An Arbitrage-free Two-factor Model of the Term Structure of Interest Rates

37 Pages·1998·0.29 MB·English

An Arbitrage-free Two-factor Model of the Term
Structure of Interest Rates: A Multivariate
1
Binomial Approach
2 3 4
Sandra Peterson Richard C. Stapleton Marti G. Subrahmanyam
May 5, 1998
1
Preliminary. Please do not quote.
2
Department of Accounting and Finance, The Management School, Lancaster University, Lan-
caster LA1 4YX, UK. Tel:(44)524{593637,Fax:(44)524{847321.
3
Department of Accounting and Finance, Strathclyde University, Glasgow,UK. Tel:(44)524{381
172, Fax:(44)524{846874, e{mail:[email protected]
4
Leonard N. Stern School 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
We build a no-arbitrage model of the term structure, using two stochastic factors on each
date, the short-term interest rate and the premium of the forward rate over the short-term
interest rate. The model can be regarded as an extension to two factors of the lognormal
interest rate model of Black-Karasinski. It allows for mean reversion in the short rate
and in the forward premium. The method is computationally e(cid:14)cient for several reasons.
First, interest rates are de(cid:12)ned on a bankers’ discount basis, as linear functions of zero-
coupon bond prices, enabling us to use the no-arbitrage condition to compute bond prices
without resorting to iterative methods. Second, the multivariate-binomial methodology
of Ho-Stapleton-Subrahmanyam is extended so that a multiperiod tree of rates with the
no-arbitrage property can be constructed using analytical methods. The method 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 number
of term structures is simpli(cid:12)ed by using a limited number of ’bucket rates’ in each term
structure scenario. In addition to these computational advantages, a key feature of the
model is that it is consistent with the observed term structure of volatilities implied by
the prices of interest rate caps and (cid:13)oors. We illustrate the use of the model by pricing
American-style and Bermudan-style options on interest rates. Option prices for realistic
examples using forty time periods are shown to be computable in seconds.
A two-factor model of term structure:::::::::::::::::::::::::::::::::::::::::::::::::::1
1 Introduction
Perhapsthemostimportantanddi(cid:14)cultproblemfacingpractitionersinthe(cid:12)eldofinterest
ratederivativesinrecentyearshasbeentobuildinter-temporalmodelsofthetermstructure
of interest rates, that are both analytically sound and computationally e(cid:14)cient. These
models are required, both to help in the pricing, and in the overall risk management of a
book of interest rate derivatives. Although many alternative models have been suggested
in the literature and implemented in practice, there are serious disadvantages with most
of them. For example, Gaussian models of interest rates, which have the advantage of
analytical tractability, have the drawbacks of allowing for negative interest rates, as well as
failing to 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. On the other hand, the Black model which is widely used to value European-style
interest rate caps and (cid:13)oors, is not strictly in line with the de(cid:12)nition of the contracts and
also is not founded on an explicit model of the term structure of interest rates.
Since the work of Ho and Lee (1986), it has been widely recognized that term-structure
modelsmust possess the no-arbitrage property. In this context, a no-arbitrage model is one
where the forward price of a bond is the expected value of the one-period-ahead spot bond
price, under the risk-neutral measure. Building models that possess this property has been
a major pre-occupation of both academics and practitioners in recent years. One model
that achieves thisobjective ina one-factor context isthe modelproposedbyBlack, Derman
and Toy (1990)(BDT), and extended by Black and Karasinski (1991)(BK). In essence, the
model which we buildin this paper can be thought of as a two-factor extension of this type
of model. In our model, interest rates are lognormal and are generated by two stochastic
factors. The general approach we take is similar to that of Hull and White (1994)(HW),
where the conditional mean of the short rate depends on the short rate and an additional
stochastic factor, which can be interpreted as the forward premium. In contrast to HW,
and in line with BK, we build a model where the conditional variance of the short rate is a
functionof time. It follows that the model can be calibrated to the observed term structure
ofinterestratevolatilitiesimpliedbyinterestratecaps/(cid:13)oors. Essentially,theaimhereisto
buildatermstructuremodelwhichcanbeappliedtovalueAmerican-stylecontingentclaims
on interest rates, which is consistent with the market prices of European-style contingent
claims.
An alternative approach to building a no-arbitrage term structure for pricing interest rate
derivativeshas beenpursuedby Heath, Jarrow andMorton (1992)(HJM). Inthisapproach,
assumptions are made about the volatility of the forward interest rates. Since the forward
rates are related in a no-arbitrage model to the future spot rates, there is a fairly close
A two-factor model of term structure:::::::::::::::::::::::::::::::::::::::::::::::::::2
relationshipbetween this approach and the one we are taking. In fact, the HJM 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-HW
approach has the advantage of requiring as inputs the volatilities of the short rate and of
longer bond yields which are more directly observable from market data on the pricing of
caps, (cid:13)oors and swaptions.
We would like any model of the stochastic term structure to have a number of desirable
properties. Apartfromsatisfyingtheno-arbitrageproperty,wewanttheoutputofthemodel
to match the inputs i.e., the conditional volatilities of the variables and the mean-reversion
oftheshortrate andthepremiumfactor . We alsorequirethattheshortterminterestrates
belognormal,sothat theyareboundedfrombelowbyzero andskewedto theright. From a
computationalperspective,werequirethestatespacetobenon-explosive,i.e. re-combining,
so that a reasonably large span of time-periodscan be covered. The complexitiescaused by
these modelrequirements are discussedinsection 2. We introducea number of new aspects
intoourmodelthatallowustosolvetheserequirements. Themostimportantsimpli(cid:12)cation
arises from modeling the bankers’ discount interest rate. We then extend and adapt the
recombining binomial methodology of Ho, Stapleton and Subrahmanyam (1995)(HSS) to
modellognormalrates, ratherthanprices. Thenewcomputationaltechniquesarediscussed
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 multiperiod tree of rates is
builtusinga modi(cid:12)cationof theHSSmethodology. Insection 6, we presentsome numerical
examples of the output of our model, apply the model to the pricing of American-style
and Bermudan-style options and discuss the computational e(cid:14)ciency of our 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 of
interestrates. Someofthesefeaturesarerequirementsfortheoreticalconsistencyandothers
are necessary for tractability in implementation. Keeping in mind the latter requirements,
it is important to recognize that the principal purpose of buildinga model of the evolution
ofthetermstructureistopriceinterest rateoptionsgenerallyand, inparticular,thosewith
pathdependentpayo(cid:11)s. Thesimplestexamplesofoptionsthatneedtobevaluedusingsuch
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 important
that the model output is a probability distribution of the term structure of interest rates
A two-factor model of term structure:::::::::::::::::::::::::::::::::::::::::::::::::::3
at each point in time. A realistic model should be able to project the term structure for
ten 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 each
term structure point at each node of the tree, then we need to be able to interpolate, where
necessary, to obtain requiredinterest rates or bond prices. As inthe no-arbitrage models of
HL, HJM, HW, BDT, and BK, we (cid:12)rst build the risk-neutral or martingale distribution of
the short-term interest rate, since other maturity rates and bond prices can be computed
from the short-term rate.
The second and crucial requirement is that the interest rate process be arbitrage-free. In
the context of term structure models, the no-arbitrage requirement, in e(cid:11)ect, means that
theone-periodforwardpriceofabondofanymaturityistheexpectedvalue,undertherisk-
neutralmeasure, oftheone-period-aheadspotpriceofthebond. SinceHL,thisrequirement
has been well-understood, within the context of single factor models, and is a property
satis(cid:12)ed by the HW, BDT, and BK models. However, the requirement is more demanding
in the two-factor setting as shown by HJM, Du(cid:14)e and Kan (1993) and by Stapleton and
Subrahmanyam (1997). In a two-factor model in which the factors are themselves interest
rates, the no-arbitrage condition restricts the behavior 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 computation of bond prices at a node by taking simple
expectations of subsequent bond prices under the risk-neutral measure.
The third requirement is that the term structure model should be consistent with the
current term structure and with the term structure of volatilities implicit in the price of
European-style interest rate caps and (cid:13)oors. Models that are consistent with the current
termstructurehavebeencommonintheliteraturesincetheworkofHoandLee(1986)(HL).
For example, the HJM, HW, BDT and BK models are all of this type. The second part of
therequirementisrathermoredi(cid:14)culttoaccommodate, sinceifvolatilitiesarenotconstant
over time,thetreeofratesmaybenon-recombiningasinsomeimplementationsoftheHJM
model, leading to an explosion in the number of states. The HW implementation of a two-
factor model, in Hulland White (1994), speci(cid:12)callyexcludes time dependent volatility. BK
on the other hand accommodate both time varying volatility and mean reversion of the
short rate withina one-factor modelby varying the meaning of the time steps inthe model.
This procedure is di(cid:14)cult to extend to a two-factor case.
At a computational level, itis necessary that the state space of the modeldoes not explode,
producing so many states that the computations become infeasible. Even in a single-factor
model, this fourth requirement means that the tree of interest rates or bond prices must
recombine. This is a property of the binomial models of HL, BDT, and BK, and also
of the trinomial model of HW. A number of computational methods have been suggested
A two-factor model of term structure:::::::::::::::::::::::::::::::::::::::::::::::::::4
to guarantee this property, including the use of di(cid:11)erent time steps and state-dependent
probabilities. Inthecontext ofatwo-factor model,therequirementiseven moreimportant.
2
Inourbivariate-binomialmodelwerequirethatthenumberofstatesisnomorethan(n+1) ,
after n time steps. This is the bivariate analog 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 (cid:12)fth require-
mentforourtwo-factormodelisthattheinterestratesarelognormallydistributed. Itiswell
known that the class of Gaussian models, where the interest rate is normally distributed,
are analytically tractable, allowing closed-form solutions for bond prices. However, apart
from admitting the signi(cid:12)cant probabilityof negative interest rates, they are not consistent
with the skewness that is considered important, at least for some currencies. Furthermore,
such a model would be inconsistent with the widely-usedBlack model to value interest rate
options, which assumes that the short-term rate is lognormally distributed. Of the models
intheliterature,theBDTandBKmodelsexplicitlyassumethattheshortrateislognormal.
The HJM andHW multi-factor modelsare general enough to allow forlognormalrates, but
at the expense of computational complexity.
In addition to these (cid:12)ve requirements, there is an overall necessity that the model be
computable for realistic scenarios, e(cid:14)ciently and with reasonable speed. In this context,
we aim to compute option prices in a matter of seconds rather than minutes. To achieve
this we need a number of modelling innovations, compared to the techniques used in prior
models. These methodological innovations are discussed in the next section.
3 Particular features of the methodology
Thedynamicsofthetermstructureofinterestratescanbemodeledintermsofoneofthree
alternative variables: zero-coupon bond prices, interest rates, or forward interest rates. If
the objective of the exercise is to price contingent claims on interest rates, it is su(cid:14)cient
to model forward rates, as demonstrated by HJM. However, there are some problems with
adopting 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 a
large number of time-steps becomes practically infeasible. Second, it is di(cid:14)cult to estimate
the volatilityinputsforthemodeldirectlyfrommarket data. Usually, theimpliedvolatility
data are obtained from the market prices of options on Euro- currency futures, caps, (cid:13)oors
and swaptions, which cannot be easily transformed into the volatility inputs required to
build the forward rates in the HJM model.
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We choose to model interest rates rather than prices because existing methodologies, intro-
duced by HSS, can be employed to approximate a process witha log-binomialprocess. One
major problem arises in modellingrates rather than prices, however, and that concerns the
no- arbitrage property. Under the risk-neutral measure, forward bond prices are related
to one-period-ahead spot 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 cumbersome. We overcome this problem by modelling interest rates de(cid:12)ned
on a bankers’ discount basis, as suggested in Stapleton and Subrahmanyam (1993) and
Stapleton and Subrahmanyam (1997). In this case, the short term interest rate is a linear
function of the price of a zero-coupon bond of the same maturity. Further, we assume that
1
the three-month rate, de(cid:12)ned on a bankers’ discount basis, is lognormally distributed.
We choose here to model interest rates, and then derive bond prices, forward prices and
forward rates, as required, from the spot rate process. We prefer to directly model the
short rate, which we interpret here as the three-month interest rate, since it is used to
determine the payo(cid:11)s on many contracts such as interest rate caps, (cid:13)oors and swaptions.
One advantage of doing so is that implied volatilities from caplet/(cid:13)oorlet (cid:13)oor prices may
be used to determine the volatility of the short-rate process, in a fairly straightforward
manner.
As in HW, and Stapleton and Subrahmanyam (1997), we model the short rate, under the
risk-neutral measure, as a two-dimensional AR process. In particular, we assume that the
logarithmoftheshortrate, followssuchaprocesswherethesecondfactor isanindependent
shock to the forward premium. The short rate itself and the premium factor each mean
revert, at di(cid:11)erent rates, allowing for quite general shifts and tilts in the term structure.
Also, in contrast to HW, we assume time dependent volatility functions for both the short
rate and the premium factor. Stapleton and Subrahmanyam (1997) explore the properties
of this model in detail. Taking conditional expectations of the process they show that the
termstructureoffuturesratesisgivenbyalog-linearmodelinanytwofuturesrates. Inthis
model, the bond prices and forward rates for all maturities can be computed by backward
induction,usingtheno-arbitrageproperty. Sincetheinterestrateprocessistherisk-neutral
distribution forward bond prices and interest rates (de(cid:12)ned as above as linear functions of
zero-bond prices) are expectations of one- period-ahead bond prices under this measure.
This property permits the rapid computation of bond prices of all maturities by backward
induction, at each point in time.
1
Notethatwe onlyassumethattheshort-terminterestrateis lognormal. Iftheshort-terminterestrate
is represented by the three-month rate, we would expect a price in the region of 0.95-0.99. It follows that
theprobability of a negative price is negligible.
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The principal computational problem is to build a tree of interest rates, which has the
property that the conditional expectation of the rate at any point depends on the rate
itself and the premium factor. A methodology available in the literature, which allows
the buildingof a multivariate tree, approximating a lognormal process with non-stationary
variances and covariances, is described in Ho, Stapleton and Subrahmanyam (1995)(HSS).
In HSS, the expectation of a variable depends on its current value, but not on the value of
a second stochastic variable. However, as we show in section 5, the methodology is easily
extendedtothismoregeneralcase. TheHSSmethodologyisitselfageneralizationtotwo or
more variables of the method advocated by Nelson and Ramaswamy (1990), who devised a
methodofbuildinga’simple’orre-combiningbinomialtreeforasinglevariable. Essentially,
the HSS method relies on (cid:12)xing the conditional probabilities on the tree to accommodate
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 model changes in
the short rate so as to re(cid:13)ect the second, premium factor. This is the key, in a two-factor
model, to maintainingtheno-arbitrage property, whileavoiding anexplosioninthe number
of states. Using our extension of the HSS methodology allows us to model the bivariate
distribution of short rate and the premium factor, with n+1 states for each variable 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
property issatis(cid:12)edand the tree isconsistent withthe given volatilitiesand mean reversion
of 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 normally
employed in these models. The tree is built around the current term structure and the
calculation proceeds by moving forward period-by-period. This is expensive in computing
time, and could become prohibitively so, in the case of multiple factors. To avoid this
problem,we devisea newdynamicmethod of implementationof the HSStree, whichallows
us to compute the multivariate tree in a matter of seconds for up to eighty periods. This
method uses the feature of HSS which allows a variation in the density of the tree over
any given time step. A forward, dynamic procedure is used whereby a two-period tree with
changing density is converted into the required multi-period tree. For example, when the
eightieth time step is computed, the program computes a two-period tree with a density
over the(cid:12)rstperiodofseventy-nineandadensityover thesecondperiodofone. Thisallows
us to compute the tree nodes and the conditional probabilities analytically, and without
recourse to iterative methods.
In practice much of the skill in building realistic models rests in deciding exactly what to
compute. Potentially, inatreecoveringeighty timesteps,wecouldcomputebondpricesfor
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between one and eighty maturities at each node of the tree. Not only is this a vast number
of bond prices, but also, most of the bond prices will not be required for the solution of
any given option valuation problem. We assume here that it will be su(cid:14)cient to compute
bond prices for maturities one year apart from each other. Intermediate maturity prices, if
required,canalwaysbecomputedbyinterpolation. Hence,inoureighty-timestepexamples,
where each time step is a quarter of a year, we compute at most twenty bond prices. This
savingreducesthenumberofcalculationsbyalmostseventy-(cid:12)ve percent. Forlargenumbers
of time steps, it can turn an almost infeasible computational task into one that can be
accomplished withina reasonable time frame. For example, with three hundred time steps,
the number of bond price calculations can be reduced from approximately twenty-seven
million 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 be
the case that the computation time is excessive for a given problem. For example, for a
Bermudan-style bond option that is exercisable every year for the (cid:12)rst six years of the
underlying bond’s twenty year life, we only require bond prices at the end of each of the
(cid:12)rst six years. One computational advantage of the HSS methodology, is that the binomial
density 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(cid:14)cient accuracy
in the computations, while the number of bond and option price calculations is minimized.
4 The Two-factor Model
As in Stapleton and Subrahmanyam (1997), we assume that, under the risk- neutral mea-
sure, the logarithm of the short-term interest rate, for loans of maturity m, follows the
process
(cid:0)
d lnr =[(cid:18)r(t) alnr+ln(cid:25)]dt+(cid:27)r(t)dz1 (1)
where
(cid:0)
d ln(cid:25) =[(cid:18)(cid:25)(t) bln(cid:25)]dt+(cid:27)(cid:25)(t)dz2
In the above equations d lnr is the change in the logarithm of the short rate, (cid:18)r(t) is a
time-dependent constant term that determines the mean, a is the speed at which the short
rate mean reverts, (cid:25) is the forward premium factor and (cid:27)r(t) is the instantaneous volatility
A two-factor model of term structure:::::::::::::::::::::::::::::::::::::::::::::::::::8
of the short rate. The forward premium factor itself follows a di(cid:11)usion process with mean
(cid:18)pi, mean reversion b and instantaneous volatility (cid:27)(cid:25)(t). Although this structure is broadly
similarto HW, note that we do not restrict the volatilities: (cid:27)r and (cid:27)(cid:25) to be constant. Also,
(cid:0)
we assume that rt =(1 Bt)=m, where m isa (cid:12)xedmaturity of the short rate and Bt is the
priceof a m-year, zero-coupon bondat timet. dz1 anddz2 are standardBrownianmotions.
2
In discrete form, equation (1) can be written
(cid:0) (cid:0) (cid:0) (cid:0)
lnrt+1 (cid:22)t+1 =(lnrt (cid:22)t)(1 a)+ln(cid:25)t (cid:22)(cid:25);t+"t+1 (3)
where
(cid:0) (cid:0) (cid:0)
ln(cid:25)t (cid:22)(cid:25);t =(ln(cid:25)t(cid:0)1 (cid:22)(cid:25);t(cid:0)1)(1 b)+(cid:23)t;
and (cid:22)t = E(lnrt) is the unconditional expectation of lnrt, and (cid:22)(cid:25);t = E(ln(cid:25)t) under the
risk-neutral measure. In equation (3), "t and (cid:23)t are mean-zero, independent, normally
distributed shocks. Since the short-term interest rate is de(cid:12)ned on a bankers’ discount
basis, the interest rate at time t is
(cid:0)
rt =(1 Bt;t+1)=m; (4)
whereBt;t+1 isthevalueofaone-periodzero-couponbondandmisthelengthofoneperiod
measured in years.
4.1 Mean and Volatility Inputs for the Model
One important requirementof any (cid:12)nancialmodelis that the parameters shouldbe observ-
able, or at least capable of being estimated, from market data. In the case of the model
in equation (2) the parameters are the expected values of the short-term interest rate, (cid:22)t
2
Indiscrete form equation (1) is
(cid:0) (cid:0)
lnrt+1 lnrt=(cid:18)r(t) alnrt+ln(cid:25)t+"t+1; (2)
where
ln(cid:25)t(cid:0)ln(cid:25)t(cid:0)1=(cid:18)(cid:25)(t)(cid:0)bln(cid:25)t(cid:0)1+(cid:23)t;
Taking expectations andsubtracting from equation (2)immediatelyyieldsequation (3).