A Two-factor Lognormal Mo del of the Term
Structure and the Valuation of American-Style
Options on Bonds
1 2 3
Sandra Peterson Richard C. Stapleton Marti G. Subrahmanyam
February 25, 1999
1
Department of Accounting and Finance, The ManagementScho ol, Lancaster University, Lan-
caster LA1 4YX, UK. Tel: 44 524{593637, Fax: 44 524{847321.
2
Department of Accounting and Finance, Strathclyde University, Glasgow, UK. Tel: 44 524{381
172, Fax: 44 524{846 874, e{mail:[email protected]
3
Leonard N. Stern Scho ol of Business, New York University, Management Education Center, 44
West 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 Mo del of the Term Structure and
the Valuation of American-Style Options on Bonds
We build a no-arbitrage mo del of the term structure, using two sto chastic factors, the short-
term interest rate and the premium of the forward rate over the short-term interest rate.
The mo del extends the lognormal interest rate mo del of Black and Karasinski 1991 to
two factors. It allows for mean reversion in the short rate and in the forward premium.
The metho d is computationally ecient for several reasons. First, we use Lib or futures
prices, enabling us to satisfy the no-arbitrage condition without resorting to iterative meth-
o ds. Second, the multivariate-binomial metho dology of Ho, Stapleton and Subrahmanyam
1995 is extended so that a multip erio d tree of rates with the no-arbitrage prop erty can
b e constructed using analytical metho ds. The metho d uses a recombining two-dimensional
binomial lattice of interest rates that minimizes the number of states and term structures
over time. Third, the problem of computing a large numb er of term structures is simpli ed
by using a limited numb er of 'bucket rates' in each term structure scenario. In addition to
these computational advantages, a key feature of the mo del is that it is consistent with the
observed term structure of volatilities implied by the prices of interest rate caps and o ors.
We illustrate the use of the mo del by pricing American-style and Bermudan-style options
on b onds. Option prices for realistic examples using forty time p erio ds are shown to be computable in seconds.
Atwo-factor lognormal mo del of the term structure 1
1 Intro duction
One of the most imp ortant and dicult problems facing practitioners, in the eld of in-
terest rate derivatives in recentyears, has b een to build inter-temp oral mo dels of the term
structure of interest rates that are b oth analytically sound and computationally ecient.
These mo dels are required b oth to help in the pricing and in the overall risk managementof
a book of interest rate derivatives. Although many alternative mo dels have b een suggested
in the literature and implemented in practice, there are serious disadvantages with most of
them. For example, Gaussian mo dels of interest rates, whichhave the advantage of analyt-
ical tractability, have the drawbackof p ermitting negative interest rates, as well as failing
to take into account the p ossibilityof skewness in the distribution of interest rates. Also,
many of the term-structure mo dels used in practice are restricted to one sto chastic factor.
Since the work of Ho and Lee 1986 , it has b een widely recognized that term-structure
mo dels must p ossess the no-arbitrage prop erty. In this context, a no-arbitrage mo del is one
in which the forward price of a b ond is the exp ected value of the one-p erio d-ahead sp ot b ond
price, under the risk-neutral measure. Building mo dels that p ossess this prop erty has b een
a ma jor pre-o ccupation of b oth academics and practitioners in recent years. One mo del
that achieves this ob jective in a one-factor context is the mo del prop osed by Black, Derman
and Toy 1990 BDT , and extended by Black and Karasinski 1991 BK . In essence, the
mo del whichwe build in this pap er is a two-factor extension of this typ e of mo del. In our
mo del, interest rates are lognormal and are generated bytwo sto chastic factors. The general
approachwe take is similar to that of Hull and White 1994 HW , where the conditional
mean of the short rate dep ends on the short rate and an additional sto chastic factor, which
can be interpreted as the forward premium. In contrast to HW, and in line with BK, we
build a mo del where the conditional variance of the short rate is a function of time. It
follows that the mo del can be calibrated to the observed term structure of interest rate
volatilities implied byinterest rate caps/ o ors. Essentially, the aim here is to build a term
structure mo del which can b e applied to value American-style contingent claims on interest
rates, which is consistent with the observed market prices of Europ ean-style contingent
claims.
Our approach to building a no-arbitrage term structure for pricing interest rate derivatives is
consistent with the general framework prop osed by Heath, Jarrow and Morton 1992 HJM .
In the HJM formulation, assumptions are made ab out the volatility of the forward interest
rates. Since the forward rates are related in a no-arbitrage mo del to the future sp ot interest
rates, there is a close relationship between this approach and the one we are taking. In
fact, the HJM forward-rate volatilities can be thought of as the outputs of our mo del.
If the parameters of the HJM mo del are known, this represents a satisfactory alternative
Atwo-factor lognormal mo del of the term structure 2
approach. However, the BDT-BK-HW approach has the advantage of requiring as inputs the
volatilities of the short rate and of longer b ond yields which are more directly observable
from market data on the pricing of caps, o ors and swaptions. Our mo del, which is an
extension of BK, has similar advantages.
We would like any mo del of the sto chastic term structure to have a number of desirable
prop erties. Apart from satisfying the no-arbitrage prop erty,wewant the mo del inputs to b e
consistent with the observed conditional volatilities of the variables and the mean-reversion
of the short rate and the premium factor. We also require that the short term interest rates
b e lognormal, so that they are b ounded from b elowby zero and skewed to the right. From a
computational p ersp ective, we require the state space to b e non-explosive, i.e. re-combining,
so that a reasonably large span of time-p erio ds can b e covered. The complexities caused by
these mo del requirements are discussed in section 2.
Weintro duce a numb er of new asp ects into our mo del that allow us to solve these require-
ments. Following Miltersen, Sandmann and Sondermann 1997 and Brace, Gatarek and
Musiela 1997 BGM we mo del the Lib or rate. We then extend and adapt the recombining
binomial metho dology of Nelson and Ramaswamy 1990 and Ho, Stapleton and Subrah-
manyam 1995 HSS to mo del lognormal rates, rather than prices. The new computational
techniques are discussed in detail in section 3. Section 4 presents the basic two-factor mo del,
and discusses some of its principal characteristics. In section 5, we explain how the multi-
p erio d tree of rates is built using a mo di cation of the HSS metho dology. In section 6, we
present some numerical examples of the output of our mo del, apply the mo del to the pricing
of American-style and Bermudan-style options and discuss the computational eciency of
our metho dology. Section 7 presents our conclusions.
2 Requirements of the mo del
There are several desirable features of any multi-factor mo del of the term structure of
interest rates. Some of these features are requirements of theoretical consistency and others
are necessary for tractability in implementation. Keeping in mind the latter requirements,
it is imp ortant to recognize that the principal purp ose of building a mo del of the evolution
of the term structure is to price interest rate options generally, and, in particular, those
with path-dep endent payo s. The simplest examples of options that need to be valued
using such a mo del are American-style and Bermudan-style options on an interest rate.
First, since we wish to b e able to price any term-structure dep endent claim, it is imp ortant
that the mo del output is a probability distribution of the term structure of interest rates
Atwo-factor lognormal mo del of the term structure 3
at each p oint in time. A realistic mo del should be able to pro ject the term structure for
ten or twenty years, at least on a quarterly basis. With the order of forty or eighty sub-
p erio ds, the computational task is substantial and complex. If we do not compute each
term structure p oint at each no de of the tree, then we need to b e able to interp olate, where
necessary, to obtain required interest rates or b ond prices. As in the no-arbitrage mo dels of
HL, HJM, HW, BDT, and BK, we rst build the risk-neutral or martingale distribution of
the short-term interest rate, since other maturity rates and b ond prices can be computed
from the short-term rate.
The second and crucial requirement is that the interest rate pro cess be arbitrage-free. In
the context of term structure mo dels, the no-arbitrage requirement, in e ect, means that
the one-p erio d forward price of a b ond of any maturity is the exp ected value, under the risk-
neutral measure, of the one-p erio d-ahead sp ot price of the b ond. Since HL, this requirement
has b een well-understo o d, within the context of single factor mo dels, and is a prop erty
satis ed by the HW, BDT, and BK mo dels. However, the requirement is more demanding
in the two-factor setting as shown by HJM, Due and Kan 1993 . In a two-factor mo del
in which the factors are themselves interest rates, the no-arbitrage condition restricts the
b ehavior of the factors themselves as well as the b ehavior of b ond prices. However, the no-
arbitrage prop erty is also an advantage in a computational sense, allowing the computation
of b ond prices at a no de by taking simple exp ectations of subsequent b ond prices under the
risk-neutral measure.
The third requirement is that the term structure mo del should be consistent with the
current term structure and with the term structure of volatilities implicit in the price of
Europ ean-style interest rate caps and o ors. Mo dels that are consistent with the current
term structure have b een common in the literature since the work of Ho and Lee 1986 HL .
For example, the HJM, HW, BDT and BK mo dels are all of this typ e. The second part of
the requirement is rather more dicult to accommo date, since if volatilities are not constant
over time, the tree of rates may b e non-recombining, as in some implementations of the HJM
mo del, leading to an explosion in the numb er of states. The HW implementation of a two-
factor mo del, in Hull and White 1994 , sp eci cally excludes time dep endentvolatility. BK,
on the other hand, accommo date b oth time varying volatility and mean reversion of the
short rate within a one-factor mo del by varying the size of the time steps in the mo del.
This pro cedure is dicult to extend to a two-factor case.
At a computational level, it is necessary that the state space of the mo del do es not explo de,
pro ducing so many states that the computations b ecome infeasible. Even in a single-factor
mo del, this fourth requirement means that the tree of interest rates or b ond prices must
recombine. This is a prop erty of the binomial mo dels of HL, BDT, and BK, and also
of the trinomial mo del of HW. A number of computational metho ds have b een suggested
Atwo-factor lognormal mo del of the term structure 4
to guarantee this prop erty, including the use of di erent time steps and state-dep endent
probabilities. In the context of a two-factor mo del, the requirementiseven more imp ortant.
In our bivariate-binomial mo del we require that the number of states is no more than
2
n +1 , after n time steps. This is the bivariate generalization of the \simple" recombining
one-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 mo del is that the interest rates are lognormally distributed. Based
on the work of Vasicek 1977 , it is well known that the class of Gaussian mo dels, where
the interest rate is normally distributed, are analytically tractable, allowing closed-form so-
lutions for b ond prices. However, apart from admitting the p ossibilityof negative interest
rates, these mo dels are not consistent with the rightskewness that is considered imp ortant,
at least for some currencies. Furthermore, such a mo del would be inconsistent with the
widely-used Black mo del to value interest rate options, which assumes that the short-term
rate is lognormally distributed. Of the mo dels in the literature, the BDT and BK mo dels
explicitly assume that the short rate is lognormal. The HJM and HW multi-factor mo d-
els are general enough to allow for lognormal rates, but at the exp ense of computational
complexity.
In addition to these ve requirements, there is an overall necessity that the mo del be
computable for realistic scenarios, eciently and with reasonable sp eed. In this context,
we aim to compute option prices in a matter of seconds rather than minutes. To achieve
this we need a numb er of mo delling innovations, compared to the techniques used in prior
mo dels. These metho dological innovations are discussed in the next section.
3 Features of the metho dology
3.1 The sto chastic pro cess for interest rates
The dynamics of the term structure of interest rates can b e mo deled in terms of one of three
alternativevariables: zero-coup on b ond prices, sp ot interest rates, or forward interest rates.
If the ob jective of the exercise is to price contingent claims on interest rates, it is sucient
to mo del forward rates, as demonstrated by HJM. However, there are some problems with
adopting this approach in a multi-factor setting. First, from a computational p ersp ective,
for general forward rate pro cesses, the tree may b e non-recombining, which implies that a
large number of time-steps b ecomes practically infeasible. Second, it may be dicult to
estimate the volatility inputs for the mo del directly from market data. Usually, the implied
Atwo-factor lognormal mo del of the term structure 5
volatility data are obtained from the market prices of options on Euro-currency futures,
caps, o ors and swaptions, which cannot be easily transformed into the volatility inputs
required to build the forward rates in the HJM mo del.
We cho ose to mo del interest rates rather than prices b ecause existing metho dologies, in-
tro duced by HSS, can b e employed to approximate a pro cess with a log-binomial pro cess.
One ma jor problem arises in mo delling rates rather than prices, however, and that concerns
the no-arbitrage prop erty. Under the risk-neutral measure, forward b ond prices are related
to one-p erio d-ahead sp ot prices, but the relationship for interest rates is more complex, as
shown by HJM. This non-linearity makes the implementation of the binomial lattice much
more cumb ersome. However, if the interest rate mo delled is the Lib or rate, the no-arbitrage
condition can b e satis ed using Lib or futures rates. The Lib or futures rate is the exp ected
value, under the risk-neutral measure, of the sp ot Lib or rate. Hence we calibrate the mo del
to the current term structure using the futures Lib or rates.
We cho ose here to mo del interest rates, and then derive b ond prices, forward prices and
forward rates, as required, from the sp ot rate pro cess. We prefer to directly mo del the
short rate, which we interpret here as the three-month interest rate, since it is used to
determine the payo s on many contracts such as interest rate caps, o ors and swaptions.
One advantage of doing so is that implied volatilities from caplet/ o orlet o or prices may
b e used to determine the volatility of the short-rate pro cess fairly directly.
As in HW, we mo del the short rate, under the risk-neutral measure, as atwo-dimensional
AR pro cess. In particular, we assume that the logarithm of the short rate, follows such
a pro cess where the second factor is an indep endent sho ck to the forward premium. The
short rate itself and the premium factor each mean revert, at di erent rates, allowing for
quite general shifts and tilts in the term structure. Also, in contrast to HW, we assume
1
time dep endentvolatility functions for b oth the short rate and the premium factor . In this
mo del, the b ond prices and forward rates for all maturities can b e computed by backward
induction, using the no-arbitrage prop erty. Since the interest rate pro cess is de ned under
the risk-neutral measure, forward b ond prices are exp ectations of one-p erio d-ahead b ond
prices under this measure. This prop erty p ermits the rapid computation of b ond prices of
all maturities by backward induction, at each p oint in time.
1
By taking conditional exp ectations of the pro cess it can b e shown that the term structure of futures
rates is given by a log-linear mo del in anytwo futures rates. The pro of relies on the fact that the futures
rate is the exp ectation, under the risk-neutral measure, of the future sp ot rate, given the linear relationship
between the interest rate and the price.
Atwo-factor lognormal mo del of the term structure 6
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 exp ectation of the rate at any p oint dep ends on the rate itself and
the premium factor. A metho dology available in the literature, which allows the building
of a multivariate tree, approximating a lognormal pro cess with non-stationary variances
and covariances, is describ ed in Ho, Stapleton and Subrahmanyam 1995 HSS . The HSS
metho dology is itself a generalization to two or more variables of the metho d advo cated
by Nelson and Ramaswamy 1990 , who devised a metho d of building a \simple" or re-
combining binomial tree for a single variable. In HSS, the exp ectation of a variable dep ends
on its current value, but not on the value of a second sto chastic variable. However, as we
show in section 5, the metho dology is easily extended to this more general case. Essentially,
the HSS metho d relies on xing the conditional probabilities on the tree to accommo date
the mean reversion of the interest rates, the changing volatilities of the variables and the
covariances of the variables. In the case of interest rates, it is crucial to mo del changes in
the short rate so as to re ect the second, premium factor. This is the key, in a two-factor
mo del, to maintaining the no-arbitrage prop erty, while avoiding an explosion in the number
of states. Using our extension of the HSS metho dology allows us to mo del the bivariate
distribution of short rate and the premium factor, with n + 1 states for eachvariable after
2
n time steps, and a total of n +1 term structures after n time steps. This is achieved by
allowing the probabilities to vary in such a manner as to guarantee that the no-arbitrage
prop erty is satis ed and the tree is consistent with the given volatilities and mean reversion
of the pro cess.
One problem with extending the typical interest rate tree building metho ds of HW, BDT,
and BK to two or more factors arises from the forward induction metho dology normally
employed in these mo dels. The tree is built around the current term structure and the
calculation pro ceeds by moving forward p erio d-by-p erio d. This is exp ensive in computing
time, and could b ecome prohibitively so, in the case of multiple factors. To avoid this
problem, we employ a dynamic metho d of implementation of the HSS tree, which allows
us to compute the multivariate tree in a matter of seconds for up to eighty p erio ds. This
metho d uses the feature of HSS which allows a variation in the density of the tree over
any given time step. A forward, dynamic pro cedure is used wherebya two-p erio d tree with
changing density is converted into the required multi-p erio d tree. For example, when the
eightieth time step is computed, the program computes a two-p erio d tree with a density
over the rst p erio d of seventy-nine, and a densityover the second p erio d of one. This allows
us to compute the tree no des and the conditional probabilities analytically, and without
recourse to iterative metho ds.
Atwo-factor lognormal mo del of the term structure 7
3.3 Improving computational eciency
In practice muchofthe skill in building realistic mo dels rests on deciding exactly what to
compute. Potentially, in a tree covering eighty time steps, we could compute b ond prices for
between one and eighty maturities at each no de of the tree. Not only is this a vast number
of b ond prices, but also, most of the b ond prices will not be required for the solution of
any given option valuation problem. We assume here that it will b e sucientto compute
b ond prices for maturities one year apart from each other. Intermediate maturity prices, if
required, can always b e computed byinterp olation. Hence, in our eighty-time step examples,
where each time step is a quarter of a year, we compute at most twenty b ond prices. This
saving reduces the numb er of calculations by almost seventy- ve p ercent. For large numb ers
of time steps, it can turn an almost infeasible computational task into one that can be
accomplished within a reasonable time frame. For example, with three hundred time steps,
the number of b ond price calculations can be reduced from approximately twenty-seven
million to approximately one million.
In spite of the computational savings that are made byhaving a recombining tree metho d-
ology and reducing the number of b ond prices that need to be calculated, it may still be
the case that the computation time is excessive for a given problem. For example, for a
Bermudan-style b ond option that is exercisable every year for the rst six years of the
underlying b ond's twenty year life, we only require b ond prices at the end of each of the
rst six years. One computational advantage of the HSS metho dology, is that the binomial
density can b e altered so that this problem is reduced to a seven-p erio d problem with dif-
ferential density numb ers of time steps . The binomial density ensures sucient accuracy
in the computations, while the numb er of b ond and option price calculations is minimized.
4 The Two-factor Mo del
4.1 No-arbitrage prop erties of the mo del
We assume that, under the risk-neutral measure, the logarithm of the short-term interest
rate, for loans of maturity m, follows the pro cess
d ln r =[ t aln r +ln ]dt + t dz 1
r r 1 where
Atwo-factor lognormal mo del of the term structure 8
d ln =[ t bln ]dt + t dz
2
In the ab ove equations d ln r is the change in the logarithm of the short rate, t is a
r
time-dep endent constant term that allows the mo del to be calibrated to the current term
structure, a is the sp eed at which the short rate mean reverts, is a sho ck to the conditional
mean of the pro cess. We refer to as the forward premium factor, since it is crucial in
determining the forward rates in the mo del. t is the instantaneous volatility of the
r
short rate. The forward premium factor itself follows a di usion pro cess with mean ,
mean reversion b and instantaneous volatility t . Note that, in this notation and
r
t t
refer to the unconditional logarithmic standard deviation of the variable, over the p erio d
0 t, on a non-annualised basis. Although this structure is broadly similar to the mo del of
Hull and White 1994 , note that wedo not restrict the volatilities: t and t to be
r
constantover time, although they are assumed to b e non-sto chastic. Also, we assume that
the short-term interest rate is de ned on a simple, Lib or basis, i.e. r = [ 1=B 1]=m,
t t;t+m
where m is a xed maturity of the short rate and B is the price of a m-year, zero-coup on
t;t+m
b ond at time t. dz and dz are standard Brownian motions.
1 2
In discrete form, equation 1 can b e written as
ln r ln r = t aln r +ln +" ; 2
t t 1 r t 1 t 1 t
where
ln ln = t bln + ;
t t 1 t 1 t
and where " and are indep endently distributed, normal variables, and the mean and
t t
unconditional standard deviation of the logarithm of r and are , and ,
t t r r
t t t t
resp ectively.
First, we solve the mo del, to determine the constants t and t . We have, for any
r
lognormal pro cess of the form of equation 2 :
Lemma 1 Suppose that the short-term interest rate fol lows the process in equation 2 .
Then
ln r = [ln r ] 1 a +ln +" 3
t r t 1 r t 1 t
t
t 1 t 1
where
ln = [ln ] 1 b + ;
t t 1 t
t
t 1
with
2
=ln[E r ] =2;
r 0 t
t
r t
Atwo-factor lognormal mo del of the term structure 9
and
2
=ln[E ] =2:
0 t
t
t
Proof
Taking the unconditional exp ectation of equation 2
= t a + ;
r r r r
t
t 1 t 1 t 1
= t a
t
t 1 t 1
Then, substituting for t and t in 2 yields 3 .
r
Also since r and are lognormal, it follows from the moment generating function of the
t t
normal distribution that
2
=2 E r = exp +
0 t r
t r
t
2
=2 E = exp +
0 t
t
t
Hence, taking logarithms yields the statement in the lemma.2
Lemma 1 is essential in the calibration of the mo del. We have to cho ose the means of
r
t
the short rate to b e consistent with the absence of arbitrage and with the term structure of
interest rates at t =0. We achieve this by assuming, as given, a set of futures rates [f ].
0;t
These are Lib or futures rates for m-month usually three-month money. Assuming for the
moment that these futures rates are given, we build a mo del of r in which the exp ected
t
value of r , under the risk-neutral measure, is the futures rate, [f ]; 8t; 0 t 0;t back from the terminal date T ,we then compute zero b ond prices assuming that the forward price, under the risk-neutral measure, is the exp ected value of the one-p erio d-ahead sp ot price. This ensures that the no-arbitrage prop erty of the mo del is guaranteed in every state, and at each date. The no-arbitrage prop erties of the mo del are established in the following prop osition. First, we de ne the no-arbitrage prop erty, precisely. We have, as in Pliska 1997 the following: De nition [The No-Arbitrage Prop erty] A term-structure mo del satis es the no-arbitrage prop erty, if and only if the zero-coup on b ond prices satisfy B = B E B ; 0 s s;t s;s+1 s s+1;t Atwo-factor lognormal mo del of the term structure 10 Wenow show that, if the mo del is suitably calibrated, then the mo del is arbitrage free. We rst state without pro of a result from Cox, Ingersoll and Ross 1981 whichwe require in the pro of of the no-arbitrage prop erty that follows: Lemma 2 The futures price F of an asset with spot price S at time t is the value of an 0;t t asset which pays S =B B :::B at time t. t 0;1 1;2 t 1;t Proof See CIR 1981 , prop osition 2.2 Prop osition 1 The No-Arbitrage Prop erty Suppose that the Libor rate, r fol lows t the process: ln r ln r = t aln r +ln +" ; t t 1 r t 1 t 1 t where ln ln = t bln + ; t t 1 t 1 t under the risk-neutral measure, with E =1;8t, where " and are independently dis- t t t tributed, normal variables. Then, if the model is calibrated so that 2 =2 ; 8t =ln f r 0;t t r t where f is the futures Libor rate at time 0, for delivery at time t and if the forward price 0;t at t,of a t+k+1 maturity zero-coupon bond, for delivery at t +1 is given by B = E B ; 8t; k t t;t+1;t+k +1 t+1;t+k +1 then the no-arbitrage property holds. Proof From lemma 1 ln r = [ln r ] 1 a +ln +" t r t 1 r t 1 t t t 1 t 1 where ln = [ln ] 1 b + ; t t 1 t t t 1 and where 2 =ln[E r ] =2 r 0 t t r t Atwo-factor lognormal mo del of the term structure 11 and 2 =ln[E ] =2: 0 t t t From the calibration of the mo del 2 =ln f =2 r 0;t t r t and hence it follows that f = E r . It then follows that F = B B . 0;t 0 t 0;t 0 t;t+m Using the prop erty of exp ectations, we can write F = E E :::E B 0;t 0 1 t 1 t;t+m and ! B t;t+m : F = B E B E :::B E 0;t 0;1 0 1;2 1 t 1;t t 1 B B :::B 0;1 1;2 t 1;t B t;t+m at time t. From lemma 2, this equation gives the value of an asset paying B B :::B 0;1 1;2 t 1;t Hence, the value of the b ond itself must b e given by B = B E B E :::B E B : 4 0;t 0;1 0 1;2 1 t 1;t t 1 t;t+m From sp ot-forward parity, and the condition of the prop osition B t;t+k +1 B = = E B 5 t t;t+1;t+k +1 t+1;t+k +1 B t;t+1 Successive substitution of the condition 5 in 4 then yields B = B E B ; 0 0;t 0;1 0 1;t Hence, the calibration of the mo del ensures that the no-arbitrage equation holds for s =0. Finally, 5 guarantees that it holds for s>0.2 Prop osition 1 guarantees that the mo del is arbitrage free, if it is calibrated correctly to futures rates. The implication, in prop osition 1, is that the futures rate is a martingale, under the risk-neutral measure. Hence we can easily calibrate the mo del to the given term structure of futures rates, and thereby guarantee that the no-arbitrage prop erty holds. Atwo-factor lognormal mo del of the term structure 12 4.2 Regression Prop erties of the Mo del The two-factor mo del of the term structure, describ ed ab ove, has the Hull-White charac- teristic that the conditional mean of the short rate is sto chastic. Since the forward rate directly dep ends on the conditional mean, this induces an imp erfect correlation b etween the short rate and the forward rate. In this section we establish the regression prop erties of the mo del, using the covariances of the short rate and premium pro cess. These prop erties are required as inputs for the building of a binomial approximation mo del of the term structure. In the following lemma, we de ne the covariance of the logarithm of the short rate and the premium factor as . The pro cess assumed in Lemma 1 has the following prop erties:- r ; t t Lemma 3 Assume that ln r = [ln r ] 1 a +ln +" t r t 1 r t 1 t t t 1 t 1 where ln = [ln ] 1 b + ; t t 1 t t t 1 with E =1, 8t. 0 t Then, 1. the multiple regression r r t t 1 ln = + ln + ln +" 6 r r r t 1 t t t t E r E r 0 t 0 t 1 has coecients 2 2 2 + + =2 = r r r t t t r r t t 1 t 1 = 1 a r t =1 r t 2. the regression ln = + ln + 7 t t 1 t t t has coecients 2 2 =[ + 1 b ]=2 t t t 1 = 1 b t Atwo-factor lognormal mo del of the term structure 13 3. the conditional variance of ln r is given by t 2 2 2 2 var " = 1 a 1 a ; t 1 t r ; r r t 1 t 1 t t 1 t 1 where denotes the annualized covariance of the logarithms of the short rate and r; the premium factor. Proof First, we derive the following covariances from equation 3 2 + ; = 1 a r ; r ;r t t t t+1 r t 2 = 1 b + 1 a 1 b r ; r ; t t t 1 t 1 t 1 and 2 ; = 1 b ; t t+1 t 2 = 1 a + : r ; r ; t t 1 t 1 t 1 t 1 Now from the multiple regression r r t 1 t = + ln + ln +" 8 ln r r r t 1 t t t t E r E r 0 t t 1 the regression co ecients are 2 r ;r r ; r ; t t t 1 t 1 t 1 t 1 t 1 = r t 2 2 2 r ; r t 1 t 1 t 1 t 1 2 r ;r r ;r r ; t t t 1 r t 1 t 1 t 1 t 1 = r t 2 2 2 r ; r t 1 t 1 t 1 t 1 Substituting the covariances, and simplifying yields = 1 a 9 r t and =1 10 r t Atwo-factor lognormal mo del of the term structure 14 From the lognormalityof r and we can write equation 3 as t t 2 2 ln r ln[E r ] + =2 = fln r ln[E r ] + =2g 1 a t 0 t t 1 0 t 1 r r t t 1 n o 2 + ln ln[E ] + =2 + " t 1 0 t 1 t t 1 Re-arranging yields r t 2 2 2 ]=2 1 a + + = [ ln r r t t t E r 0 t r t 1 + ln 1 a +ln +" t 1 t E r 0 t 1 Given 8 , 9 , and 10 , we have as stated in the lemma. Similarly, with E =1, r 0 t t wehave ln = +ln 1 b + ; t t 1 t t and = E [ln ] 1 b [ln[E ] 0 t 0 t t 2 2 ]=2 + 1 b =[ t t t 1 Finally, the variance of " , given t,is t r r t t 1 2 var " = var ln var ln t 1 t 0 0 r;t E r E r 0 t 0 t 1 2 var [ln ] cov [ln r ; ln ] 0 t 1 r r t 1 t 1 t t r t or, 2 2 2 2 var " = 1 a 1 a : t 1 t r ; r r t 1 t 1 t t 1 t 1 2 Note that we require the multiple regression co ecients ; ; and in order to build r r r t t t the binomial approximation of the multi-variate pro cess, using the metho d of Ho, Stapleton and Subrahmanyam 1995 . From part 1 of the lemma, the co ecients simply re ect the r mean-reversion of the short rate. The co ecients are all unity, re ecting the one-to-one r Atwo-factor lognormal mo del of the term structure 15 relationship between , the forward premium factor and the exp ected sp ot rate. The r co ecients re ect the drift of the lognormal distribution, which dep ends on the variances of the variables. Part 2 of the lemma shows that the regression relation for is a simple t regression, where the co ecients re ect the constant mean reversion of the premium factor. Lastly, part 3 of the lemma gives an expression for the conditional variance of the logarithm of the short rate. 4.3 Determining the Volatility Inputs of the Mo del In order to build the mo del outlined ab ove, we need the parameters of the premium pro cess, as well as those for the short rate pro cess itself. The result in Lemma 3, part 3 gives the relationship of the conditional volatility of the short rate to the unconditional volatilities of the 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, p erhaps from caplet/ o orlet volatilities and that the mean reversion is given. The premium pro cess, , t on the other hand, determines the extent to which the rst forward rate di ers from the sp ot rate in the mo del. Since the premium factor is not directly observable, we need to b e able to estimate the mean and volatility of the premium factor from the b ehavior of forward or futures rates. In order to discuss this, we rst establish the following general result:- Lemma 4 Assume that ln r = [ln r ] 1 a +ln +" t r t 1 r t 1 t t t 1 t 1 where ln = [ln ] 1 b + ; t t 1 t t t 1 with E =1, 8t, then the conditional volatility of is given by 0 t t 2 2 2 0:5 t =[ t 1 a t ] x r 2 where x = E r and var [ln x ] = t . t t+1 t 1 t x Proof Taking the conditional exp ectation of equation 3 at t E [ln r ] = [ln r ] 1 a +ln : t t+1 r t r t t t t+1 Atwo-factor lognormal mo del of the term structure 16 Given x = E r and using the lognormal prop ertyof r , t t t+1 t+1 2 ln x = E [ln r ] + t +1 =2 t t t+1 r 2 = t+1 =2+ + [ln r ] 1 a + [ln ] 1 b + r t r t 1 t r t t+1 t 1 Hence 2 2 t =var [ln x ] = 1 a var [ln r ]+var [ ] t 1 t t 1 t t 1 t x or 2 2 2 2 t = t + 1 a t x r and the statement in the lemma follows.2 Lemma 4 relates the volatility of the premium factor to the volatility of the conditional exp ectation of the short rate. To apply the result in the current context, we rst assume that the short rate follows the pro cess assumed in the lemma under the risk-neutral pro cess. We then use the fact that the unconditional exp ectation of the t + 1 th rate is f = E r , t;1 t t+1 i.e. the rst futures or forward rate is the exp ected value of the next p erio d sp ot rate. This implication of no-arbitrage leads to 2 lnf = + [lnr ] 1 a +ln 1 b + + t =2: 11 t;1 r t r t 1 t t r t+1 It follows that the conditional logarithmic variance of the rst futures rate is given by the relation 2 2 2 2 t = 1 a t + t : 12 f r Hence, the volatility of the premium factor is p otentially observable from the volatility of the rst futures rate. This in turn could b e estimated empirically or implied from options on the futures rate. Atwo-factor lognormal mo del of the term structure 17 5 The Multivariate-Binomial Approximation of the Pro cess 5.1 The HSS approximation The general approachwe take to building a bivariate-binomial lattice, representing a discrete approximation of the pro cess in equation 1 , is to construct separate re-combining binomial trees for the short-term interest rate and the forward-premium factors. The no-arbitrage prop erty and the covariance characteristics of the mo del are then captured bycho osing the conditional probabilities at each no de of the tree. Wenow outline our metho d for approximating the two-factor pro cess interest rate pro cess, describ ed ab ove. We use three typ es of inputs: the unconditional means E r , t =1; :::; T , 0 t of the short-term rate, the volatilities of " i.e. the conditional volatility of the short rate, t given the previous short rate and the one-p erio d forward premium factor, denoted t and r the 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 pro cess in 3 is then approximated using an adaptation of the metho dology describ ed in Ho, Stapleton and Subrahmanyam 1995 HSS . HSS showhow to construct a multip erio d multivariate- binomial approximation to a joint-lognormal distribution of M variables with a re-combining binomial lattice. However, in the present case, we need to mo dify the pro cedure, allowing the exp ected value of the interest rate variable to dep end up on the premium factor. That is, we need to mo del the twovariables r and , where r dep ends up on . Furthermore, t t t t 1 in the present context, we need to implement a multi-p erio d pro cess for the evolution of the interest rate, whereas HSS only implement a two-p erio d example of their metho d. In this section, these mo di cations and the multi-p erio d algorithm are presented in detail. We divide the total time p erio d into T p erio ds of equal length of m years, where m is the maturity p erio d, in years, of the short-term interest rate. Over each of the p erio ds from t to t +1 we denote the numb er of binomial time steps, termed the binomial density,byn. t Note that, in the HSS metho d, n can vary with t allowing the binomial tree to have a ner t density, if required for accurate pricing, over a sp eci ed p erio d. This might b e required, for example if the option exercise price changes b etween two dates, increasing the likeliho o d of the option b eing exercised, or for pricing barrier options. We use the following result, adapted from HSS: Lemma 5 Suppose that X fol lows a lognormal process, where E X = 1; 8t. Let the t 0 t conditional logarithmic standard deviation of X be t . If X is approximated by a log- t x t binomial distribution with binomial density N = N + n and if the proportionate up and t t 1 t Atwo-factor lognormal mo del of the term structure 18 down movements, u and d are given by t t 2 p d = t 1 + exp 2 t 1=n x t u = 2 d t t and the conditional probability of an up-move at node r of the lattice is given by E X + N r ln u n + r ln d t 1 t t 1 t t t q = x t n [ln u ln d ] t t t then the unconditional mean and conditional volatility of the approximatedprocess approach ^ their true values, i.e., E X ! 1 and ^ ! as n !1. 0 t x x t t Proof If E X = 18t, then we obtain the result as a sp ecial case of HSS 1995 , Theorem 0 t 1.2 5.2 Computing the no dal values In this section, we rst describ e how the vectors of the short-term rates and the premium factor are computed. We approximate the pro cess for the short-term interest rate, r , with t a binomial pro cess, whichmoves up or down from its exp ected value, by the multiplicative factors d and u . Following HSS, equation 7 , these are given by r r t t 2 p d = r t 1 + exp 2 t 1=n r t u = 2 d r r t t We then build a separate tree of the forward premium factor . The up-factors and down- factors in this case are given by 2 p d = t 1 + exp 2 t 1=n t u = 2 d t t Atwo-factor lognormal mo del of the term structure 19 At no de j at time t, the interest rates r and premium factors are calculated from the t t equations N j j t r = u d E r ; 13 t;j 0 t r r t t N j j t = u d ; t;j t t j = 0; 1; :::; N : t P where N = n . In general, there are N +1 no des, i.e., states of r and , since b oth t t t t t t 2 binomial trees are re-combining. Hence, there are N +1 states after t time steps. t 5.3 Computing the conditional probabilities As in HW, in general, the covariance of the two variables may be re ected by varying the conditional probabilities in the binomial pro cess. Since the trees of the rates and the forward premium are b oth re-combining, the time-series prop erties of eachvariable must also b e captured by adjusting the conditional probabilities of moving up or down the tree, as in HSS and in Nelson and Ramaswamy 1991 . Since, increments in the premium variable are indep endentof r , this is the simplest variable to deal with. Using the results of lemma 3, we t compute the conditional probability using HSS, equation 10 . In this case the probability of a up-move, given that is at no de j ,is t 1 + ln N j lnu j + n lnd t 1;j t 1 t t t t t 14 q = t n lnu lnd t t t where = 1 b t 2 2 = + =2 t t t t 1 2 and where b is the co ecient of mean reversion of , and is the unconditional logarithmic t variance of over the p erio d 0 t . The key step in the computation is to x the conditional probability of an up-movementin the rate r , given the outcome of r , the mean reversion of r , and the value of the premium t t 1 factor . In discussing the multi-p erio d, multi-factor case, HSS present the formula for t 1 the conditional probability when a variable x dep ends up on x and a contemp oraneous 2 1 Atwo-factor lognormal mo del of the term structure 20 variable, y . Again using the regression prop erties derived in lemma 3, and adjusting HSS, 2 equation 13 to the present case, we compute the probability + ln r =E r + ln N j lnu j + n lnd r r t 1;j t 1 r t 1;j t 1 r t r t t t t t 15 q = r t n lnu lnd t r r t t where = 1 a r t =1 r t 2 2 2 ]=2 + + =[ r r r t t r t r t t t 1 Then, by lemma 5, the pro cess converges to a pro cess with the given mean and variance inputs. 5.4 The multip erio d algorithm HSS 1995 provides the equations for the computation of the no dal values of the variables, and the asso ciated conditional probabilities, in the case of two p erio ds t and t +1. Ecient implementation requires the following pro cedure for the building of the T p erio d tree. The metho d is based on forward induction. First, compute the tree for the case where t=1. This gives the no dal values of the variables and the conditional probabilities, for the rst two p erio ds. Then, treat the rst two p erio ds as one new p erio d, but with a binomial density equal to the sum of the rst two binomial densities. The computations are then made for p erio d three no dal values and conditional probabilities. Note that the equations for the up and down movements of the variables always require the conditional volatilities of the variables to compute the vectors of no dal values. The following steps are followed:- 1. Using equation 13 , compute the [n x1]vectors of the no dal outcomes of r , using 1 1 1 inputs 1 , E r , 1 , E and binomial density n . Also compute the [ n + n x1] r 1 1 1 1 2 vectors r , using inputs 2 , E r , 2 , E and binomial density n . Assume the 2 2 r 2 2 2 probability of an up-move in r is 0.5 and then compute the conditional probabilities q 1 1 using equation 14 with t=1. Then compute the conditional probabilities q , q , using r 2 2 equations 14 and 15 , with t=2. 2. Using equation 13 , compute the [ N + n x1]vectors r , using inputs 3 , E r , 2 3 3 3 r 3 3 , E r and binomial density, n . Then compute the conditional probabilities q , q r 3 3 r 3 3 using equations 14 and 15 with t=3. Atwo-factor lognormal mo del of the term structure 21 3. Continue the pro cedure until the nal p erio d T . In implementing the ab ove pro cedure, we rst complete step 1, using t = 1 and t = 2, and with the given binomial densities n and n . To e ect step 2, we then rede ne the p erio d 1 2 from t = 0 to t = 2 as p erio d 1 and the p erio d 3 as p erio d 2 and re-run the pro cedure with a binomial densities n = n + n and n = n . This algorithm allows the multip erio d 1 2 3 1 2 lattice to b e built by rep eated application of equations 13 , 14 and 15 . 5.5 A summary of the approximation metho d We will summarize the metho dology by using a two-p erio d and a three-p erio d example. Figure 1 shows the recombining no des for the two-factor pro cess in the two-p erio d case. The interest rate go es up to r or down to r at t = 1. The forward premium factor 1;0 1;1 go es up to or down to at t =1, with probability q . In the second p erio d, there 1;0 1;1 1 are just 3 no des of the interest rate tree, together with 3 p ossible premium factor values. There are 9 p ossible states, and the probability of an r value materialising is q . Note 2 r 2 that this probability dep ends on the level of the premium factor and of the interest rate at time t =1. The recombining prop erty of the lattice, which is crucial for the computability of the lattice, is emphasised in Figure 2, where we show the pro cess for the interest rate over p erio ds t = 2 and t =3. After two p erio ds there are 3 interest rate states and 9 states representing all the p ossible combinations of the interest rate and premium factor. The interest rate then go es to 4 p ossible states at time t = 3 and there are 16 states representing all the p ossible combinations of rates and premium factor. Note that the probability of reaching an interest rate state at t = 3, dep ends on b oth the interest rate and the premium factor at t =2. It is these probabilities that allow the no-arbitrage prop ertyof the mo del to b e ful lled. In the mo del, 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 densityof 2 n= 1 there are t +1 term structures generated by the binomial approximation, at time t. 6 Mo del Validation and Examples of Inputs and Outputs This section shows the results from several numerical examples and examines the two factor term structure mo del describ ed in previous sections in detail. Firstly,we show an example of howwell the binomial approximation converges to the mean and unconditional volatility inputs. Secondly, we show that a two-factor term structure mo del can b e run in a sp eedy Atwo-factor lognormal mo del of the term structure 22 and ecient manner. Thirdly,we discuss the input and output for an eight- p erio d example, showing illustrative output of zero-coup on b ond prices, and conditional volatilities. Finally we show the output from running a forty-eight quarter mo del, including the pricing of Europ ean and Bermudan style options on coup on b onds. 6.1 Convergence of Mo del Statistics to Exogenous Data Inputs The rst test of the two-factor mo del is how quickly the mean and variance of the short rates generated converge to the exogenous data. Table 1 shows an example of a twenty-p erio d mo del, where the input mean of the sp ot rate is 5, with a 10 conditional volatility. There is no mean reversion and the premium has a volatility of 1. Note rst that for a binomial density of 1, the accuracy of the binomial approximation deteriorates for later p erio ds. This is due to the premium factor increasing with maturity and the diculty of coping 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 density to 2, 3, and 4 resp ectively. By comparing di erent binomial densities in a given row of the table we observe the convergence of the binomial approximation to the exogenous inputs as the density increases. Even for the 20 p erio d case, high accuracy is achieved by increasing the binomial densityto4. Table 1 here 6.2 Computing Time The most imp ortant feature of the two-factor mo del prop osed in this pap er is the compu- tation time. With two sto chastic factors rather than one, the computation time can easily increase dramatically. In table 2, we illustrate the eciency of our mo del by showing the time taken to compute the zero-coup on b ond prices and option prices. With a binomial density of 1, the 48-p erio d mo del takes 12.3 seconds. Doubling the number of p erio ds in- creases the computer time by a factor of six. There is a trade-o between the number of p erio ds, the binomial density of each p erio d, and the computation time for the mo del. This is illustrated by the second line in the table, showing the e ect of using a binomial density of 2. Again the computation time increases more than prop ortionately as the density in- Atwo-factor lognormal mo del of the term structure 23 creases. The time taken for the 28-p erio d mo del, when the binomial density is 2, is roughly the same as that for the 48-p erio d mo del with a densityof1. Table 2 here 6.3 Numerical Example: An Eight-Perio d Bond This subsection shows a numerical example of the input and output of the two-factor term structure mo del, in a simpli ed eight quarter example . It illustrates the large amount of data pro duced by the mo del, even in this small scale case. The input is shown rst in Table 3. We assume a rising curve of futures rates, starting at 5 and rising to 6. These are used to x the means of the short rate for the various p erio ds. The second row shows the conditional volatilities assumed for the short rate. These start at 14 and fall through time to 12. We then assume a constant mean reversion of the short rate, of 10, and constant conditional volatilities and mean reversion of the premium factor, of 2 and 40 resp ectively. Table 3 here Tables 4 and 5 show a selection of the basic output of the mo del. For a binomial density of one, there are 4 states at time 1, 9 states at time 2, 16 states at time 3, and so on. In each state the mo del computes the whole term structure of zero-b ond prices. In Table 4, we show just the longest b ond price, paying one unit at p erio d 8. These are shown for the 4 states at time 1, in the rst blo ck of the table. The subsequent blo cks show the 9 prices at time 2, the 16 prices at time 3, and so on. Table 4 here One of the most imp ortant features of the metho dology is the way that the no-arbitrage prop erty is preserved, by adjusting the conditional probabilities at each no de in the tree of rates. In Table 5, we show the probabilityof 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 blo ck of the table is the set of probabilities conditional on b eing in one of four p ossible states at time 1. The second blo ck shows the conditional probabilities at time 2, in the 9 p ossible states, and so on. Atwo-factor lognormal mo del of the term structure 24 Table 5 here 6.4 An Example of an Option on a Coup on-Bond An imp ortant application of the mo del is to price and hedge contingent claims such as options with American and path-dep endent features. A go o d example is a Bermudan option on a coup on b ond. We illustrate our metho dology by pricing a six-year option on a twelve- year coup on b ond. The option has the Bermudan feature that it is exercisable, at par, at the end of eachyear up to the option maturityinyear six. We rst build the tree of rates, assuming the data detailed in the notes to Table 6. Note that the mo del uses 48 quarterly time p erio ds, to cover the twelve-year life of the coup on b ond. Table 6 here Table 6 shows that the Bermudan option on the coup on b ond is worth considerably more than the Europ ean option. Also there is a small p ositive e ect of valuing the options with the binomial tree with a density of two. Illustrative output of the price of the underlying coup on b ond is shown in Table 7. Table 7 here 7 Conclusions In this pap er wehave presented a mo del of the term structure of interest rates which can b e regarded as a two-factor extension of the Black-Karasinski lognormal-rate mo del. However, in this mo del, we assume that the short-term Lib or interest rate follows a lognormal pro cess. Wehave shown that, by calibrating to the current term structure of futures rates, the mo del is arbitrage-free in the sense of Ho and Lee 1986 and Pliska 1997 . The mo del 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 develop ed a re-combining bivariate-binomial tree, which has a non-explo ding numb er of no des. Atwo-factor lognormal mo del of the term structure 25 Wehave applied the mo del to the valuation of American-style and Bermudan-style options on coup on b onds. Wehave shown that prices are computable in seconds for examples with a realistic number of p erio ds. Also, prices in the two-factor mo del exceed those in the one-factor mo del, 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 typ e mo del develop ed here and the Heath-Jarrow-Morton multifactor mo del needs to be explored further. Sp eci cally,we in- tend to generate forward-rate volatilities as outputs of our mo del in an extension of the analysis in the pap er. Second, it is not clear to what extentatwo-factor mo del, as opp osed to a one-factor mo del, is necessary for the accurate valuation of sp eci c interest-rate related contingent claims. We intend to investigate the prop erties of the mo del, in a subsequent pap er, by applying it to a range of American-style, Bermudan-style and exotic options on b onds and interest rates. Third, the application of mo dels of this typ e to derive risk man- agement measures for interest-rate dep endent claims should be studied further. We hop e to use the framework presented in this pap er to address these issues in further research. Atwo-factor lognormal mo del of the term structure 26 References 1. 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Vasicek, O., 1977 , \An Equilibrium Characterization of the Term Structure", Jour- nal of Financial Economics, 5, 177{188. Atwo-factor lognormal mo del of the term structure 28 Figure 1: A Recombining Two-factor Pro cess for the Short-term Interest Rate Two-p erio d case ;r 2;0 2;0 * q 2 P * P r P 2;0 P P Pq ;r 2;1 2;0 H * > H 7 H H H Hj q r 2 H H H H H Hj ;r ;r 2;2 2;0 1;0 1;0 1 q 1 @ r 1;0 @ @ P ;r 2;0 2;1 P * P @ P P P @ P P Pq @ P P * H P r 2;1 H @ P P ;r H 1;1 1;0 P H @ P P H @R Pq Hj ;r 1 * 2;1 2;1 H @ H H ;r @ 1;0 1;1 H H Hj @ 1 H H S @ H H S @ H Hj ;r @R 2;2 2;1 S P P P S P P r 1;1 P S P P Pq S Z ;r 2;0 2;2 ;r * S 1;1 1;1 Z Z S Z S Z H * S Z H H Z S H Sw Z H ;r Z~ Hj 2;1 2;2 * H H r 2;2 H H H Hj H H H H H Hj ;r 2;2 2;2 t =1: 4 states t=2: 9 states [1] The probabilityofmoving, for example, to r given ;r isq , de ned in Equation 15 . 2;0 1;0 1;0 r 2 [2] The probability of moving, for example, to ;r given r and is q , de ned in 1;0 1;0 1;0 1;0 1 Equation 14 . Atwo-factor lognormal mo del of the term structure 29 Figure 2: A Recombining Two-factor Pro cess for the Short-term Interest Rate Three-p erio d case ;r 3;0 3;0 * H * H Hj ;r 3;1 3;0 H * * H Hj r 3;0 *H H H H Hj Hj ;r 3;2 3;0 H * H H Hj H H Hj ;r 3;3 3;0 ;r ;r 3;0 3;1 2;0 2;0 * * Z Z H * H * r 2;0 Z H H Hj ;r Hj ;r 3;1 3;1 Z 2;1 2;0 X H X H * * * X Z H X H X Xz Hj Z~ r H Hj 3;1 : * H H H * H H H Hj Hj Hj ;r ;r 3;2 3;1 2;2 2;0 H * H H Hj ;r 2;0 2;1 H * H J @ Hj ;r 3;3 3;1 r * J @ H 2;1 H @R ;r Hj J 2;1 2;1 H * @ @ H J Hj ;r 3;0 3;2 H @ @ * J H Hj @R ;r @ 2;2 2;1 J H H * H J H @ J Hj ;r ;r H 3;1 3;2 2;0 2;2 X * H X * H J * @ X J H A X H X Hj Hj @RJ^ Xz r J : 3;2 H * H H A * > H H H J Hj Hj Hj ;r ;r J^ A 3;2 3;2 2;1 2;2 H * * H H J H A Hj Hj r 2;2 H H J A H H Hj Hj ;r ;r J 3;3 3;2 2;2 2;2 A @ J A ;r 3;0 3;3 @ * J A @ J A * H @ H J A ;r 3;1 3;3 Hj H * * @ J H A @R Hj J^ AU r 3;3 H H * H H Hj Hj ;r 3;2 3;3 * H H Hj H H Hj ;r 3;3 3;3 t =2: 9 states t =3: 16 states [1] The probabilityofmoving, for example, to r given ;r isq , de ned in Equation 15 . 3;0 2;0 2;0 r 2 [2] The probability of moving, for example, to ;r given r and is q , de ned in 3;0 3;1 3;1 2;0 3 Equation 14 . Atwo-factor lognormal mo del of the term structure 30 Table 1: Convergence of Term Structure Mo del Binomial Density 1 2 3 4 Perio d [0, 1 ] mean 5.0 5.0 5.0 5.0 volatility 10.00 10.00 10.00 10.00 [0, 2 ] mean 4.9999 4.99997 4.99998 4.99998 volatility 9.97 9.99 9.99 9.99 [0, 3 ] mean 4.9998 4.99992 4.99994 4.99996 volatility 9.95 9.97 9.99 9.99 3 [0, 4 ] mean 4.9997 4.99985 4.9999 4.99992 4 volatility 9.92 9.97 9.98 9.98 [0, 5 ] mean 4.9996 4.9997 4.9998 4.9998 volatility 9.93 9.96 9.97 9.97 [0, 10] mean 4.998 4.9992 4.9995 4.9996 volatility 9.88 9.94 9.96 9.97 [0, 20] mean 4.996 4.998 4.998 4.999 volatility 9.85 9.92 9.94 9.96 The numb ers in the table are the computed means and volatilities, in p ercent, for the short rate over p erio ds 1,2,3,4,5,10, and 20, using the output of the two-factor mo del. The means are calculated using the p ossible outcomes and the no dal 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. Atwo-factor lognormal mo del of the term structure 31 Table 2: Computing Time for Bond and Option Pricing seconds Number of Perio ds 8 12 28 48 Binomial Density1 0 0.3 2.5 12.3 Binomial Density2 0.3 1 12.7 - The table shows the time taken to compute all the zero-b ond prices, coup on b ond prices and option prices, given the tree of rates.The computer sp eed is 266 MHZ, and the pro cessor is Pentium. Atwo-factor lognormal mo del of the term structure 32 Table 3: 8-p erio d Example Input Perio d 1 2 3 4 5 6 7 8 Futures rate 5.0 5.2 5.4 5.6 5.7 5.8 5.9 6.0 Conditional 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 10.0 Conditional 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 numb ers are in p ercent. The table shows the exogenous data input for an 8-p erio d example. The short rate is the quarterly rate, so the p erio d length is quarter of one year. Input data relating to the short rate app ears in the rst three rows; data relating to the premium in the last tworows. Atwo-factor lognormal mo del of the term structure 33 Table 4: Illustrative Output of Zero-Coup on 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-coup on b ond paying one unit of currency at the end of p erio d 8. The rst set of 4 numb ers are the time 1 b ond prices B , the second set of 9 numb ers are the time 2 b ond 1;8 prices B , through to the time 5 set of 36 prices B . The 49 prices, B , and 64 prices, B , are 2;8 5;8 6;8 7;8 not shown for reasons of space. Atwo-factor lognormal mo del of the term structure 34 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 numb ers are the probabilities at time 1, the second set of9numb ers 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 erentvalues of the premium factor. Atwo-factor lognormal mo del of the term structure 35 Table 6: 48 p erio d Option Price $ Binomial Density 1 2 Bermudan Call 1.0400 1.0573 Europ ean Call 0.5910 0.6032 The ab ove table shows the value of a 6-year call option on a coup on b ond that pays $100 in 12 years time. The annual coup on rate of the b ond is 5. The interest rate tree is built assuming a futures rate of 5 for each maturity,avolatility of the short rate of 10, a co ecient of mean reversion of the short rate of 10, and volatility of the premium at 1 with 50 co ecient of mean reversion. Atwo-factor lognormal mo del of the term structure 36 Table 7: Illustrative Output of Coup on 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 coup on-b ond prices are computed at each of 25 no des at the end of year 1, shown in the rst blo ck in the table. 91 prices are computed at the end of year 2, shown in the second blo ckofthe table. The coup on b ond has an annual coup on 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.