2-factor black karasinski interest rate model.pdf

A Two-factor Lognormal Model of the Term

Structure and the Valuation of American-Style

Options on Bonds

Sandra Peterson1 Richard C. Stapleton2 Marti G. Subrahmanyam3

February 25, 1999

1Department of Accounting and Finance, The Management School, Lancaster University, Lan-caster LA1 4YX, UK. Tel:(44)524{593637, Fax:(44)524{847321.

2Department of Accounting and Finance, Strathclyde University, Glasgow, UK. Tel:(44)524{381172, Fax:(44)524{846 874, e{mail:[email protected]

3Leonard N. Stern School of Business, New York University, Management Education Center, 44West 4th Street, Suite 9{190, New York, NY10012{1126, USA. Tel: (212)998{0348, Fax: (212)995{4233, e{mail:[email protected].

Abstract

A Two-factor Lognormal Model of the Term Structure and

the Valuation of American-Style Options on Bonds

We build a no-arbitrage model of the term structure, using two stochastic factors, the short-term interest rate and the premium of the forward rate over the short-term interest rate.The model extends the lognormal interest rate model of Black and Karasinski (1991) totwo factors. It allows for mean reversion in the short rate and in the forward premium.The method is computationally e�cient for several reasons. First, we use Libor futuresprices, enabling us to satisfy the no-arbitrage condition without resorting to iterative meth-ods. Second, the multivariate-binomial methodology of Ho, Stapleton and Subrahmanyam(1995) is extended so that a multiperiod tree of rates with the no-arbitrage property canbe constructed using analytical methods. The method uses a recombining two-dimensionalbinomial lattice of interest rates that minimizes the number of states and term structuresover time. Third, the problem of computing a large number of term structures is simpli�edby using a limited number of 'bucket rates' in each term structure scenario. In addition tothese computational advantages, a key feature of the model is that it is consistent with theobserved term structure of volatilities implied by the prices of interest rate caps and oors.We illustrate the use of the model by pricing American-style and Bermudan-style optionson bonds. Option prices for realistic examples using forty time periods are shown to becomputable in seconds.

A two-factor lognormal model of the term structure 1

1 Introduction

One of the most important and di�cult problems facing practitioners, in the �eld of in-terest rate derivatives in recent years, has been to build inter-temporal models of the termstructure of interest rates that are both analytically sound and computationally e�cient.These models are required both to help in the pricing and in the overall risk management ofa book of interest rate derivatives. Although many alternative models have been suggestedin the literature and implemented in practice, there are serious disadvantages with most ofthem. For example, Gaussian models of interest rates, which have the advantage of analyt-ical tractability, have the drawback of permitting negative interest rates, as well as failingto take into account the possibility of skewness in the distribution of interest rates. Also,many of the term-structure models used in practice are restricted to one stochastic factor.

Since the work of Ho and Lee (1986), it has been widely recognized that term-structuremodels must possess the no-arbitrage property. In this context, a no-arbitrage model is onein which the forward price of a bond is the expected value of the one-period-ahead spot bondprice, under the risk-neutral measure. Building models that possess this property has beena major pre-occupation of both academics and practitioners in recent years. One modelthat achieves this objective in a one-factor context is the model proposed by Black, Dermanand Toy (1990)(BDT), and extended by Black and Karasinski (1991)(BK). In essence, themodel which we build in this paper is a two-factor extension of this type of model. In ourmodel, interest rates are lognormal and are generated by two stochastic factors. The generalapproach we take is similar to that of Hull and White (1994)(HW), where the conditionalmean of the short rate depends on the short rate and an additional stochastic factor, whichcan be interpreted as the forward premium. In contrast to HW, and in line with BK, webuild a model where the conditional variance of the short rate is a function of time. Itfollows that the model can be calibrated to the observed term structure of interest ratevolatilities implied by interest rate caps/ oors. Essentially, the aim here is to build a termstructure model which can be applied to value American-style contingent claims on interestrates, which is consistent with the observed market prices of European-style contingentclaims.

Our approach to building a no-arbitrage term structure for pricing interest rate derivatives isconsistent with the general framework proposed by Heath, Jarrow andMorton (1992)(HJM).In the HJM formulation, assumptions are made about the volatility of the forward interestrates. Since the forward rates are related in a no-arbitrage model to the future spot interestrates, there is a close relationship between this approach and the one we are taking. Infact, the HJM forward-rate volatilities can be thought of as the outputs of our model.If the parameters of the HJM model are known, this represents a satisfactory alternative


approach. However, the BDT-BK-HW approach has the advantage of requiring as inputs thevolatilities of the short rate and of longer bond yields which are more directly observablefrom market data on the pricing of caps, oors and swaptions. Our model, which is anextension of BK, has similar advantages.

We would like any model of the stochastic term structure to have a number of desirableproperties. Apart from satisfying the no-arbitrage property, we want the model inputs to beconsistent with the observed conditional volatilities of the variables and the mean-reversionof the short rate and the premium factor. We also require that the short term interest ratesbe lognormal, so that they are bounded from below by zero and skewed to the right. From acomputational perspective, we require the state space to be non-explosive, i.e. re-combining,so that a reasonably large span of time-periods can be covered. The complexities caused bythese model requirements are discussed in section 2.

We introduce a number of new aspects into our model that allow us to solve these require-ments. Following Miltersen, Sandmann and Sondermann (1997) and Brace, Gatarek andMusiela (1997)(BGM) we model the Libor rate. We then extend and adapt the recombiningbinomial methodology of Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrah-manyam (1995)(HSS) to model lognormal rates, rather than prices. The new computationaltechniques are discussed in detail in section 3. Section 4 presents the basic two-factor model,and discusses some of its principal characteristics. In section 5, we explain how the multi-period tree of rates is built using a modi�cation of the HSS methodology. In section 6, wepresent some numerical examples of the output of our model, apply the model to the pricingof American-style and Bermudan-style options and discuss the computational e�ciency ofour methodology. Section 7 presents our conclusions.

2 Requirements of the model

There are several desirable features of any multi-factor model of the term structure ofinterest rates. Some of these features are requirements of theoretical consistency and othersare necessary for tractability in implementation. Keeping in mind the latter requirements,it is important to recognize that the principal purpose of building a model of the evolutionof the term structure is to price interest rate options generally, and, in particular, thosewith path-dependent payo�s. The simplest examples of options that need to be valuedusing such a model are American-style and Bermudan-style options on an interest rate.

First, since we wish to be able to price any term-structure dependent claim, it is importantthat the model output is a probability distribution of the term structure of interest rates


at each point in time. A realistic model should be able to project the term structure forten or twenty years, at least on a quarterly basis. With the order of forty or eighty sub-periods, the computational task is substantial and complex. If we do not compute eachterm structure point at each node of the tree, then we need to be able to interpolate, wherenecessary, to obtain required interest rates or bond prices. As in the no-arbitrage models ofHL, HJM, HW, BDT, and BK, we �rst build the risk-neutral or martingale distribution ofthe short-term interest rate, since other maturity rates and bond prices can be computedfrom the short-term rate.

The second and crucial requirement is that the interest rate process be arbitrage-free. Inthe context of term structure models, the no-arbitrage requirement, in e�ect, means thatthe one-period forward price of a bond of any maturity is the expected value, under the risk-neutral measure, of the one-period-ahead spot price of the bond. Since HL, this requirementhas been well-understood, within the context of single factor models, and is a propertysatis�ed by the HW, BDT, and BK models. However, the requirement is more demandingin the two-factor setting as shown by HJM, Du�e and Kan (1993). In a two-factor modelin which the factors are themselves interest rates, the no-arbitrage condition restricts thebehavior of the factors themselves as well as the behavior of bond prices. However, the no-arbitrage property is also an advantage in a computational sense, allowing the computationof bond prices at a node by taking simple expectations of subsequent bond prices under therisk-neutral measure.

The third requirement is that the term structure model should be consistent with thecurrent term structure and with the term structure of volatilities implicit in the price ofEuropean-style interest rate caps and oors. Models that are consistent with the currentterm structure have been common in the literature since the work of Ho and Lee (1986)(HL).For example, the HJM, HW, BDT and BK models are all of this type. The second part ofthe requirement is rather more di�cult to accommodate, since if volatilities are not constantover time, the tree of rates may be non-recombining, as in some implementations of the HJMmodel, leading to an explosion in the number of states. The HW implementation of a two-factor model, in Hull and White (1994), speci�cally excludes time dependent volatility. BK,on the other hand, accommodate both time varying volatility and mean reversion of theshort rate within a one-factor model by varying the size of the time steps in the model.This procedure is di�cult to extend to a two-factor case.

At a computational level, it is necessary that the state space of the model does not explode,producing so many states that the computations become infeasible. Even in a single-factormodel, this fourth requirement means that the tree of interest rates or bond prices mustrecombine. This is a property of the binomial models of HL, BDT, and BK, and alsoof the trinomial model of HW. A number of computational methods have been suggested


to guarantee this property, including the use of di�erent time steps and state-dependentprobabilities. In the context of a two-factor model, the requirement is even more important.In our bivariate-binomial model we require that the number of states is no more than(n+1)2, after n time steps. This is the bivariate generalization of the \simple" recombiningone-variable binomial tree of Cox, Ross and Rubinstein (1979).

Based on the empirical evidence as well as on theoretical considerations, the �fth require-ment for our two-factor model is that the interest rates are lognormally distributed. Basedon the work of Vasicek (1977), it is well known that the class of Gaussian models, wherethe interest rate is normally distributed, are analytically tractable, allowing closed-form so-lutions for bond prices. However, apart from admitting the possibility of negative interestrates, these models are not consistent with the right skewness that is considered important,at least for some currencies. Furthermore, such a model would be inconsistent with thewidely-used Black model to value interest rate options, which assumes that the short-termrate is lognormally distributed. Of the models in the literature, the BDT and BK modelsexplicitly assume that the short rate is lognormal. The HJM and HW multi-factor mod-els are general enough to allow for lognormal rates, but at the expense of computationalcomplexity.

In addition to these �ve requirements, there is an overall necessity that the model becomputable for realistic scenarios, e�ciently and with reasonable speed. In this context,we aim to compute option prices in a matter of seconds rather than minutes. To achievethis we need a number of modelling innovations, compared to the techniques used in priormodels. These methodological innovations are discussed in the next section.

3 Features of the methodology

3.1 The stochastic process for interest rates

The dynamics of the term structure of interest rates can be modeled in terms of one of threealternative variables: zero-coupon bond prices, spot interest rates, or forward interest rates.If the objective of the exercise is to price contingent claims on interest rates, it is su�cientto model forward rates, as demonstrated by HJM. However, there are some problems withadopting this approach in a multi-factor setting. First, from a computational perspective,for general forward rate processes, the tree may be non-recombining, which implies that alarge number of time-steps becomes practically infeasible. Second, it may be di�cult toestimate the volatility inputs for the model directly from market data. Usually, the implied


volatility data are obtained from the market prices of options on Euro-currency futures,caps, oors and swaptions, which cannot be easily transformed into the volatility inputsrequired to build the forward rates in the HJM model.

We choose to model interest rates rather than prices because existing methodologies, in-troduced by HSS, can be employed to approximate a process with a log-binomial process.One major problem arises in modelling rates rather than prices, however, and that concernsthe no-arbitrage property. Under the risk-neutral measure, forward bond prices are relatedto one-period-ahead spot prices, but the relationship for interest rates is more complex, asshown by HJM. This non-linearity makes the implementation of the binomial lattice muchmore cumbersome. However, if the interest rate modelled is the Libor rate, the no-arbitragecondition can be satis�ed using Libor futures rates. The Libor futures rate is the expectedvalue, under the risk-neutral measure, of the spot Libor rate. Hence we calibrate the modelto the current term structure using the futures Libor rates.

We choose here to model interest rates, and then derive bond prices, forward prices andforward rates, as required, from the spot rate process. We prefer to directly model theshort rate, which we interpret here as the three-month interest rate, since it is used todetermine the payo�s on many contracts such as interest rate caps, oors and swaptions.One advantage of doing so is that implied volatilities from caplet/ oorlet oor prices maybe used to determine the volatility of the short-rate process fairly directly.

As in HW, we model the short rate, under the risk-neutral measure, as a two-dimensionalAR process. In particular, we assume that the logarithm of the short rate, follows sucha process where the second factor is an independent shock to the forward premium. Theshort rate itself and the premium factor each mean revert, at di�erent rates, allowing forquite general shifts and tilts in the term structure. Also, in contrast to HW, we assumetime dependent volatility functions for both the short rate and the premium factor1. In thismodel, the bond prices and forward rates for all maturities can be computed by backwardinduction, using the no-arbitrage property. Since the interest rate process is de�ned underthe risk-neutral measure, forward bond prices are expectations of one-period-ahead bondprices under this measure. This property permits the rapid computation of bond prices ofall maturities by backward induction, at each point in time.

1By taking conditional expectations of the process it can be shown that the term structure of futures

rates is given by a log-linear model in any two futures rates. The proof relies on the fact that the futures

rate is the expectation, under the risk-neutral measure, of the future spot rate, given the linear relationship

between the interest rate and the price.


3.2 Building a multivariate tree for interest rates

The principal computational problem is to build a tree of interest rates, which has the prop-erty that the conditional expectation of the rate at any point depends on the rate itself andthe premium factor. A methodology available in the literature, which allows the buildingof a multivariate tree, approximating a lognormal process with non-stationary variancesand covariances, is described in Ho, Stapleton and Subrahmanyam (1995)(HSS). The HSSmethodology is itself a generalization to two or more variables of the method advocatedby Nelson and Ramaswamy (1990), who devised a method of building a \simple" or re-combining binomial tree for a single variable. In HSS, the expectation of a variable dependson its current value, but not on the value of a second stochastic variable. However, as weshow in section 5, the methodology is easily extended to this more general case. Essentially,the HSS method relies on �xing the conditional probabilities on the tree to accommodatethe mean reversion of the interest rates, the changing volatilities of the variables and thecovariances of the variables. In the case of interest rates, it is crucial to model changes inthe short rate so as to re ect the second, premium factor. This is the key, in a two-factormodel, to maintaining the no-arbitrage property, while avoiding an explosion in the numberof states. Using our extension of the HSS methodology allows us to model the bivariatedistribution of short rate and the premium factor, with n+ 1 states for each variable aftern time steps, and a total of (n+1)2 term structures after n time steps. This is achieved byallowing the probabilities to vary in such a manner as to guarantee that the no-arbitrageproperty is satis�ed and the tree is consistent with the given volatilities and mean reversionof the process.

One problem with extending the typical interest rate tree building methods of HW, BDT,and BK to two or more factors arises from the forward induction methodology normallyemployed in these models. The tree is built around the current term structure and thecalculation proceeds by moving forward period-by-period. This is expensive in computingtime, and could become prohibitively so, in the case of multiple factors. To avoid thisproblem, we employ a dynamic method of implementation of the HSS tree, which allowsus to compute the multivariate tree in a matter of seconds for up to eighty periods. Thismethod uses the feature of HSS which allows a variation in the density of the tree overany given time step. A forward, dynamic procedure is used whereby a two-period tree withchanging density is converted into the required multi-period tree. For example, when theeightieth time step is computed, the program computes a two-period tree with a densityover the �rst period of seventy-nine, and a density over the second period of one. This allowsus to compute the tree nodes and the conditional probabilities analytically, and withoutrecourse to iterative methods.


3.3 Improving computational e�ciency

In practice much of the skill in building realistic models rests on deciding exactly what tocompute. Potentially, in a tree covering eighty time steps, we could compute bond prices forbetween one and eighty maturities at each node of the tree. Not only is this a vast numberof bond prices, but also, most of the bond prices will not be required for the solution ofany given option valuation problem. We assume here that it will be su�cient to computebond prices for maturities one year apart from each other. Intermediate maturity prices, ifrequired, can always be computed by interpolation. Hence, in our eighty-time step examples,where each time step is a quarter of a year, we compute at most twenty bond prices. Thissaving reduces the number of calculations by almost seventy-�ve percent. For large numbersof time steps, it can turn an almost infeasible computational task into one that can beaccomplished within a reasonable time frame. For example, with three hundred time steps,the number of bond price calculations can be reduced from approximately twenty-sevenmillion to approximately one million.

In spite of the computational savings that are made by having a recombining tree method-ology and reducing the number of bond prices that need to be calculated, it may still bethe case that the computation time is excessive for a given problem. For example, for aBermudan-style bond option that is exercisable every year for the �rst six years of theunderlying bond's twenty year life, we only require bond prices at the end of each of the�rst six years. One computational advantage of the HSS methodology, is that the binomialdensity can be altered so that this problem is reduced to a seven-period problem with dif-ferential density (numbers of time steps). The binomial density ensures su�cient accuracyin the computations, while the number of bond and option price calculations is minimized.

4 The Two-factor Model

4.1 No-arbitrage properties of the model

We assume that, under the risk-neutral measure, the logarithm of the short-term interestrate, for loans of maturity m, follows the process

d ln(r) = [�r(t)� aln(r) + ln(�)]dt + �r(t)dz1 (1)

where


d ln(�) = [��(t)� bln(�)]dt+ ��(t)dz2

In the above equations d ln(r) is the change in the logarithm of the short rate, �r(t) is atime-dependent constant term that allows the model to be calibrated to the current termstructure, a is the speed at which the short rate mean reverts, � is a shock to the conditionalmean of the process. We refer to � as the forward premium factor, since it is crucial indetermining the forward rates in the model. �r(t) is the instantaneous volatility of theshort rate. The forward premium factor itself follows a di�usion process with mean ��,mean reversion b and instantaneous volatility ��(t). Note that, in this notation �rt and ��trefer to the unconditional logarithmic standard deviation of the variable, over the period0� t, on a non-annualised basis. Although this structure is broadly similar to the model ofHull and White (1994), note that we do not restrict the volatilities: �r(t) and ��(t) to beconstant over time, although they are assumed to be non-stochastic. Also, we assume thatthe short-term interest rate is de�ned on a simple, Libor basis, i.e. rt = [(1=Bt;t+m)�1]=m,wherem is a �xed maturity of the short rate and Bt;t+m is the price of am-year, zero-couponbond at time t. dz1 and dz2 are standard Brownian motions.

In discrete form, equation (1) can be written as

ln(rt)� ln(rt�1) = �r(t)� aln(rt�1) + ln(�t�1) + "t; (2)

whereln(�t)� ln(�t�1) = ��(t)� bln(�t�1) + �t;

and where "t and �t are independently distributed, normal variables, and the mean andunconditional standard deviation of the logarithm of rt and �t are �rt , �rt and ��t, ��trespectively.

First, we solve the model, to determine the constants �r(t) and ��(t). We have, for anylognormal process of the form of equation (2):

Lemma 1 Suppose that the short-term interest rate follows the process in equation (2).

Then

ln(rt)� �rt = [ln(rt�1)� �rt�1 ](1� a) + ln(�t�1)� ��t�1 + "t (3)

where

ln(�t)� ��t = [ln(�t�1)� ��t�1 ](1� b) + �t;

with

�rt = ln[E0(rt)]� �2rt=2;


and

��t = ln[E0(�t)]� �2�t=2:

Proof

Taking the unconditional expectation of equation (2)

�rt � �rt�1 = �r(t)� a�rt�1 + ��t�1 ;

��t � ��t�1 = ��(t)� a��t�1

Then, substituting for �r(t) and ��(t) in (2) yields (3).

Also since rt and �t are lognormal, it follows from the moment generating function of thenormal distribution that

E0(rt) = exp(�rt + �2rt=2)

E0(�t) = exp(��t + �2�t=2)

Hence, taking logarithms yields the statement in the lemma.2

Lemma 1 is essential in the calibration of the model. We have to choose the means �rt ofthe short rate to be consistent with the absence of arbitrage and with the term structure ofinterest rates at t = 0. We achieve this by assuming, as given, a set of futures rates [f0;t].These are Libor futures rates for m-month (usually three-month) money. Assuming for themoment that these futures rates are given, we build a model of rt in which the expectedvalue of rt, under the risk-neutral measure, is the futures rate, [f0;t];8t; 0 < t < T . Workingback from the terminal date T , we then compute zero bond prices assuming that the forwardprice, under the risk-neutral measure, is the expected value of the one-period-ahead spotprice. This ensures that the no-arbitrage property of the model is guaranteed in every state,and at each date. The no-arbitrage properties of the model are established in the followingproposition. First, we de�ne the no-arbitrage property, precisely. We have, as in Pliska(1997) the following:

De�nition [The No-Arbitrage Property]

A term-structure model satis�es the no-arbitrage property, if and only if the zero-couponbond prices satisfy

Bs;t = Bs;s+1Es(Bs+1;t); 0 � s < t � T:


We now show that, if the model is suitably calibrated, then the model is arbitrage free. We�rst state without proof a result from Cox, Ingersoll and Ross (1981) which we require inthe proof of the no-arbitrage property that follows:

Lemma 2 The futures price F0;t of an asset with spot price St at time t is the value of an

asset which pays St=B0;1B1;2:::Bt�1;t at time t.

Proof

See CIR (1981), proposition 2.2

Proposition 1 (The No-Arbitrage Property) Suppose that the Libor rate, rt follows

the process:

ln(rt)� ln(rt�1) = �r(t)� aln(rt�1) + ln(�t�1) + "t;

where

ln(�t)� ln(�t�1) = ��(t)� bln(�t�1) + �t;

under the risk-neutral measure, with E(�t) = 1;8t, where "t and �t are independently dis-

tributed, normal variables. Then, if the model is calibrated so that

�rt = ln(f0;t)� �2rt=2 ;8t

where f0;t is the futures Libor rate at time 0, for delivery at time t and if the forward price

at t, of a t+ k + 1 maturity zero-coupon bond, for delivery at t+ 1 is given by

Bt;t+1;t+k+1 = Et(Bt+1;t+k+1); 8t; k

then the no-arbitrage property holds.

Proof

From lemma 1

ln(rt)� �rt = [ln(rt�1)� �rt�1 ](1� a) + ln(�t�1)� ��t�1 + "t

whereln(�t)� ��t = [ln(�t�1]� ��t�1)(1� b) + �t;

and where�rt = ln[E0(rt)]� �2rt=2


and��t = ln[E0(�t)]� �2�t=2:

From the calibration of the model

�rt = ln(f0;t)� �2rt=2

and hence it follows that f0;t = E0(rt). It then follows that F0;t = B0(Bt;t+m).

Using the property of expectations, we can write

F0;t = E0(E1(:::Et�1(Bt;t+m)))

and

F0;t = B0;1E0(B1;2E1(:::Bt�1;tEt�1

Bt;t+m

B0;1B1;2:::Bt�1;t

!)):

From lemma 2, this equation gives the value of an asset paying�

Bt;t+m

B0;1B1;2:::Bt�1;t

�at time t.

Hence, the value of the bond itself must be given by

B0;t = B0;1E0(B1;2E1(:::Bt�1;tEt�1(Bt;t+m))): (4)

From spot-forward parity, and the condition of the proposition

Bt;t+1;t+k+1 =Bt;t+k+1

Bt;t+1

= Et(Bt+1;t+k+1) (5)

Successive substitution of the condition (5) in (4) then yields

B0;t = B0;1E0(B1;t); 0 < t � T:

Hence, the calibration of the model ensures that the no-arbitrage equation holds for s = 0.Finally, (5) guarantees that it holds for s > 0.2

Proposition 1 guarantees that the model is arbitrage free, if it is calibrated correctly tofutures rates. The implication, in proposition 1, is that the futures rate is a martingale,under the risk-neutral measure. Hence we can easily calibrate the model to the given termstructure of futures rates, and thereby guarantee that the no-arbitrage property holds.


4.2 Regression Properties of the Model

The two-factor model of the term structure, described above, has the Hull-White charac-teristic that the conditional mean of the short rate is stochastic. Since the forward ratedirectly depends on the conditional mean, this induces an imperfect correlation between theshort rate and the forward rate. In this section we establish the regression properties of themodel, using the covariances of the short rate and premium process. These properties arerequired as inputs for the building of a binomial approximation model of the term structure.In the following lemma, we de�ne the covariance of the logarithm of the short rate and thepremium factor as �rt;�t. The process assumed in Lemma 1 has the following properties:-

Lemma 3 Assume that


where

ln(�t)� ��t = [ln(�t�1)� ��t�1 ](1� b) + �t;

with E0(�t) = 1, 8t.

Then,

1. the multiple regression

ln

�rt

E0(rt)

�= �rt + �rt ln

�rt�1

E0(rt�1)

�+ rt ln(�t�1) + "t (6)

has coe�cients

�rt = (��2rt + �rt�2rt�1

+ rt�2�t�1

)=2

�rt = (1� a)

rt = 1

2. the regression

ln(�t) = ��t + ��t ln(�t�1) + �t (7)

has coe�cients

��t = [��2�t + �2�t�1(1� b)]=2

��t = (1� b)


3. the conditional variance of ln(rt) is given by

vart�1("t) = �2rt � (1� a)2�2rt�1 � �2�t�1 � (1� a)�rt�1;�t�1;

where �r;� denotes the annualized covariance of the logarithms of the short rate and

the premium factor.

Proof

First, we derive the following covariances from equation (3)

�rt+1;rt = (1� a)�2rt + �rt;�t;

�rt;�t = (1� b)�2�t�1 + (1� a)(1 � b)�rt�1;�t�1

and��t+1;�t = (1� b)�2�t ;

�rt;�t�1 = (1� a)�rt�1;�t�1 + �2�t�1 :

Now from the multiple regression

ln

�rt

E0(rt)

�= �rt + �rt ln

�rt�1

E(rt�1)

�+ rt ln(�t�1) + "t (8)

the regression coe�cients are

�rt =�rt;rt�1�

2�t�1

� �rt;�t�1�rt�1;�t�1

�2rt�1�2�t�1

� (�rt�1;�t�1)2

rt =�rt;rt�1�

2rt�1

� �rt;rt�1�rt�1;�t�1

�2rt�1�2�t�1

� (�rt�1;�t�1)2

Substituting the covariances, and simplifying yields

�rt = (1� a) (9)

and rt = 1 (10)


From the lognormality of rt and �t we can write equation (3) as

ln(rt)� ln[E0(rt)] + �2rt=2 = fln(rt�1)� ln[E0(rt�1)] + �2rt�1=2g(1 � a)

+ ln(�t�1)�nln[E0(�t�1)] + �2�t�1=2

o+ "t

Re-arranging yields

ln

�rt

E0(rt)

�= [��2rt + �2rt(1� a) + �2�t ]=2

+ ln

�rt�1

E0(rt�1)

�(1� a) + ln(�t�1) + "t

Given (8), (9), and (10), we have �rt as stated in the lemma. Similarly, with E0(�t) = 1,we have

ln(�t) = ��t + ln(�t�1)(1� b) + �t;

and��t = E0[ln(�t)]� (1� b)[ln[E0(�t)]

��t = [��2�t + (1� b)�2�t�1 ]=2

Finally, the variance of "t, given t, is

vart�1("t) = var0

�ln

�rt

E0(rt)

�� 2r;tvar0

�ln

�rt�1

E0(rt�1)

��

� 2rtvar0[ln(�t�1)]� �rt rtcov[ln(rt�1); ln(�t�1)]

or,vart�1("t) = �2rt � (1� a)2�2rt�1 � �2�t�1 � (1� a)�rt�1;�t�1 :

2

Note that we require the multiple regression coe�cients �rt ; �rt ; and rt in order to buildthe binomial approximation of the multi-variate process, using the method of Ho, Stapletonand Subrahmanyam (1995). From part 1 of the lemma, the �r coe�cients simply re ect themean-reversion of the short rate. The r coe�cients are all unity, re ecting the one-to-one


relationship between �, the forward premium factor and the expected spot rate. The �rcoe�cients re ect the drift of the lognormal distribution, which depends on the variancesof the variables. Part 2 of the lemma shows that the regression relation for �t is a simpleregression, where the �� coe�cients re ect the constant mean reversion of the premiumfactor. Lastly, part 3 of the lemma gives an expression for the conditional variance of thelogarithm of the short rate.

4.3 Determining the Volatility Inputs of the Model

In order to build the model outlined above, we need the parameters of the premium process,as well as those for the short rate process itself. The result in Lemma 3, part 3 gives therelationship of the conditional volatility of the short rate to the unconditional volatilities ofthe short rate, the volatility of the premium factor, and mean reversion of the short rate.We assume that the unconditional volatilities of the short rate are available, perhaps fromcaplet/ oorlet volatilities and that the mean reversion is given. The premium process, �t,on the other hand, determines the extent to which the �rst forward rate di�ers from thespot rate in the model. Since the premium factor is not directly observable, we need to beable to estimate the mean and volatility of the premium factor from the behavior of forwardor futures rates. In order to discuss this, we �rst establish the following general result:-

Lemma 4 Assume that


where

ln(�t)� ��t = [ln(�t�1)� ��t�1 ](1� b) + �t;

with E0(�t) = 1, 8t, then the conditional volatility of �t is given by

��(t) = [�2x(t)� (1� a)2�2r (t)]0:5

where x = Et(rt+1) and vart�1[ln(xt)] = �2x(t).

Proof

Taking the conditional expectation of equation (3) at t

Et[ln(rt+1)]� �rt+1 = [ln(rt)� �rt ](1� a) + ln(�t)� ��t :


Given xt = Et(rt+1) and using the lognormal property of rt+1,

ln(xt) = Et[ln(rt+1)] + �2r(t+ 1)=2

= �2r (t+ 1)=2 + �rt+1 + [ln(rt)� �rt ](1� a) + [ln(�t�1)� ��t�1 ](1� b) + �t

Hence�2x(t) = vart�1[ln(xt)] = (1� a)2vart�1[ln(rt)] + vart�1[�t]

or�2x(t) = �2�(t) + (1� a)2�2r (t)

and the statement in the lemma follows.2

Lemma 4 relates the volatility of the premium factor to the volatility of the conditionalexpectation of the short rate. To apply the result in the current context, we �rst assumethat the short rate follows the process assumed in the lemma under the risk-neutral process.We then use the fact that the unconditional expectation of the t+1 th rate is ft;1 = Et(rt+1),i.e. the �rst futures (or forward) rate is the expected value of the next period spot rate.This implication of no-arbitrage leads to

lnft;1 = �rt+1 + [lnrt � �rt ](1� a) + ln�t�1(1� b) + �t + �2r (t)=2: (11)

It follows that the conditional logarithmic variance of the �rst futures rate is given by therelation

�2f (t) = (1� a)2�2r (t) + �2�(t): (12)

Hence, the volatility of the premium factor is potentially observable from the volatility ofthe �rst futures rate. This in turn could be estimated empirically or implied from optionson the futures rate.


5 The Multivariate-Binomial Approximation of the Process

5.1 The HSS approximation

The general approach we take to building a bivariate-binomial lattice, representing a discreteapproximation of the process in equation (1), is to construct separate re-combining binomialtrees for the short-term interest rate and the forward-premium factors. The no-arbitrageproperty and the covariance characteristics of the model are then captured by choosing theconditional probabilities at each node of the tree.

We now outline our method for approximating the two-factor process interest rate process,described above. We use three types of inputs: the unconditional means E0(rt), t = 1; :::; T ,of the short-term rate, the volatilities of "t ( i.e. the conditional volatility of the short rate,given the previous short rate and the one-period forward premium factor, denoted �r(t) andthe conditional volatilities of the premium, denoted ��(t)), estimates of the mean reversion,a, of the short rate, and the mean reversion, b, of the premium factor. The process in (3)is then approximated using an adaptation of the methodology described in Ho, Stapletonand Subrahmanyam (1995) (HSS). HSS show how to construct a multiperiod multivariate-binomial approximation to a joint-lognormal distribution ofM variables with a re-combiningbinomial lattice. However, in the present case, we need to modify the procedure, allowingthe expected value of the interest rate variable to depend upon the premium factor. Thatis, we need to model the two variables rt and �t, where rt depends upon �t�1. Furthermore,in the present context, we need to implement a multi-period process for the evolution ofthe interest rate, whereas HSS only implement a two-period example of their method. Inthis section, these modi�cations and the multi-period algorithm are presented in detail.

We divide the total time period into T periods of equal length of m years, where m is thematurity period, in years, of the short-term interest rate. Over each of the periods from t

to t+ 1 we denote the number of binomial time steps, termed the binomial density, by nt.Note that, in the HSS method, nt can vary with t allowing the binomial tree to have a �nerdensity, if required for accurate pricing, over a speci�ed period. This might be required, forexample if the option exercise price changes between two dates, increasing the likelihood ofthe option being exercised, or for pricing barrier options.

We use the following result, adapted from HSS:

Lemma 5 Suppose that Xt follows a lognormal process, where E0(Xt) = 1;8t. Let the

conditional logarithmic standard deviation of Xt be �x(t). If Xt is approximated by a log-

binomial distribution with binomial density Nt = Nt�1+nt and if the proportionate up and


down movements, ut and dt are given by

dt =2

1 + exp(2�x(t)p1=nt)

ut = 2� dt

and the conditional probability of an up-move at node r of the lattice is given by

qxt =Et�1(Xt) + (Nt�1 � r) ln(ut)� (nt + r) ln(dt)

nt[ln(ut)� ln(dt)]

then the unconditional mean and conditional volatility of the approximated process approach

their true values, i.e., E0(Xt)! 1 and �xt ! �xt as n!1.

Proof If E0(Xt) = 18t, then we obtain the result as a special case of HSS(1995), Theorem1.2

5.2 Computing the nodal values

In this section, we �rst describe how the vectors of the short-term rates and the premiumfactor are computed. We approximate the process for the short-term interest rate, rt, witha binomial process, which moves up or down from its expected value, by the multiplicativefactors drt and urt. Following HSS, equation (7), these are given by

drt =2

1 + exp(2�r(t)p1=nt)

urt = 2� drt

We then build a separate tree of the forward premium factor �. The up-factors and down-factors in this case are given by

d�t =2

1 + exp(2��(t)p1=nt)

u�t = 2� d�t


At node j at time t, the interest rates rt and premium factors �t are calculated from theequations

rt;j = u(Nt�j)rt

djrtE0(rt); (13)

�t;j = u(Nt�j)�t

dj�t ;

j = 0; 1; :::; Nt :

where Nt =P

t nt. In general, there are Nt + 1 nodes, i.e., states of rt and �t, since bothbinomial trees are re-combining. Hence, there are (Nt + 1)2 states after t time steps.

5.3 Computing the conditional probabilities

As in HW, in general, the covariance of the two variables may be re ected by varyingthe conditional probabilities in the binomial process. Since the trees of the rates and theforward premium are both re-combining, the time-series properties of each variable must alsobe captured by adjusting the conditional probabilities of moving up or down the tree, as inHSS and in Nelson and Ramaswamy (1991). Since, increments in the premium variable areindependent of rt, this is the simplest variable to deal with. Using the results of lemma 3, wecompute the conditional probability using HSS, equation (10). In this case the probabilityof a up-move, given that �t�1 is at node j, is

q�t =��t + ��t ln�t�1;j � (Nt�1 � j)lnu�t � (j + nt)lnd�t

nt (lnu�t � lnd�t)(14)

where��t = (1� b)

��t = (��2�t + ��t�2�t�1

)=2

and where b is the coe�cient of mean reversion of �, and �2�t is the unconditional logarithmicvariance of � over the period (0� t).

The key step in the computation is to �x the conditional probability of an up-movement inthe rate rt, given the outcome of rt�1, the mean reversion of r, and the value of the premiumfactor �t�1. In discussing the multi-period, multi-factor case, HSS present the formula forthe conditional probability when a variable x2 depends upon x1 and a contemporaneous


variable, y2. Again using the regression properties derived in lemma 3, and adjusting HSS,equation (13) to the present case, we compute the probability

qrt =�rt + �rt ln(rt�1;j=E(rt�1)) + rt ln�t�1;j � (Nt�1 � j)lnurt � (j + nt)lndrt

nt(lnurt � lndrt)(15)

where�rt = (1� a)

rt = 1

�rt = [��2rt + �rt�2rt�1

+ rt�2�t]=2

Then, by lemma 5, the process converges to a process with the given mean and varianceinputs.

5.4 The multiperiod algorithm

HSS(1995) provides the equations for the computation of the nodal values of the variables,and the associated conditional probabilities, in the case of two periods t and t+1. E�cientimplementation requires the following procedure for the building of the T period tree. Themethod is based on forward induction. First, compute the tree for the case where t=1. Thisgives the nodal values of the variables and the conditional probabilities, for the �rst twoperiods. Then, treat the �rst two periods as one new period, but with a binomial densityequal to the sum of the �rst two binomial densities. The computations are then made forperiod three nodal values and conditional probabilities. Note that the equations for theup and down movements of the variables always require the conditional volatilities of thevariables to compute the vectors of nodal values. The following steps are followed:-

1. Using equation (13), compute the [n1 x 1] vectors of the nodal outcomes of r1, �1 usinginputs �r(1), E(r1), ��(1), E(�1) and binomial density n1. Also compute the [(n1+n2) x 1]vectors r2, �2 using inputs �r(2), E(r2), ��(2), E(�2) and binomial density n2. Assume theprobability of an up-move in r1 is 0.5 and then compute the conditional probabilities q�1using equation (14) with t=1. Then compute the conditional probabilities qr2 , q�2, usingequations (14) and (15), with t=2.

2. Using equation (13), compute the [(N2+n3) x 1] vectors r3, �3 using inputs �r(3), E(r3),�r(3), E(r3) and binomial density, n3. Then compute the conditional probabilities qr3 , q�3using equations (14) and (15) with t=3.


3. Continue the procedure until the �nal period T .

In implementing the above procedure, we �rst complete step 1, using t = 1 and t = 2, andwith the given binomial densities n1 and n2. To e�ect step 2, we then rede�ne the periodfrom t = 0 to t = 2 as period 1 and the period 3 as period 2 and re-run the procedurewith a binomial densities n�1 = n1+n2 and n�2 = n3. This algorithm allows the multiperiodlattice to be built by repeated application of equations (13), (14) and (15).

5.5 A summary of the approximation method

We will summarize the methodology by using a two-period and a three-period example.Figure 1 shows the recombining nodes for the two-factor process in the two-period case.The interest rate goes up to r1;0 or down to r1;1 at t = 1. The forward premium factorgoes up to �1;0 or down to �1;1 at t = 1, with probability q�1 . In the second period, thereare just 3 nodes of the interest rate tree, together with 3 possible premium factor values.There are 9 possible states, and the probability of an r2 value materialising is qr2 . Notethat this probability depends on the level of the premium factor and of the interest rate attime t = 1. The recombining property of the lattice, which is crucial for the computabilityof the lattice, is emphasised in Figure 2, where we show the process for the interest rateover periods t = 2 and t = 3. After two periods there are 3 interest rate states and 9 statesrepresenting all the possible combinations of the interest rate and premium factor. Theinterest rate then goes to 4 possible states at time t = 3 and there are 16 states representingall the possible combinations of rates and premium factor. Note that the probability ofreaching an interest rate state at t = 3, depends on both the interest rate and the premiumfactor at t = 2. It is these probabilities that allow the no-arbitrage property of the modelto be ful�lled. In the model, the term structure at time t is determined by the two factors,one representing the short rate and the premium factor. Thus, with a binomial density ofn = 1 there are (t+ 1)2 term structures generated by the binomial approximation, at timet.

6 Model Validation and Examples of Inputs and Outputs

This section shows the results from several numerical examples and examines the two factorterm structure model described in previous sections in detail. Firstly, we show an exampleof how well the binomial approximation converges to the mean and unconditional volatilityinputs. Secondly, we show that a two-factor term structure model can be run in a speedy


and e�cient manner. Thirdly, we discuss the input and output for an eight- period example,showing illustrative output of zero-coupon bond prices, and conditional volatilities. Finallywe show the output from running a forty-eight quarter model, including the pricing ofEuropean and Bermudan style options on coupon bonds.

6.1 Convergence of Model Statistics to Exogenous Data Inputs

The �rst test of the two-factor model is how quickly the mean and variance of the short ratesgenerated converge to the exogenous data. Table 1 shows an example of a twenty-periodmodel, where the input mean of the spot rate is 5%, with a 10% conditional volatility.There is no mean reversion and the premium has a volatility of 1%. Note �rst that fora binomial density of 1, the accuracy of the binomial approximation deteriorates for laterperiods. This is due to the premium factor increasing with maturity and the di�culty ofcoping with the increased premium, by adjusting the conditional probabilities.

One way to increase the accuracy of the approximation is to increase the binomial density.In the last three columns of the table we show the e�ect of increasing the binomial densityto 2, 3, and 4 respectively. By comparing di�erent binomial densities in a given row of thetable we observe the convergence of the binomial approximation to the exogenous inputs asthe density increases. Even for the 20 period case, high accuracy is achieved by increasingthe binomial density to 4.

Table 1 here

6.2 Computing Time

The most important feature of the two-factor model proposed in this paper is the compu-tation time. With two stochastic factors rather than one, the computation time can easilyincrease dramatically. In table 2, we illustrate the e�ciency of our model by showing thetime taken to compute the zero-coupon bond prices and option prices. With a binomialdensity of 1, the 48-period model takes 12.3 seconds. Doubling the number of periods in-creases the computer time by a factor of six. There is a trade-o� between the number ofperiods, the binomial density of each period, and the computation time for the model. Thisis illustrated by the second line in the table, showing the e�ect of using a binomial densityof 2. Again the computation time increases more than proportionately as the density in-


creases. The time taken for the 28-period model, when the binomial density is 2, is roughlythe same as that for the 48-period model with a density of 1.

Table 2 here

6.3 Numerical Example: An Eight-Period Bond

This subsection shows a numerical example of the input and output of the two-factor termstructure model, in a simpli�ed eight quarter example . It illustrates the large amount ofdata produced by the model, even in this small scale case. The input is shown �rst in Table3. We assume a rising curve of futures rates, starting at 5% and rising to 6%. These areused to �x the means of the short rate for the various periods. The second row shows theconditional volatilities assumed for the short rate. These start at 14% and fall throughtime to 12%. We then assume a constant mean reversion of the short rate, of 10%, andconstant conditional volatilities and mean reversion of the premium factor, of 2% and 40%respectively.

Table 3 here

Tables 4 and 5 show a selection of the basic output of the model. For a binomial densityof one, there are 4 states at time 1, 9 states at time 2, 16 states at time 3, and so on. Ineach state the model computes the whole term structure of zero-bond prices. In Table 4,we show just the longest bond price, paying one unit at period 8. These are shown for the4 states at time 1, in the �rst block of the table. The subsequent blocks show the 9 pricesat time 2, the 16 prices at time 3, and so on.

Table 4 here

One of the most important features of the methodology is the way that the no-arbitrageproperty is preserved, by adjusting the conditional probabilities at each node in the treeof rates. In Table 5, we show the probability of an up move in interest rate given a state,where the state is de�ned by the short rate and the premium factor. In the �rst block ofthe table is the set of probabilities conditional on being in one of four possible states attime 1. The second block shows the conditional probabilities at time 2, in the 9 possiblestates, and so on.


Table 5 here

6.4 An Example of an Option on a Coupon-Bond

An important application of the model is to price and hedge contingent claims such asoptions with American and path-dependent features. A good example is a Bermudan optionon a coupon bond. We illustrate our methodology by pricing a six-year option on a twelve-year coupon bond. The option has the Bermudan feature that it is exercisable, at par, atthe end of each year up to the option maturity in year six. We �rst build the tree of rates,assuming the data detailed in the notes to Table 6. Note that the model uses 48 quarterlytime periods, to cover the twelve-year life of the coupon bond.

Table 6 here

Table 6 shows that the Bermudan option on the coupon bond is worth considerably morethan the European option. Also there is a small positive e�ect of valuing the options withthe binomial tree with a density of two. Illustrative output of the price of the underlyingcoupon bond is shown in Table 7.

Table 7 here

7 Conclusions

In this paper we have presented a model of the term structure of interest rates which can beregarded as a two-factor extension of the Black-Karasinski lognormal-rate model. However,in this model, we assume that the short-term Libor interest rate follows a lognormal process.We have shown that, by calibrating to the current term structure of futures rates, the modelis arbitrage-free in the sense of Ho and Lee (1986) and Pliska (1997).

The model was implemented by using a multivariate-binomial tree approach. By extend-ing previous work of Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrah-manyam (1995), we have developed a re-combining bivariate-binomial tree, which has anon-exploding number of nodes.


We have applied the model to the valuation of American-style and Bermudan-style optionson coupon bonds. We have shown that prices are computable in seconds for examples witha realistic number of periods. Also, prices in the two-factor model exceed those in theone-factor model, calibrated to the same data.

A number of related research issues remain to be resolved in future work. First, the re-lationship between the Hull-White, Black-Karasinski type model developed here and theHeath-Jarrow-Morton multifactor model needs to be explored further. Speci�cally, we in-tend to generate forward-rate volatilities as outputs of our model in an extension of theanalysis in the paper. Second, it is not clear to what extent a two-factor model, as opposedto a one-factor model, is necessary for the accurate valuation of speci�c interest-rate relatedcontingent claims. We intend to investigate the properties of the model, in a subsequentpaper, by applying it to a range of American-style, Bermudan-style and exotic options onbonds and interest rates. Third, the application of models of this type to derive risk man-agement measures for interest-rate dependent claims should be studied further. We hopeto use the framework presented in this paper to address these issues in further research.


References

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2. Black, F., E. Derman, and W. Toy (1990), \A One-Factor Model of Interest Rates andits Application to Treasury Bond Options", Financial Analysts' Journal, 46, 33{39.

3. Black, F., (1976), \The Pricing of Commodity Contracts", Journal of Financial Eco-nomics, 3, 167{179.

4. Black, F. and P. Karasinski (1991), \Bond and Option Pricing when Short Rates areLognormal", Financial Analysts Journal, 47, 52{59

5. Brace, A., Gatarek, D. and Musiela, M. (1997), \The Market Model of Interest RateDynamics", Mathematical Finance, 7, 127-155.

6. Cox, J.C., J.E. Ingersoll, and S.A. Ross (1981), \The Relationship between ForwardPrices and Futures Prices", Journal of Financial Economics, 9, 1981, 321-46.

7. Du�e, D. and R. Kan (1993), \A Yield -Factor Model of Interest Rates," WorkingPaper, Graduate School of Business, Stanford University, September.

8. Heath, D., R.A. Jarrow, and A. Morton (1990a), \Bond Pricing and the Term Struc-ture of Interest Rates: A Discrete Time Approximation," Journal of Financial and

Quantitative Analysis, 25, 4, December, 419{440.

9. Heath, D., R.A. Jarrow, and A. Morton (1990b), \Contingent Claim Valuation witha Random Evolution of Interest Rates," Review of Futures Markets, 9, 55{75.

10. Heath, D., R.A. Jarrow, and A. Morton (1992), \Bond Pricing and the Term Structureof Interest Rates: A New Methodology for Contingent Claims Valuation," Economet-

rica, 60, 1, January, 77{105.

11. Hull, J. and A. White (1994), \Numerical Procedures for Implementing Term Struc-ture Models II: Two-Factor Models," Journal of Derivatives, Winter, 37{48.

12. Ho, T.S.Y. and S.B. Lee (1986), \Term Structure Movements and Pricing of InterestRate Claims," Journal of Finance, 41, December, 1011{1029.

13. Ho. T.S., R.C. Stapleton, and M.G. Subrahmanyam (1995), \Multivariate BinomialApproximations for Asset Prices with Non- Stationary Variance and Covariance Char-acteristics," Review of Financial Studies, 8, 1125-1152.


14. Nelson, D.B. and K. Ramaswamy (1990), \Simple Binomial Processes as Di�usionApproximations in Financial Models," Review of Financial Studies, 3, 393-430.

15. Pliska, S.R., Introduction to Mathematical Finance, Blackwell, 1977

16. Stapleton, R.C. and M.G. Subrahmanyam (1993), \Analysis and Valuation of InterestRate Options," Journal of Banking and Finance, 17, December, 1079{1095.

17. Stapleton, R.C. and M.G. Subrahmanyam (1997) \Arbitrage Restrictions and Multi-Factor Models of the Term Structure of Interest Rates," working paper

18. Vasicek, O., (1977), \An Equilibrium Characterization of the Term Structure", Jour-nal of Financial Economics, 5, 177{188.


Figure 1: A Recombining Two-factor Process for the Short-term Interest Rate (Two-periodcase)

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Table 1: Convergence of Term Structure Model

Binomial Density 1 2 3 4

Period

[0, 1 ] mean 5.0 5.0 5.0 5.0volatility 10.00 10.00 10.00 10.00

[0, 2 ] mean 4.9999 4.99997 4.99998 4.99998volatility 9.97 9.99 9.99 9.99

[0, 3 ] mean 4.9998 4.99992 4.99994 4.99996volatility 3 9.95 9.97 9.99 9.99

[0, 4 ] mean 4 4.9997 4.99985 4.9999 4.99992volatility 9.92 9.97 9.98 9.98

[0, 5 ] mean 4.9996 4.9997 4.9998 4.9998volatility 9.93 9.96 9.97 9.97

[0, 10] mean 4.998 4.9992 4.9995 4.9996volatility 9.88 9.94 9.96 9.97

[0, 20] mean 4.996 4.998 4.998 4.999volatility 9.85 9.92 9.94 9.96

The numbers in the table are the computed means and volatilities, in percent, for the short rate over

periods 1,2,3,4,5,10, and 20, using the output of the two-factor model. The means are calculated

using the possible outcomes and the nodal probabilities. The volatilities are the annualized standard

deviations of the logarithm of the short rate. The binomial density refers to the denseness of the

binomial tree of the short rate and the premium factor, over each sub interval. The input parameters

in this case are a constant mean of 5%, and conditional volatility of 10% with no mean reversion of

the short rate. The premium factor has a volatility of 1%, a mean of 1, and no mean reversion.


Table 2: Computing Time for Bond and Option Pricing (seconds)

Number of Periods 8 12 28 48

Binomial Density 1 0 0.3 2.5 12.3Binomial Density 2 0.3 1 12.7 -

The table shows the time taken to compute all the zero-bond prices, coupon bond prices and option

prices, given the tree of rates.The computer speed is 266 MHZ, and the processor is Pentium.


Table 3: 8-period Example Input

Period 1 2 3 4 5 6 7 8

Futures rate 5.0 5.2 5.4 5.6 5.7 5.8 5.9 6.0Conditional volatility (r) 14.0 14.0 13.5 13.0 13.0 12.5 12.5 12.0

Mean reversion (r) 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0Conditional volatility (�) 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

Mean reversion (�) 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0

All numbers are in percent. The table shows the exogenous data input for an 8-period example.

The short rate is the quarterly rate, so the period length is quarter of one year. Input data relating

to the short rate appears in the �rst three rows; data relating to the premium in the last two rows.


Table 4: Illustrative Output of Zero-Coupon Bond Prices

0.9016606 0.9053589

0.9121947 0.9155353

0.9074258 0.9105445 0.9135762

0.9175077 0.9203237 0.9230596

0.9265372 0.9290763 0.9315416

0.9162768 0.9187132 0.9210944 0.9234185

0.9251191 0.9273254 0.9294793 0.9315810

0.9330661 0.9350606 0.9370067 0.9389042

0.9401996 0.9420003 0.9437568 0.9454675

0.9282870 0.9299746 0.9316477 0.9332908 0.9349043

0.9355540 0.9370946 0.9386109 0.9400998 0.9415616

0.9421182 0.9435164 0.9448895 0.9462374 0.9475607

0.9480383 0.9493036 0.9505457 0.9517651 0.9529554

0.9533705 0.9545143 0.9556373 0.9567394 0.9578075

0.9429089 0.9437140 0.9446889 0.9456537 0.9466083 0.9475538

0.9485605 0.9494144 0.9502989 0.9511741 0.9520400 0.9528975

0.9537454 0.9545554 0.9553573 0.9561506 0.9569353 0.9577125

0.9584548 0.9591885 0.9599150 0.9606336 0.9613444 0.9620482

0.9626968 0.9633611 0.9640187 0.9646692 0.9653126 0.9658693

0.9665154 0.9671165 0.9677114 0.9682999 0.9688820 0.9693053

All prices are for a zero-coupon bond paying one unit of currency at the end of period 8. The �rst

set of 4 numbers are the time 1 bond prices B1;8, the second set of 9 numbers are the time 2 bond

prices B2;8, through to the time 5 set of 36 prices B5;8. The 49 prices, B6;8, and 64 prices, B7;8, are

not shown for reasons of space.


Table 5: Illustrative Output of Conditional Probabilities

0.5530794674 0.4087419001

0.5932346449 0.4488970776

0.6622172425 0.5086479906 0.3550787388

0.6514562049 0.4978869530 0.3443177012

0.6406951673 0.4871259154 0.3335566636

0.7780765989 0.6126741059 0.4472716129 0.2818691198

0.7548745812 0.5894720882 0.4240695952 0.2586671021

0.7316725635 0.5662700705 0.4008675775 0.2354650844

0.7084705458 0.5430680528 0.3776655597 0.2122630667

0.8091566018 0.6377849460 0.4664132901 0.2950416343 0.1236699785

0.8248723262 0.6535006704 0.4821290145 0.3107573587 0.1393857029

0.8405880506 0.6692163947 0.4978447389 0.3264730831 0.1551014272

0.8563037749 0.6849321191 0.5135604633 0.3421888074 0.1708171516

0.8720194993 0.7006478435 0.5292761877 0.3579045318 0.1865328760

1.0000000000 0.8614994841 0.6744916140 0.4874837439 0.3004758739 0.1134680038

1.0000000000 0.8248200186 0.6378121486 0.4508042785 0.2637964084 0.0767885384

0.9751484233 0.7881405532 0.6011326831 0.4141248131 0.2271169430 0.0401090730

0.9384689578 0.7514610878 0.5644532177 0.3774453477 0.1904374776 0.0034296075

0.9017894924 0.7147816224 0.5277737523 0.3407658822 0.1537580122 0.0000000000

0.8651100270 0.6781021569 0.4910942869 0.3040864168 0.1170785467 0.0000000000

All the probabilities are conditional probabilities of an up-move in the interest rate, given the short

rate and the premium factor. The �rst set of 4 numbers are the probabilities at time 1, the second set

of 9 numbers are the conditional probabilities at time 2, through to the time 5 set of 36 conditional

probabilities. The 49 probabilities at time 6, and the 64 probabilities at time 7, are not shown for

reasons of space. In each case the columns show the probabilities for di�erent (increasing to the

right) values of the short rate. The rows show the values (increasing downwards) for di�erent values

of the premium factor.


Table 6: 48 period Option Price ($)

Binomial Density 1 2

Bermudan Call 1.0400 1.0573European Call 0.5910 0.6032

The above table shows the value of a 6-year call option on a coupon bond that pays $100 in 12 years

time. The annual coupon rate of the bond is 5%. The interest rate tree is built assuming a futures

rate of 5% for each maturity, a volatility of the short rate of 10%, a coe�cient of mean reversion of

the short rate of 10%, and volatility of the premium at 1% with 50% coe�cient of mean reversion.


Table 7: Illustrative Output of Coupon Bond Prices

0.9406288 0.9482179 0.9557422 0.9632041 0.9706163

0.9605314 0.9679123 0.9752284 0.9824820 0.9896855

0.9797344 0.9869074 0.9940156 1.0010610 1.0080570

0.9982522 1.0052180 1.0121200 1.0189590 1.0257490

1.0161010 1.0228610 1.0295580 1.0361940 1.0427790

0.8786865 0.8932577 0.9039789 0.9118041 0.9195621 0.9272566 0.9349106 0.9452104 0.9561665

0.8995920 0.9138273 0.9242790 0.9318979 0.9394492 0.9469364 0.9543821 0.9644060 0.9765066

0.9198181 0.9337106 0.9438899 0.9513011 0.9586445 0.9659235 0.9731604 0.9829078 0.9955054

0.9393716 0.9529170 0.9628220 0.9700252 0.9771606 0.9842317 0.9912601 1.0007310 1.0131950

0.9582615 0.9714571 0.9810874 0.9880830 0.9950113 1.0018750 1.0086970 1.0178940 1.0300090

0.9764993 0.9893436 0.9987003 1.0054900 1.0122120 1.0188710 1.0254870 1.0344120 1.0461810

0.9944811 1.0065910 1.0156760 1.0222610 1.0287800 1.0352370 1.0416500 1.0503060 1.0617320

1.0121860 1.0232150 1.0320300 1.0384140 1.0447330 1.0509890 1.0572030 1.0655940 1.0766820

1.0299100 1.0393630 1.0477810 1.0539660 1.0600870 1.0661470 1.0721650 1.0802960 1.0910510

The coupon-bond prices are computed at each of 25 nodes at the end of year 1, shown in the �rst

block in the table. 91 prices are computed at the end of year 2, shown in the second block of the

table. The coupon bond has an annual coupon of 5% and a par value of 1 unit. Prices are also

computed at the end of year 3, year 4, year 5, and year 6, but are not shown here for reasons of

space.

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