two ways. It canEXHIBIT 4DISCOUNT BONDWITH A PRINCIPAL OF 100 FOR MODELV A L U E OF THREE-MONTHP T IO N O N A B N - Y E A RIN EXHIBITNumber of Strike Price*Time Steps 0.96 0.98 1.00 1.02 1.04

5 1.9464 1.0077 0.2890 0.0397 0.001210 1.9499 1.0077 0.2968 0.0370 0.001620 1.9512 1.0085 0.3000 0.0366 0.001540 1.9536 1.0107 0.3022 0.0369 0.001460 1.9529 1.0099 0.3020 0.0367 0.0014

Analytic 1.9532 1.0101 0.3023 0.0367 0.0014*The strike price is expressed as a proportion of the forward bondprice.Note: The t-year zero rate is 0.08 - 0.05e4~'*'.

where dz, is the Wiener process driving the DM risk-freerate, and p and (5 are functions of Z. The work of C o x ,Ingersoll, and Ross [1985] and others shows that the pro-cess for Z* has the form:

dZ*=[p(Z*)- (Z*)]Z*dt+~(Z*)z*dz,where the risk premium,h, is a function of Z*.6

We first apply this result to the case where the vari-able under consideration is the DM price of a DM discountbond. Define P as the value of this variable from the view-point of a risk-neutral DM investor and P* as its value fromthe viewpoint of a risk-neutral USD investor. From theperspective of a risk-neutral DM investor, the variable isthe price of a traded security so that:

dP =r,Pdt +opPdz, (A-1)where oP s the volatility of P, and other variables are asdefined in the body of the article. Hence:'

dP* = [rl - bp]P*dt + bpP*dz2 (A-2)44 NUME RI CAL PROCEDURES FOR IMPLEMENTING TERM STRUCTURE MODELS 11: TWO-FACTOR MODELS WINTER 1994


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The risk-neutral process for the exchange rate, X, providmgfrom the viewpoint of a risk-neutral USD investor is

Bt-& +1 = OdX = (rl - rg)Xdt + oxXdzx

where dz, is a Wiener process.The variable XP* is the price in USD of the DM

C,- C +B =01 1A, - e(t)m+ ,o?m2+ _CT;AC~+L L

bond. The drift of XP* in a risk-neutral USD world musttherefore be rlXP*. From Equations (A-2) and (A-3), thisd d t can also be written as

po,o,ABc = 0

It follows that

h =PXOX

The solution to Equation (B-1) that satisfies the boundarycondition B(t, t) =0 is

When moving from a risk-neutd DM investor to a risk-ment of p,~,.the beginning of Appendix A to the variable f(r2). We areassuming that:

The Solution to Equation (B-2) that is consistent with thsneutral USD investor there is a market price of risk adjust- and the c( t*, = is

We can now apply the general result gven for Z a t e-a(T - t) -C(t, T) = a(a - b)1 e-b(T - t) 1+ -b(a - b) ab

Hence By direct substitution, the solution to Equation (B-3) for Aand 8 that satisfies the boundary conditions A(t, T) =A(0, T) when t =0 and A(t, T) =1 when t =T isdf2(r;) = [e,(t) - pxo,02 - af2(r;)ldl + o,dz,

APPENDIX B logA(t, T) = log- p(o' T, + B(t, T)F(O, T) +P(0, t)In ths appendut we show that the complete yield

curve can be calculated analytically from r and u in themodel considered in Section 111when f(r) =r.

The differential equation satisfied by a discountbond price, f , is

N O , t)B(t, T) +6 O , T)dT -J&T' T)dT

f, + [e(t)+ u - arlf, - buf, +1 1,ol"fm + ,o;f"" + polo,fm - rf = 0L L

e(t) = Ft(O, t) + aF(0, t) + $,(O, t) + a+@, t)where F(t, T) is the instantaneous forward rate for time Tas seen at time t, and

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This reduces to

where

-2aT 2at -11 - e (e 1)(a+b)T (a+b)t -e [eY1 = (a +b)(a - b) 2a(a - b)1

ab 2y1 + C(t, T) - C(0, T) + -B(t, T)21 e-a(T-t) - e-aT]2 a a2-B(O, V2 + - -

.-(a+b)t - 1 e-2at - 1+Y3 = -( a - b)(a + b) 2a(a - b)

2+a e-at a - 'I1-C(O, T)2 + Y22

The volatility function oP s given by0; =J;{G:[B(T, T) - B(2, t)I2 +

u = 1 [e-aT - e-at]a(a - b)and

v = 1 [e-b= - -bt]b(a - b)The first component of 0: is

The second is

2 a + bThe third is

APPENDIX CIn t l u s appendix we e x p i n how the tree for the two-factor model discussed in Section I11 is fitted to the initialterm structure. We assume that a three-clmensional tree for yand u has been constructed on the assumption that e(t)=0

46 NUMERICAL PROCEDURES FOR IMPLEMENTING TERM STRUCT URE MODELS 11 TWO-FACTORMODELS WINTER 1994


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using the procedure in Section 11, that the spacing between y-values on the tree is Ay, and that the spacing between the u-values on the tree is Au. From Equation (4) when y =jAyand u =Mu, the short rate, r is given by the initial tree to be

r = gjAy --,Fa]where g is the inverse hnct ion off .As in Hull and White [1994], we work forward

through the tree calculating the c l ~hat must be added tothe ys so that the tree is perfectly consistent with the initialterm structure. Once the appropriatea or time iAt, ai, asbeen calculated, it is used to calculate Arrow-Debreu pricesfor the nodes at time iAt. (The Arrow-Debreu price for anode is the present value of a security that pays off $1 if thenode is reached and zero otherwise.) These are then usedto calculateai+,nd so on.

Define Qij,k s the present value of a security thatpays off $1 if y =ai+jAy and u =kAu at itime iAt, andzero otherwise. Qo,o,o= 1. Simdarly to Equation (6) inHull and White [1994],

where the summation is taken over all values o f j and k attime mAt. When f(r) =r so that g(r) = r thrs :can be solvedanalytxcally for a,:

In

am=logCj,kQmj,kexp{jAY - uu/(b - a)) - l0gp,+l

Atother situations, a one-dimensional Newton-Raphson

search is required.The Q s are updated as the tree is consgructed using

exp{-g[a, + j*Ay - kAu/(b - a)]At}where in the summation j* and k* are set equal to all possi-

ble pairs of values of j and k at time mat. The variableq(j, k, j*, k*) is the probability of a transition from node(m, *, k*) to (m +1, , k).ENDNOTES

Exhibit 1 is the same as Exhibit 3 in Hull andWhite [1994]. Fuller details of how it is developed aregiven in that article.

2Using the notation in Hull and White [1994],t h s is the amount by whch a, should be adjusted once thetree has been constructed. Since the r on the tree is the At-period rate rather than the instantaneous rate, the adjust-ment is exact only in the limit as At tends to zero.

3The procedure described here can (with appro-priate mo&fications) be used to construct a three-&men-sional tree for any two correlated variables from two treesthat describe the movements of each variable separately.The two trees can be binomial or trinomial.

4There is no loss of generhty in assuming that thereversion level of u is zero and that its initial value is zero.For example, if u reverts to some level c, u* = u - ctreverts to 0. We can define u* as the second factor andabsorb the difference between u and u* in e(t).

5These parameters are chosen to p roduce thedesired pattern. Many other different volatdty patterns canbe achieved.

6h s the difference between the market price ofthe DM interest rate risk from the perspective of a risk-neu-tral DM investor and the market price of DM interest raterisk from the perspective of a risk-neutral USD investor.

Note that r2 in Equation (A-1) becomes r; inEquation (A-2). Ths is because the r2 in (A-2) is, strictlyspeakmg, r2(P). The Cox, Ingersoll, and Ross result out-lined a t the beginning of the appendix shows that thisbecomes r2(P*)or r l in Equation (A-2).

Black, F. and P. Karasinslu. Bond and Option Pricingwhen Short Rates are Lognormal. Financ ia l Ana lys t sJournal, July-August 1991, pp. 52-59.Cox, J.C., J.E. Ingersoll, and S.A. Ross. An IntertemporalGeneral Eqdbrium Model of Asset Prices. Econornetrica,53 (1985), pp. 363-384.Heath, D., R. Jarrow, and A. Morton. Bond Pricing andthe Term Structure of Interest Rates: A NewMethodology. Econornetrica, 60, 1 (1992), pp. 77-105.



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Ho, T.S.Y., and S.-B. Lee. Term Structure Move-ments and the Pricing of Interest Rate ContingentClaims. Journal of Finance, 41 (December 1986), pp.1011-1029.Hull, J., and A . White. Numerical Procedures forImplementing Term Structure Models I: Single-FactorModels.Journal ofDerivatives, 2, 1 (Fall 1994), pp. 7-16.

- Pricing Interest Rate Derivative Securities. Review$Financial Studies, 3, 4 1990), pp. 573-592.Jamshihan, F. An Exact Bond Option Pricing Formula.Journal $Finance, 44 (March 1989), pp. 205-209.Wei, J. Valuing Dkfierential Swaps. Journal ofDerivatives,1, 3 (Spring 1994), pp. 64-76.

ERRATUM The last equation on page 13 (unnumbered)should read

T h e f i rs t part of this article, NumericalProcedures for Implementing Term Structure ModelsI: Single-Factor Models, published in the Fall 1994issue of The Journal ofDerivatives, contained an equa-tion that was stated incorrectly.

log[j:lmQ m , j ei-.) logp(0, m + 1)Ata, =


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