The jump component of the volatility structure of interest rate futures markets: An international comparison

THE JUMP COMPONENT

OF THE VOLATILITY

STRUCTURE OF INTEREST

RATE FUTURES MARKETS:AN INTERNATIONAL

COMPARISON

CARL CHIARELLATHUY-DUONG TO*

We propose a generalization of the Shirakawa (1991) model to capture thejump component in fixed-income markets. The model is formulated underthe Heath, Jarrow, & Morton (1992) framework, and allows the presence ofa Wiener noise and a finite number of Poisson noises, each associated witha time deterministic volatility function. We derive the evolution of thefutures price and use this evolution to estimate the model parameters viathe likelihood transformation technique of Duan (1994). We apply themethod to the short-term futures contracts traded on CME, SFE, LIFFE,

The authors would like to acknowledge the helpful comments from participants and discussants atthe 2002 Quantitative Method in Finance Conference, the 15th Australasian Finance, and BankingConference, and the 13th Annual Asia–Pacific Futures Research Symposium. All remaining errorsare our own responsibility.*Correspondence author, School of Finance and Economics, University of Technology, Sydney, 1-59Quay Street, Broadway NSW 2007, Australia; e-mail: [email protected]

Received February 2003; Accepted June 2003

� Carl Chiarella is a professor in the School of Finance and Economics at the Universityof Technology, Sydney in Sydney, Australia.

� Thuy-Duong To is a Ph.D. candidate at the School of Finance and Economics at theUniversity of Technology, Sydney in Sydney, Australia.

The Journal of Futures Markets, Vol. 23, No. 12, 1125–1158 (2003) © 2003 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.10105

1126 Chiarella and To

and TIFFE, and find that each market is characterized by very differentbehavior. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1125–1158, 2003

INTRODUCTION

The financial market is inevitably affected by information arrivals. Asargued in Merton (1976) in relation to the jump-diffusion stock marketmodel, the release of routine trading information will result in smoothchanges in the underlying stock return process, whereas bursts of infor-mation are often reflected in price behavior as discontinuous jumps. Theexistence of jump components has been documented in the literature forthe stock market and the exchange rate market, for example, in Ball andTorous (1983, 1985), Jorion (1988), and recently in Bates (1996, 2000),Jiang (1999), and Pan (2002).

The fixed-income market is equally substantially impacted by infor-mation surprises. As cited in Das (2002), a number of researchers havefound that economic news announcements and other releases of infor-mation have an impact on the Treasury bond market. Such findingsinclude those of Hardouvelis (1988), Dwyer and Hafer (1989), and,more recently, Balduzzi, et al. (1998) and Green (1998). Therefore,jump-diffusion models for interest rates are proposed and analyzed inBabbs and Webber (1995), Naik and Lee (1995), El-Jahel, et al. (1997),Piazzesi (1999), and Das (2002), albeit the specification of the jumpmodels are different. The general models of Chacko (1998), Chacko andDas (2002), and Singleton (2001) also contain jumps.

The above-mentioned works have extensively focused on the spotrate of interest. In this article, we study jump-diffusion interest ratemodels under the general framework of Heath, Jarrow, & Morton (1992)(hereafter HJM), which starts with specification of the dynamics of theinstantaneous forward rate, under which the spot rate is one particularrate, i.e., the instantaneous forward rate with zero maturity. The HJMmodel is an arbitrage-free model that matches the current term structureby construction, and relies on no assumption about investor preferences.The model offers a parsimonious representation of the market dynamicsand requires only specification of the form of the forward rate volatilityfunction.

The HJM model in the form of a diffusion process is difficult to esti-mate due to its (in general) non-Markovian nature. When jumps areintroduced into the model, the degree of complexity is magnified. In thisarticle, we only consider time-deterministic diffusion volatility processes.Unlike previous studies on jump diffusion of the spot rate of interest,

Jump Component—Interest Rate Futures Markets 1127

1As far as we are aware, research into jump processes in interest rate models so far has exclusivelyfocused on bond and option pricing rather than futures contracts.2Björk and Landén (2002) start with a marked point process for the instantaneous forward rate, andassert a marked point process for the futures prices (which is obvious). However, they do not estab-lish a link between the two sets of parameters.

where the jump size is assumed to be drawn from a continuous distribu-tion, we use multiple jump processes, each of which is associated with aconstant jump value scaled by a time-deterministic jump volatility. Theadvantage of this approach is that if the jump size is distributed with aninfinite number of possible realizations (as in the case of a continuousdistribution such as the normal distribution), under the HJM frame-work, the market will not be complete (see Björk, et al., 1997, for adetailed discussion of this issue), and therefore contingent claim pricesare not uniquely defined. Our approach is based on Shirakawa’s (1991)extension of the HJM model, in which a Wiener noise is generalizedinto a Wiener-Poisson noise, with a finite number of possible jumps.Shirakawa (1991) assumes the existence of a sufficient number of bondsto hedge away all of the jump risks, and so obtains a unique arbitrage-free pricing measure. Our specification here is slightly more general inthat it allows the generalized noise to be maturity dependent.

Bhar et al. (2002) demonstrate that attempts to estimate HJM mod-els directly from the instantaneous forward rate specification using theshort-term futures yield as a proxy would result in nonnegligible estima-tion bias. Therefore, they first derive the evolution of the futures pricesby treating the futures contract as a derivative instrument written on theinstantaneous forward rate. In this article we extend the approach ofBhar et al. (2002) to the HJM model under jump-diffusion and showthat if the underlying market variable follows a jump-diffusion process,then the futures prices should also experience jump components. Wethen use this futures price evolution to estimate our model’s parametersusing Duan’s (1994) likelihood transformation method. Under our spec-ification, the likelihood is well defined; details will be discussed in latersections of the article.

The theoretical contribution of the article is, therefore, twofold.The first is to derive the evolution of futures prices.1 This evolution canthen be used as an input for the Björk and Landén (2002) framework todetermine the evolution of options on futures.2 The second contributionis to utilize this evolution to obtain a method to estimate the parametersfor all HJM models with time-deterministic diffusion and jump volatili-ties, under the finite jump assumption. This estimation method does notrely on the Markovianization of the system comprising the spot rate of


interest, and equivalently the bond price, which may not always bepossible.

We then estimate the jump-diffusion HJM model for four majorfutures markets around the world, namely the Chicago MercantileExchange (CME), the London International Financial and FuturesExchange (LIFFE), the Tokyo International Financial Futures Exchange(TIFFE), and the Sydney Futures Exchange (SFE). The empirical analy-sis not only supports the existence of jump components in the CMEfutures market, as predicted by previous studies of jumps utilizing U.S.data, but also finds that the jump component is an important feature ofthe LIFFE market and the TIFFE market in its very low interest periodof 1996–2001. However, the mechanism by which a surprise in informa-tion impacts on markets differs among the exchanges. The shocks haveequal impacts along the yield curve in the CME, whereas they die outgradually along the yield curve in the LIFFE and TIFFE. Neither thesmall SFE market nor the TIFFE during 1989–1994 experience jumpcomponents; nevertheless, the Wiener volatility in the TIFFE market ismuch higher.

The rest of the article is organized as follows. The next section dis-cusses the class of HJM-jump models with which we are working. Theevolution of futures prices are derived in section following this. Thisevolution is subsequently used to estimate the model’s parameters viathe likelihood transformation method proposed by Duan (1994), as pre-sented in the Likelihood Transformation Technique section. Data andmodels considered are described in the section after that. We thendiscuss the empirical finding, and finally, we conclude the article. Alltechnical details are relegated to the appendices.

MODEL FRAMEWORK

Consider a financial market characterized by a probability space (�,F,P).Assume that the probability space carries a standard Wiener process W,and M Poisson processes respectively associated withintensities We further assume that the Poissonprocesses are independent of each other and of the Wiener process.

The dynamic evolution of the instantaneous T-maturity forward ratef(t, T) (for ) is assumed to be governed by the stochasticdifferential equation

(1)df(t, T) � a(t, T) dt � s(t, T) dW(t) � aM

m�1dm(t, T)gm dNm(t)

t � T � ��

l1(t), l2(t), . . . , lM(t).N1, N2, . . . , NM,


3The reason we impose a finite marked space is to ensure market completeness. Details will bediscussed later in the section.4Here, is the drift under the Shirakawa (1991) model.5For some results on Markovianization of the HJM models under jump-diffusion, see Chiarella andNikitopoulos Sklibosios (2002) and Chiarella and Kwon (2002).

a*(t, T)

where a(t, T) and the s(t, T) are, respectively, the drift and the Wienerdiffusion coefficient for the instantaneous forward rate to maturity T,whereas is the constant jump size that occurs at theNm Poisson jump time. The corresponding jump volatility function isdm(t, T).

This model extends the HJM framework by allowing the evolution ofthe instantaneous forward rate to include discrete jump components inaddition to the diffusion process. Each of the noise terms is then scaledby the corresponding volatility functions. The process is, therefore, aspecial case of the marked point process introduced by Björk et al.(1997), where the marked space is finite.3 If we recall one of the veryfirst models that include jump components for forward rate specifica-tion, that of Shirakawa (1991) for which the dynamics of the forwardrate are given by4

(2)

we can see that our specification in Equation (1) allows for the“generalized noise term” to be maturity dependent. Instead of generaliz-ing the Wiener noise increment dW(t) into a maturity independentWiener-Poisson noise increment

we generalize the Wiener noise increment into the maturity dependentWiener-Poisson noise increment

This generalization has the advantage of giving us greater flexibility tocapture empirically the jump components. It has the disadvantage ofmaking the Markovianization of the associated spot rate or bond pricedifficult, if not impossible.5 However, we do not suffer from this disad-vantage because our work is based on the futures prices, and our manip-ulations to obtain an expression for futures prices do not requireMarkovianization of the system dynamics.

dW(t) � aM

m�1

dm(t, T)

s(t, T) gm(dNm(t) � lm dt)

dW(t) � aM

m�1gm(dNm(t) � lm dt)

df(t, T) � a*(t, T) dt � s(t, T) cdW(t) � aM

m�1gm(dNm(t) � lm dt) d

gm(m � 1, . . . , M)


Under the historical martingale measure P, the instantaneousT-maturity forward rate Equation (1) evolves according to the stochasticintegral equation

(3)

from which the evolution of the instantaneous spot rate can be derivedby setting T � t, thus

By application of the generalized Ito’s lemma (see Appendix A), thedynamics for P(t, T), the price at time t of a bond maturing at time T,may be expressed as

where

and

The specification Equation (1) limits the jump space to be finite.Björk et al. (1997) then prove (in theorem 5.6, p. 235) that the market iscomplete (i.e., every contingent claim can be replicated by a self-financingportfolio) if the following two conditions hold: (1) for each t, the functionss(t, T) and dm(t, T) are analytic in the T-variable, and (2) for each t, thefunctions

(4)S(t, T),�eDm(t,T) � 1,�m � 1, . . . , M

Dm(t, T) � ��T

t

dm(t, s)gm ds, for m � 1, . . . , M

S(t, T) � ��T

t

s(t, s) ds

A(t, T) � ��T

t

a(t, s) ds

� aM

m�1P(t�, T)[eDm(t,T) � 1] dNm(t)

dP(t, T) � P(t, T) c r(t) � A(t, T) �12

S(t, T)2 d dt � P(t, T)S(t, T) dW(t)

� aM

m�1�

t

0

dm(u, t)gm dNm(u)

r(t) � f(0, t) � �t

0

a(u, t) du � �t

0

s(u, t) dW(u)

� aM

m�1�

t

0

dm(u, T) gm dNm(u)

f(t, T) � f(0, T) � �t

0 a(u, T) du � �

t

0 s(u, T) dW(u)


of the argument T are linearly independent. Furthermore, for each twe can choose the distinct bond maturities arbitrarily, apart from a finitenumber of values on every compact interval. If all functions above aredeterministic and analytic also in the t-variable, then the maturities canbe chosen to be the same for every t.

The completeness of the market implies that the martingale meas-ure is unique (see Corollary 4.8 of Björk et al., 1997, p. 229). Under thisequivalent martingale measure P, the bond dynamics are given by

where is a standard Wiener process and Nm is a Poisson process associ-ated with intensity for which can be interpreted as themarket price of jump risk (see Shirakawa, 1991). The Poisson processesNm remain independent of each other and of the Wiener process .

Björk et al. (1997) (Proposition 3.13, p. 222) show that to eliminatearbitrage opportunities, the relation

(5)

must hold under P.

EVOLUTION OF FUTURES PRICESUNDER HJM WITH JUMPS

Let F(t, TF, TB) be the price at time t of a futures contract maturing attime TF(� t). The contract is written on a pure discount instrument thathas a face value of $1 and matures at time TB(� TF). Let the price of thisinstrument at time TF be P(TF, TB) and the logarithm value of the pricebe V(TF, TB).

Because futures contracts are marked to market, it is shown in Cox,Ingersoll, & Ross (1981) that the futures price is a martingale under theequivalent measure , so that

� �P~

t [exp(V(TF, TB)) ƒFt]

� �P~

t [P(TF, TB) ƒFt]

F(t, TF, TB) � �P~

t [F(TF, TF, TB) ƒFt]

P~

P~

� a~(t, T)

a(t, T) � s(t, T) �T

t

s(t, s) ds � aM

m�1dm(t, T) gmeDm(t,T) cm(t)

W~

m � 1, . . . , M,cm,W~

� aM

m�1P(t�, T)[eDm(t,T) � 1](dNm(t) � cm dt)

dP(t, T) � P(t, T) r(t) dt � P(t, T) S(t, T) dW~(t)

P~


6The reader is reminded that at time t, the integrals from 0 to t of and dNm(t) are realizedquantities.

dW�

The log-price V(TF, TB) under is given by

By an application of the stochastic Fubini theorem

Therefore,6

(6)

due to the independence of the Wiener and the Poisson processes.The final task is to carry out the expectation operations. The first

expectation in Equation (6) is a standard result. In fact, Musiela et al.(1992) have used this method to derive the futures price evolution whenthe market jump components are absent from the market. The second

� qM

m�1�P

�

t c exp a��TF

t�

TB

TF

dm(u, s) gm ds dNm(u)b d� �P

�

t c exp a��TF

t�

TB

TF

s(u, s) ds dW~(u)b d

� �t

0�

TB

TF

s(u, s) ds dW~

(u) � aM

m�1�

t

0�

TB

TF

dm(u, s)gm ds dNm(u) d� exp c��

TB

TF

f(0, s) ds � �TF

0�

TB

TF

a~(u, s) ds du

F(t, TF, TB) � �P�

t [exp(V(TF, TB)) ƒFt]

� �TF

0�

TB

TF

s(u, s)ds dW~

(u) � aM

m�1�

TF

0�

TB

TF

dm(u, s)gm ds dNm(u)

V(TF, TB) � ��TB

TF

f(0, s) ds � �TF

0�

TB

TF

a~(u, s) ds du

� �TB

TF

�TF

0

s(u, s) dW~

(u) ds � aM

m�1�

TB

TF

�TF

0

dm(u, s)gm dNm(u) ds

� ��TB

TF

f(0, s) ds � �TB

TF

�TF

0

a�(u, s) du ds

V(TF, TB) � ��TB

TF

f(TF, s) ds

P~


expectation, which involves the increments of Poisson processes, isevaluated in Appendix B. The final result is

(7)

From this stochastic integral equation for F(t, TF, TB), we can derivethe corresponding stochastic differential equation

(8)

where

We have treated the futures contract as a derivative instrument writ-ten on the forward rates to derive the evolution for the futures price.This evolution is important for two practical reasons. First, because

a(F)(t, # ) � �aM

m�1cmd

(F)m

d(F)m (t, # ) � expa��

TB

TF

dm(t, s)gm dsb � 1

s(F)(t, # ) � ��TB

TF

s(t, s) ds

dF(t, TF, TB)

F(t�, TF, TB)� a(F)(t, # ) dt � s(F)(t, # ) dW

~(t) � a

M

m�1d(F)

m (t, # ) dNm(t)

� �t

0�

TB

TF

s(u, s) ds dW~(u) � a

M

m�1�

t

0�

TB

TF

dm(u, s)gm ds dNm(u) f�a

M

m�1cm�

t

0

c exp a��TB

TF

dm(u, s)gm dsb � 1 d du

� F(0, TF, TB)exp e�12 �

t

0

a �TB

TF

s(u, s) dsb2

du

� qM

m�1exp ecm�

TF

t

c exp a��TB

TF

dm(u, s)gm dsb � 1 d du f� exp c 1

2 �TF

t

a �TB

TF

s(u, s) dsb2

du d� �

t

0�

TB

TF

s(u, s) ds dW~(u) � a

M

m�1�

t

0�

TB

TF

dm(u, s)gm ds dNm(u) d� exp c��

TB

TF

f(0, s) ds � �TF

0�

TB

TF

a~(u, s) ds du

F(t, TF, TB)


7The quoted value in the market is not exactly F(t, TF, TB), but some value related to F(t, TF, TB),depending on the convention of each exchange.

futures prices are quoted in the market,7 the evolution of F(t, TF, TB) canbe used to estimate the parameters of the forward rate specification,which is our main focus here. Second, this evolution may be used as aninput for the Björk and Landén (2002) framework to derive the prices ofoptions on futures; such applications, however, are beyond the scopeof the present article.

THE LIKELIHOOD TRANSFORMATIONTECHNIQUE

The evolution of F(t, TF, TB) involves the parameters of the forward ratespecification. To use this evolution to estimate the model parameters,the evolution first needs to be expressed in the physical (historical)measure. Let f be the market price of the Wiener risk, the evolution ofthe futures prices becomes

(9)

where here the intensity of N(m) is the intensity under the historicalmeasure, which is lm.

To proceed with the estimation, one possible approach is to dis-cretize the stochastic differential Equation (9). However, the usual Eulerdiscretization may result in inconsistent estimators (see Lo, 1988, fordetailed discussion). Therefore, we will derive the likelihood functionfor futures prices via a state variable, under the likelihood transforma-tion technique of Duan (1994).

The method of Duan only requires one derivative instrument to esti-mate the underlying model parameters. However, in our framework, foreach underlying instrument, there are a number of futures contracts(with different maturities) traded in the market. To include all of thesefutures series in the estimation, we follow the approach of Bhar et al.(2002), who set up and derive the full information likelihood functionfor the quoted futures prices in the case of pure Wiener processes. Areview of the method will be presented in the first half of this section forcompleteness, whereas the second half of the section derives the likeli-hood function needed to handle the case of Wiener-Poisson noise.

� aM

m�1d(F)

m (t, # ) dNm(t)

dF(t, TF, TB)

F(t�, TF, TB)� (a(F)(t, # ) � fs(F) (t, # )) dt � s(F)(t, # ) dW(t)


State Variables

Assume that for each underlying pure-discount interest rate instrument,there are K futures contracts maturing at times The(observable) quoted futures price in the market is whichis linked with via a function h, i.e.,

(10)

The link between F and G depends on the quoting convention of eachexchange. For example, Eurodollar futures prices traded on the ChicagoMercantile Exchange are related to quoted prices according to

(11)

The quoting convention of other exchanges that are considered in theempirical part of this article are given in Appendix C. We are consideringthe case in which all of the futures contracts are written on the sameunderlying instrument, and therefore the time to maturity of the under-lying contract is constant for all

Let be a state variable defined by

(12)

From Equation (7) we have that satisfies the stochasticintegral equation

where

nm,k(u, # ) � ��TBk

TFk

dm(u, s) gm ds

bk(u, # ) � ��TBk

TFk

s(u, s) ds

mk(u, # ) � fbk(u, # ) �12

b2k(u, # ) � a

M

m�1cm[exp(nm,k(u, # )) � 1]

� �t

0

bk(u, # ) dW(u) � aM

m�1�

t

0

nm,k(u, # ) dNm(u)

X(t, TFk, TBk) � X(0, TFk, TBk) � �t

0

mk(u, # ) du

X(t, TFk, TBk)

� z(F(t, TFk, TBk))

X(t, TFk, TBk) � ln(F(t, TFk, TBk))

X(t, TFk, TBk)k � [0, K].TBk � TFk � t

� h(G(t, TFk, TBk))

F(t, TFk, TBk) � 1 � a1 �G(t, TFk, TBk)

100b t

F(t, TFk, TBk) � h(G(t, TFk, TBk))

F(t, TFk, TBk)G(t, TFk, TBk),

TFk(k � 1, 2, . . . , K).


To set up a maximum likelihood estimation procedure, we need tofind the density function for the state variable X, then transform itsuccessively into the density functions for F and G.

The Likelihood Transformation Formula

Let be the kth unobservable state variable occurring at time

Denote by xj the vector of unobservable state variables occurring attime tj, i.e., Denote by x the unobservable statevector of size at time tJ, i.e.,

x � vec(x0 x1 xJ)

where vec is the standard matrix operator that, when applied to a matrix,transforms the matrix into a vector by stacking the columns of the matrixon top of each other.

Denote the density function of X by

where is the parameter vector of interest.Suppose that a transformation � exists, which applied to X, pro-

duces a vector Z that is observable in the market, so that

where

z � vec(z0 z1 zJ)

Assume that this transformation is one-to-one for every U� ∏.

� vec ± Z01 Z11 p ZJ1

Z02 Z12 p ZJ2

o o ∞ oZ0K Z1K p ZJK

≤p

Z � �(X; U) : �K( J�1)�1S�K( J�1)�1

U� ∏

pX(x; U) � pX(x0, x1, . . . , xJ; U)

� vec ± X(t0, TF1, TB1) X(t1, TF1, TB1) p X(tJ, TF1, TB1)X(t0, TF2, TB2) X(t1, TF2, TB2) p X(tJ, TF2, TB2)

o o ∞ oX(t0, TFK, TBK) X(t1, TFK, TBK) p X(tJ, TFK, TBK)

≤p

K( J � 1) � 1xj � (Xj1, Xj2, . . . , XjK).

tj TFk( j � 0, 1, . . . , J).1, 2, . . . , K)(k �Xjk � X(tj, TFk, TBk)


8This assumption is not restrictive in the financial market context, because each financial instru-ment is usually determined by only one particular underlying variable. For example, a bond price ata time is determined by the interest rate of a particular maturity at that particular time.

Because � is one to one, there exists an inverse Applying the standard change of variable technique we obtain thedensity function for Z as

where J is the Jacobian of the transformation from X to Z, i.e.,

Duan (1994) proves that if the transformation is on an element-by-element basis, i.e., (and for all and then the first-derivative matrix is diagonal, therefore

Furthermore, if X is “joint-Markovian,” i.e.,

then upon substitution, the likelihood for Z can be compactly written as

where

and

Full Information Likelihood forQuoted Futures Prices

Before deriving the likelihood function, we introduce into the system ameasurement error, i.e., a deviation of market quoted value and the true

J(±(zj; U)) � qK

k�1

d°jk(Zjk; U)

dZjk�( j � 0, 1, . . . , J)

pZ(zj, tj 0 zj�1, tj�1; U) � pX(±j(zj), tj 0 ±j�1(zj�1), tj�1; U) � 0 J(±j(zj; U)) 0 pZ(z0, t0; U) � pX(±0(y0), t0; U) � 0 J(±0(z0; U)) 0pZ(z0, z1, . . . , zJ; U) � pZ(z0, t0; U)q

J

j�1pZ(zj, tj 0 zj�1, tj�1; U)

pX(x0, x1, . . . , xJ; U) � pX(x0, t0; U)qJ

j�1pX(xj, tj 0 xj�1, tj�1; U)

J(±(z; U)) � qJ

j�0q

K

k�1 d°jk(Zjk; U)

dZjk.

k � [1, K],8j � [0, J]Xjk � °jk(Zjk))Zjk � jk(Xjk)

J(±(z; U)) � ` 0±(z; U)0z�

`pZ(z, U) � pX(±(z; U)) � 0 J(±(z; U)) 0

��1 � ±(Z; U).


9In practice, this error volatility should be small in order and magnitude compared to the diffusionvolatility, so that any attempt to set up an arbitrage portfolio to trade on this uncertainty source willnot result in profits after transaction costs are taken into account.10Recall that

x � vec(x0 x1 xJ)

and for and k � 1, 2, . . . , K.j � 0, 1, . . . , JXjk � X(tj, TFk, TBk),

� vec ± X01 X11 p XJ1

X02 X12 p XJ2

o o ∞ oX0K X1K p XJK

≤p

model value. We will assume that, on average, this measurement errorbrings zero return to the trading of futures contracts. Therefore, a newWiener process which is independent of the process driving theuncertainty of the forward rate, is introduced into the evolution of

so Equation (9) becomes

We assume that there is no correlation between the return-measurementerror on contracts with different maturities (i.e., no correlation between

for Because there is no reason to believe the error inone particular contract is larger than in others, we use the same error-volatility value for all contracts.9

The evolution of our state variable X, incorporating this measure-ment error, is

(13)

Suppose that the process is sampled discrete points in time(not necessarily equally spaced apart). Due to the Markovian

nature of the system (i.e., the stochastic processes for forthe likelihood function for the joint observation x,10 for a

given parameter vector of interest is

pX(x0, x1, . . . , xJ; U) � pX(x0, t0; U)qJ

j�1pX(xj, tj 0 xj�1, tj�1; U)

U� ∏,k � 1, . . . , K),

X(t, TFk, TBk)t0, t1, . . . , tn

J � 1

� �t

0

bk(u, # ) dW(u) � �t

0

se dek(u) � aM

m�1�

t

0

nm,k(u, # ) dNm(u)

X(t, TFk, TBk) � X(0, TFk, TBk) � �t

0 amk(u, # ) �

12

s2eb du

se

k � 1, . . . , K).dek

� se dek(t) � aM

m�1d(F)

m (t, # ) dNm(t)

dF(t, TF, TB)

F(t�, TF, TB)� (a(F)(t, # ) � fs(F)(t, # )) dt � s(F)(t, # ) dW(t)

F(t, TFk, TBk),

ek,


11For a discussion of the unboundedness of the mixture of normal distributions, the reader isreferred to Kiefer (1978), Quandt and Ramsey (1978), Honoré (1998), chapter 22 of Hamilton(1994), and the references therein.

The specification Equation (13) implies that the transitional likeli-hood function for X will have a form of a (Poisson) mixture of normal dis-tributions. Our assumption that allows for only finite (in fact only one)jump size at each jump time, rather than jumps being drawn from a(normal) distribution, ensures that the transitional likelihood function isbounded, and therefore, well defined.11

The transitional likelihood function is

where and is a multivariate Gaussian density defined by

whose mean and variance are

and the matrix elements are defined by

(14)

(15)

In this article, rather than using the exact Poisson mixture ofGaussian densities, we use the approximate Bernoulli mixture of Gaussian

bj(k1k2)� �

tj

tj�1

bk1(u)bk2

(u) du�for k1

� k2

bj(kk) � �tj

tj�1

(b2k(u) � s2

e ) du

ajk � �tj

tj�1

amk(u) �12

s2e � a

M

m�1nmnmk(u)b du

�j � ± bj(11) bj(12) p bj(1K)

bj(21) bj(22) p bj(2K)

o o ∞ obj(K1) bj(K2) p bj(KK)

≤ aj � (aj1 aj2 p ajK)�

£(xj; aj, �j) � (2p)�K2 0 �j 0�1

2 expa�

12

(xj � xj�1 � aj)��1j (xj � xj�1 � aj)b

£¢t � tj � tj�1

� a

n1�0 a

n2�0 p a

nM�0ae�l1¢t

(l1¢t)n1

n1!b p ae�lM¢t

(lM¢t)nM

nM!b£(xj; aj, �j)

pX(xj, tj 0 xj�1, tj�1; U)


densities so as to obtain a practically and economically implementablemodel. The approximation is the result of ignoring all of the values oforder higher than (dt)2 in the transitional likelihood function. If the timeinterval is short (say 1 day) and the Poisson intensity is small, the twodensities are practically indistinguishable. The transitional likelihoodfunction for X under this Bernoulli approximation becomes

(16)

where

and the matrix elements are defined by

(17)

(18)

The log likelihood function for the state variable X, based on theobservation x is

(19)

Recall that there exists a transformation from X to F [see Equa-tion (12)] with inverse function z. It is clear that this transformation is onan element-by-element basis. Applying the transformation formula, thelikelihood function for F is

(20)

Applying for a second time the transformation from F to G, thequoted futures price in the market, with the inverse transformation func-tion [see Equation (10)] results in the log-likelihood function

(21)LG(U) � aJ

j�1ln(pF(H(Gj), tj 0 H(Gj�1), tj�1; U)) � a

J

j�1ln ` 0Hj(Gj; U)

0Gj`

H

LF(U) � aJ

j�1ln(pX(Z(Fj), tj 0 Z(Fj�1), tj�1; u)) � a

J

j�1ln ` 0Zj(Fj; U)

0Fj`

LX(U) � aJ

j�1ln apX(xj, tj 0 xj�1, tj�1; U)b

ajk(m) � �tj

tj � 1

amk(u) �12

s2e � nmk(u)b du�for m � 0

ajk(0) � �tj

tj�1

amk(u) �12

s2eb

du

aj(m) � (aj1(m) aj2(m) p ajK(m) )�, for m � 0, 1, . . . , M

� aM

m�1lm¢t £(xj; aj(m), �j) � a1 � a

M

m�1lm¢tb £(xj; aj(0), �j)

pX(xj, tj 0 xj�1, tj�1; U)


Finally, it should be noted that futures prices at different maturitiesare less than perfectly correlated with each other under a stochastic set-ting. Therefore, the full information maximum likelihood method afore-described, which is applied to pooled time series and cross-sectionaldata, will allow us to exploit the full information content along the yieldcurve.

MODELS AND DATA

Models

The HJM model given by Equation (1) is determined once the volatilityfunctions associated with the diffusion and jump processes are specified.

For the diffusion component, we choose the “hump-volatility” curvespecification,

which nests the exponential model (Hull & White, 1990, extendedVasicek model), the linear absolute model considered in Amin andMorton (1994), and the absolute (constant) model of Ho and Lee (1986).

For the jump component, we distinguish between two types of infor-mation surprise. The first one is of the macroeconomic type that willaffect the whole yield curve more or less equally. The second one is lessprominent and will affect some sections of the yield curve more than theothers. Therefore, we choose an exponential form for one jump volatilityand a unit volatility for the other jumps to scale up the two assumed con-stant jump intensities under the risk-neutral measure. Mathematicallymore precisely, we have set M � 2, c1 and c2 constant, d1(t, T) �

exp(�k1(T � t)), and d2(t, T) � 1.Under these specifications, both the diffusion volatility and the

jump volatility are functions of time-to-maturity T � t, but not functionsof a specific calender time t. Due to the independence of andthe the condition Equation (4) for market completeness anduniqueness of the equivalent martingale measure holds. Furthermore, atany time t, the investment opportunities in the market are spanned byM � 1 basic bonds with different maturities.

To find the analytical likelihood function for the quoted futuresprices, we need to perform the integrations in Equations (17), (18), (14),and (15), i.e., find each of the elements in the drift and covariance matrixand substitute them into the final likelihood function Equations (16), (19),(20), and (21). Details of these calculations can be found in Appendix D.

e�kmT,T e�kT

s(t, T) � [s0 � s1(T � t)] exp(�k(T � t))


Data

We carry out our empirical investigation using 3-month futures contractstraded on four major exchanges from different parts of the world. Wechoose the Eurodollar futures contract traded on the ChicagoMercantile Exchange (CME), the Sterling futures contract traded on theLondon International Financial and Futures Exchange (LIFFE), the90-day Bank Accepted Bills futures contract traded on the SydneyFutures Exchange (SFE), and the Euroyen futures contract traded onthe Tokyo International Financial Futures Exchange (TIFFE).

All of the data have been obtained from the Datastream™ database.The sample period spans 14 years, from January 1988 to December2001, except for the Euroyen futures contract traded on TIFFE, theavailable data are from September 1989. We choose our sample periodto start from 1988, because including 1987 in the sample might bias theresult in favor of finding jumps.

Because a futures contract has a relatively short life, we roll overfutures contracts along the 14-year sample period. Take the Eurodollarfutures contract on the CME as an example. Sample observations for theperiod January 2000 to December 2000 consist of six different contracts:March 2001, December 2001, September 2002, June 2003, March2004, and December 2004; whereas the sample observations for theperiod January 2001 to December 2001 consist of a different set of sixcontracts: March 2002, December 2002, September 2003, June 2004,March 2005, and December 2005 (see Fig. 1). It should be noted thatthe sets of contracts used are spaced three quarters apart to ensure suf-ficient variation in the futures prices, and the contracts are selected untilthere is no longer sufficient liquidity.

FIGURE 1

Research design—futures contracts used in sample period January 2001–December 2001.


12See Doornik (1996).13We note that our computer program seems to have difficulty finding the global maxima for l1.Even though the significance of the estimate does not change, the point estimate tends to convergeto a local maxima, depending on the starting values.

Because each exchange has a different liquidity level, the number ofcontracts used at each point in time will be different, and the time spanof each set of contracts to be rolled over will also be different. TheEurodollar contract traded on the CME is the most liquid one, andtherefore we use on average five different contracts in our estimation ateach point of time. In the SFE and LIFFE case, we use on average twocontracts, and only one contract is used in the case of the TIFFE.

EMPIRICAL RESULTS

All of our empirical work was carried out using Ox™, a matrix-orientedprogramming language.12

The Estimated Models

We ran the estimation for the 3-month futures contracts traded on thefour exchanges. In the case of the TIFFE, we broke the full sampleperiod into two subsamples, one from 1989–1994, the other from1996–2001. In 1995, in the Japanese market, there was a sharp fall ofthe interest rate from 2.5 to 0.5%. From 1996 to 2001, the level of inter-est rate remained very low, at an average level of 0.4%.

The empirical results suggested that we did not need two jumpprocesses to capture the effect of surprises on the market. When twojump processes were present together, the estimated values of theparameters associated with the jump components became insignificant.Therefore, we reestimated our model with only one jump process. Thevolatility associated with this jump takes the form of an exponentialfunction, which nests the unity case. Depending on the value of thedecay factor of the exponential function (i.e., significantly different fromzero or not), we can distinguish between the two mechanisms of spread-ing the effect of surprise shocks.

The model estimated parameters are given in Table I. The symbol *next to the estimated parameters indicates a significant value at 95%level of confidence. Table II presents the estimation results for moreparsimonious models where all of the insignificant parameters have beentaken out.13


TABLE I

Estimated Parameters for 1 Wiener–1 Poisson Model

CME SFE LIFFE TIFFE TIFFEParameter 1988–2001 1988–2001 1988–2001 1989–1994 1996–2001

0.0096* 0.0107* 0.0148* 0.0199* 0.0447*[0.0005] [0.0012] [0.0014] [0.0057] [0.0214]

0.0041* 0.0088* 2.7 � 10�8 �0.0902* �0.1027*[0.0006] [0.0034] [1.4 � 10�5] [0.0215] [0.0513]

0.2387* 0.3157* 0.0691 2.7250* 2.6559*[0.0198] [0.1022] [0.0523] [0.3405] [0.6533]

0.0049* 0.0016 �0.0118 �0.0002 0.0001*[0.0010] [0.0055] [0.0083] [0.0008] [1.1 � 10�5]

0.3445 0.3160* 5.5568* �2.4451 0.2338[0.3781] [0.0980] [1.7640] [3.4691] [0.4401]

3.9982* 0.9999 2.0742* 0.0829 4.0003*[1.9308] [11.819] [1.0587] [41.973] [0.1692]

2.0084* 0.6419 6.5594* 17.897 1.8607*[0.9628] [0.4588] [0.8519] [25.429] [0.1445]

0.0009* 0.0009* 0.0010* 0.0015* 0.0008*[1.7 � 10�5] [2.4 � 10�5] [6.8 � 10�5] [0.0001] [7.8 � 10�5]

Note. This table reports the estimated parameter values for the model with humped diffusion volatility andexponential jump volatility. The corresponding robust asymptotic standard errors are given in square parenthe-sis under the estimated values. A * next to the estimated value indicates a significance level lower than 5%.

se

c1

l1

k1

g1

k

s1

s0

In the CME, SFE, and TIFFE markets, the diffusion volatility wasfound to be humped-shaped, whereas in the LIFFE we did not find sup-porting evidence for humped or exponential shapes. The instantaneousvolatilities of the short rate are similar in CME and SFE, at about 1%,which is smaller than the 1.4% level in LIFFE. The Japanese market hasa much higher volatility, which stays at 2.1% during 1989–1994, andsoars toward a double level of 4.5% during the 1996–2001 period, whenthe interest rate remains extremely low at near 0.5% level. However, theeffects on the volatilities of a Wiener shock in the Japanese market dieout (along the yield curve) much quicker than that in the American andAustralian market. The decay factor k estimated in TIFFE is 10 timeslarger than those estimated in CME and SFE.

The three parameters g1, k1, and l1 identify the jump component.The magnitude of the jump volatility is shown via g1, and the jump inten-sity is l1, whereas k1 indicates the mechanism by which the effect ofjump components are spread across the maturities.

In the CME and LIFFE markets, both and are significantlydifferent from zero, supporting the presence of a jump noise component.

g1l1


TABLE II

Estimated Parameters for Restricted Model

CME SFE LIFFE TIFFE TIFFE1988–2001 1988–2001 1988–2001 1989–1994 1996–2001

0.0097* 0.0107* 0.0138* 0.0213* 0.0447*[0.0005] [0.0012] [0.0006] [0.0053] [0.0214]

0.0040* 0.0088* — �0.0943* �0.1028*[0.0006] [0.0034] — [0.0227] [0.0513]

0.2361* 0.3170* — 2.7695* 2.6560*[0.0210] [0.1012] — [0.3523] [0.6535]

�0.0071* — �0.0172* — 0.0001*[0.0002] — [0.0074] — [3.2 � 10�5]

— — 6.3038* — —— — [1.7454] — —

4.0000* — 3.9950* — 2.5000*[1.4913] — [1.8236] — [0.3211]

2.0010 — 5.4030* — 1.9788*[2.0197] — [2.3135] — [0.5197]

0.0009* 0.0009* 0.0010* 0.0015* 0.0008*[1.8 � 10�5] [2.4 � 10�5] [7.2 � 10�5] [0.0001] [7.8 � 10�5]

Likelihood 0.0004 — 0.0059 — 0.0000ratio stat.

Note. This table reports the estimated parameter values for the restricted jump-diffusion model. The robustasymptotic standard errors are given in square parenthesis. A * next to the estimated parameter value indicatesa significance level lower than 5%. The likelihood ratio statistics are also reported.

se

c1

l1

k1

g1

k

s1

s0

The probabilities of jump arrivals are similar in the two market, at about1.6% per day. However, the average value of the volatility shock is higherin the U.K market, and the mechanism by which a surprise in infor-mation impacts on the U.K market is different from that of the U.S.market. The insignificant value of and the likelihood ratio test for therestriction of k1 � 0 in Table II, suggest that a surprise in informationwill affect the whole of the U.S yield curve more or less equally. On theother hand, in the U.K market, a surprise in information has a largeimpact on the volatility of the short-term section of the yield curve, andthis impact dies out gradually along the yield curve toward the longermaturity section. The rate of decrease is reflected in the decay factor .Borrowing a term from the time series analysis literature, as Raj et al.(1997) have done, we define the half life of the shock as the difference inthe time to maturity of two forward rates, so that one undergoes doublethe effect of the other from the same shock, i.e.,

g1 exp�k1(T1�t) � 2g1exp�k1(T2�t)

k1

k1,


from which

As calculated above, the two forward rates need to be about 1.5 monthsapart in maturity for the size of the impact (from the same shock) on thevolatility level to decrease by a half.

In the Australian market we find that both and are not signifi-cantly different from zero, suggesting the absence of the jump compo-nent. This result should not be surprising. The Australian market is arelatively small one, which is affected substantially by what happens inthe U.S market. This market usually takes its cue from the UnitedStates, and therefore does not experience surprise shocks. The daily rou-tine information arrivals that are unique to the Australian market areadequately picked up in the Wiener volatility. Note that we cannot usethe likelihood ratio test here to check the fitness of the reduced modeldue to the problem of nonidentification of the null hypothesis (either g1

or c1 equal to zero will result in a no-jump component), and we did relyon the significance of estimated parameters in the jump-diffusion speci-fication to draw conclusions about the markets.

The Japanese market has been divided into two sample periodsbased on the level of the interest rate. The market behavior for eachperiod in terms of the volatility of the interest rate is also different. Wedo not find supporting evidence for the jump component in the first sam-ple period, when the level of the interest rate is still high. On the otherhand, this jump component is important during the second sample peri-od, when the level of the interest rate is extremely low. The informationsurprise shocks affect the whole yield curve equally, and average at onlyone basis point. The jump arrival rate is about 1% per day, which issmaller than those of the U.S and the U.K market. The second period is,therefore, characterized by a very low interest rate level (0.5%), hugeWiener volatility, which arises from routine trading information, and asmall information surprise component.

Model Validity Check

We test for model validity by the score method. The score of the jthobservation is defined as the derivative of the natural logarithm of theconditional likelihood with respect to the parameter vector U, namely

(22)hj(U) �0 ln p(Gj 0 Gj�1; U)

0U

g1l1

T2 � T1 �ln 2k1

� 0.11 years


If the model is correctly specified, the score hj(U0) (evaluated at thetrue parameter value U0) should be impossible to forecast based onany past information, such as elements of the lagged score hj�k(U0)(k � 1).

The test for serial correlation in the scores is proposed by White(1987), using the conditional moment tests of Newey (1985) andTauchen (1985). Hamilton (1996) provides an excellent summary of themethod, whose notation we use here.

First we collect in an (l � 1) vector cj(U) those elements of theH � H matrix [hj(U)][hj�1(U)]� that we want to confirm have a zero meanwhen evaluated at u0. If the model is correctly specified, then

where denotes the (2,2) subblock of the inverse of the partitionedmatrix

In our first test for the model’s validity, we gathered all of the ele-ments of the matrix [hj(U)][hj�1(U)]� in the vector cj(U). The test rejectedthe assumption of no serial correlation in the score in all of the cases. Itis clear that the model proposed here is not good enough to capture themovements of the market.

To further investigate the mis-specification, we redid the test foreach element of the matrix [hj(U)][hj�1(U)]� separately. The p-values foreach test are reported in Table III. The parameter set can be divided intotwo subsets, one consisting of s0, s1, and k, which are associated withthe Wiener noise, and the other consisting of k1, g1, l1, and c1, whichare associated with the Poisson noise. It can be seen that the problemof serial correlation is more significant between the elements in eachsubset than between the elements across subsets.

The score test clearly indicates the need for a better model specifi-cation. Some suggestions include (1) to incorporate level dependent dif-fusion volatility, (2) to utilize a more elaborate jump volatility structure,and (3) to consider multifactor specifications. We leave analysis of thesesuggestions for future research.

A � (1�J) # ° aj[hj(U)][hj(U)]� aj

[hj(U)][cj(U)]�

aj[cj(U)][hj(U)]� aj

[cj(U)][cj(U)]�¢

A22

c J�1�2a

J

j�1ct(U) d � # A22

# c J�1�2a

J

j�1ct(U) d Sd x2(l)


TABLE III

p-Value for the Score Tests(a) CME Case—Humped Diffusion Volatility—Constant Jump Volatility

Lagged Variables

0 1 1 1 1

0.007 0.004 0.004 0.026 0.025 0.026 0.5310.012 0.002 0.002 0.002 0.002 0.002 0.1050.012 0.002 0.002 0.002 0.002 0.002 0.175

0.179 0.096 0.142 0.000 0.000 0.000 0.8130.176 0.096 0.140 0.000 0.000 0.000 0.8040.179 0.096 0.142 0.000 0.000 0.000 0.812

0.034 0.005 0.012 0.058 0.059 0.058 0.002se

c1

l1

g1

k

s1

s0

seclgkss

(b) LIFFE Case—Constant Diffusion Volatility—Exponential Jump Volatility

Lagged Variables

0 1 1 1 1

0.000 0.034 0.014 0.015 0.014 0.010

0.395 0.000 0.000 0.000 0.000 0.2240.419 0.000 0.000 0.000 0.000 0.2180.417 0.000 0.000 n.a. 0.000 0.2180.419 0.000 0.000 0.000 0.000 0.218

0.222 0.209 0.170 0.170 0.170 0.198se

c1

l1

g1

k1

s0

seclgks

(c) SFE Case—Humped Diffusion Volatility—No Jump Component

Lagged Variables

0 1

0.005 0.135 0.781 0.0170.037 0.001 0.001 0.1930.327 0.001 0.001 0.692

0.077 0.022 0.078 0.000se

k

s1

s0

sekss

(d) TIFFE Case—First Sample PeriodHumped Diffusion Volatility—No Jump Component

Lagged Variables

0 1

0.068 0.038 0.012 0.0590.046 0.021 0.008 0.0250.025 0.010 0.005 0.005

0.078 0.033 0.007 0.024se

k

s1

s0

sekss

(Continued)


CONCLUSION

Under the framework of Heath, Jarrow, & Morton (1992), we have pro-posed a generalized version of the Shirakawa (1991) model to capturethe effect of jumps in the fixed-income market. The model is based on aninstantaneous forward rate specification that allows the coexistence of aWiener noise and multiple Poisson noises, each being associated with atime-deterministic volatility function. This specification with a finitenumber of jump components ensures the completeness of the marketand the uniqueness of any contingent claim prices.

Based on this specification, we derive the evolution of futures prices,and prove that if the underlying instantaneous forward rate evolution con-tains a jump component, the jump must also present itself in the evolutionof futures prices. This evolution can be used in the estimation process todetermine the model parameters. The advantage of the approach is thatwe do not need to Markovianize the system that contains the spot rate ofinterest or equivalently the bond prices, which may not always be possible.Instead, we treat the futures contract as a derivative instrument writtenon the instantaneous forward rate, and use the likelihood transformationmethod of Duan (1994) to estimate the model parameters.

The proposed approach is sufficiently general to deal with any spec-ification with time deterministic diffusion and jump volatility functions.However, to illustrate our method, we choose a humped-shape specifica-tion for our diffusion volatility, and a “binomial” approach to model

TABLE III (Continued)

(e) TIFFE Case—Second Sample PeriodHumped Diffusion Volatility—Constant Jump Volatility

Lagged Variables

0 1 1 1 1

0.000 0.001 0.105 0.050 0.050 0.050 0.4420.002 0.006 0.095 0.136 0.136 0.136 0.2010.247 0.157 0.091 0.686 n.a. 0.686 0.025

0.021 0.034 0.344 0.008 0.000 0.008 0.1050.021 0.034 0.344 0.008 n.a. n.a. 0.1050.021 0.034 0.344 0.008 n.a. 0.008 0.105

0.189 0.083 0.022 0.928 0.928 0.058 0.001

Note. This table reports the p-value for the score tests in the restricted models (i.e., the best model for eachexchange). For each test, the vector cj (U) consists of only one element, which is the element (ik)—row i, columnk—of the matrix [hj (U)] [hj�1(U)]�. The p-value for that test is reported in the (ik) position of the table. The “n.a.”denotes the case where test statistic cannot be found due to the problem of matrix noninvertibility.

se

c1

l1

g1

k

s1

s0

seclgkss


Poisson noise. We distinguish between information surprises that affectthe whole yield curve more or less equally, and information surprises thatare less prominent, and that affect some sections of the yield curve morethan others. We run our estimation for 3-month futures contracts tradedon the CME, SFE, LIFFE, and TIFFE, the four large exchanges fromdifferent parts of the world.

The empirical results suggest very different jump behavior in eachmarket. The small Australian market does not contain a jump compo-nent. On the other hand, a jump component is an important part of theU.S. and U.K. markets. The probability of jump arrivals in these twomarkets are similar; however, the mechanism by which a jump shockimpacts the markets are different. In the U.S. market, a shock has itsimpact spread out equally along the yield curve. In the U.K. market,the impact of a shock dies out gradually along the yield curve, and there-fore short-term instruments are affected more heavily than longer terminstruments. The Japanese market is split into two subsample periods.The jump component only exists in the 1996–2001 period. This period ischaracterized by an extremely low level of interest rate, very high Wienervolatility, and relatively small jump volatility.

However, model diagnostic tests indicate that the specification isstill not rich enough to capture the true market behavior. Some possibleavenues to explore to improve the model fit include a level-dependentinstantaneous forward rate volatility and extension to multifactor noiseterms. We intend to explore these issues in subsequent research.

APPENDIX A. GENERALIZED ITÔ’S LEMMA

Consider a jump diffusion process in terms of the stochastic integralequation

or in terms of the stochastic differential equation

Given a C1,2-function F(t, X), the stochastic differential equation forF is

� FX st dWt � [F(t, Xt� � dt) � F(t, Xt� )] dNt

dF(t, Xt) � cFt� � FX�at dt �12

FXX�s2t d dt

dX � at dt � at dW(t) � dt dNt

Xt � X0 � �t

0

as ds � �

t

0

ss dWs � �t

0

ds dNs


where

APPENDIX B. THE EXPECTATIONOF A “LOG-POISSON” PROCESS

With a view to finding the second expectation in Equation (6), we con-sider a variable X whose evolution can be described by the stochasticintegral equation

where N is a Poisson process with intensity . We are interested in thevalue .

The value of X(T) is just the sum of realized jump components, eachappearing at the jump time Li between t and T, i.e.,

Using a standard result for conditional expectations we have that

(B.1)

First, consider the conditional expectation

Lemma B.1. Given that N(T) � N(t) � n, the n arrival timehave the same distribution as the order statistics corresponding to n inde-pendent random variables uniformly distributed on the interval (t, T).

Proof. See theorem 2.3.1 of Ross (1996), page 67. ❏

L1, L2, . . . , Ln

�t[eX(T) 0 N(T) � N(t) � n] � �t c exp a an

i�1z(Li, # )b d

�t[eX(T)] � a

n�0�t[e

X(T) 0 N(T) � N(t) � n] �[N(T) � N(t) � n]

X(T) � aN(T)�N(t)

i�1z(Li, �)

�t[exp(X(T))]c

X(T) � �T

t

z(u, # ) dN(u)

FXX� �02F(t, Xt)

0X2

FX� �0F(t, Xt)

0X

Ft� �0F(t, Xt)

0t


The required conditional expectation thus becomes

Substitute this result back to (B.1)

APPENDIX C. QUOTING CONVENTION OFEXCHANGES FOR FUTURES CONTRACTS

Let be the price at time t of a futures contract matur-ing at time . The contract is written on a pure discount instru-ment, which has a face value of $1 and matures at time .

We are considering 3-month futures contracts, and thereforeconstant for all .

The SFE quotes a value Gjk which is linked with Fjk via

The other three exchanges quote a value Gjk, which is linked with Fjk via

where and N vary across exchanges.The CME and LIFFE have the same quoting convention, where

N � 100 and . The TIFFE’s quoting convention was the same ast � 90360

t

Fjk � 1 � a1 �Gjk

Nbt � h(Gjk)

Fjk �1

1 � (1 �Gjk

100) 90365

� h(Gjk)

k � [0, K]TBk � TFk � t

TBk(� TFk)TFk(� t)

Fjk � F(t, TFk, TBk)

� exp ec�T

t

[exp(z(u, # )) � 1] du f � e�c(T�t) ec(T�t)°

� a

n�0°n e�c(T�t)

(c(T � t))n

n!

�t[eX(T)] � a

n�0°n �[N(T) � N(t) � n]

� °n

¶ � c �T

t

exp(z(u, # ))1

T � t du d n � 5�t[exp(z(Li, # ))]6n�t[e

X(T) 0 (N(T) � N(t)) � n] � �t c expa ani�1z(Li, # )b d


the CME and LIFFE for the period prior to October 1, 1999, after whichdate N � 1,000 has been used.

APPENDIX D. FULL INFORMATIONLOG-LIKELIHOOD FUNCTION FORQUOTED FUTURES PRICES

The main task in deriving the log-likelihood function is to calculate theJacobian of the transformation and write out the drift vector and covari-ance matrix for each transition log-likelihood function. These quantitiesthen can be substituted directly to the likelihood formulae in the text[Equations (16), (19), (20), and (21)] to write out the likelihood func-tion for observable futures prices.

From Equation (12)

we have

From Equation (11)

where for CME Eurodollar futures, we find that

The calculations for LIFFE and TIFFE exchanges are very similar,the calculations for SFE is equally easy, and therefore will be omittedhere.

The variance Equation (14) is

� N2 I02 � 2N RI12 � R2 I22 � s2e(tj � tj�1)

� �tj

tj�1

a �TBk

TFk

s(u, s) dsb2

du � �tj

tj�1

s2e du

bj(kk) � �tj

tj�1

(b2k(u) � s2

e ) du

0h(Gjk; u)

0Gjk�t

100

t � 90�360

Fjk � 1 � a1 �Gjk

100b t � h(Gjk)

0z(Fjk; U)

0Fjk�

1Fjk

Xjk � ln(Fjk) � z(Fjk)


where

(D.1)

(using successive integration by part).The covariance Equation (15) is

where

and Iab are defined as in (D.1).The mean term Equation (17) is

ajk(0) � �tj

tj�1

amk(u) �12

s2eb

du

R2 �s1

k(e�kTBk2 � e�kTFk2 )

R1 �s1

k(e�kTBk1 � e�kTFk1 )

N2 � �as0

k�s1

k2 b (e�kTBk2 � e�kTFk2 ) �s1

k (TBk2

e�kTBk2 � TFk2 e�kTFk2 )

N1 � �as0

k�s1

k2 b (e�kTBk1 � e�kTFk1 ) �s1

k (TBk1

e�kTBk1 � TFk1 e�kTFk1 )

� N1N2I02 � (N1R2 � R1N2)I12 � R1R2I22

� �tj

tj�1

a �TBk1

TFk1

s(u, s) dsb a �TBk2

TFk2

s(u, s) dsb du

bj(k1k2)� �

tj

tj�1

bk1(u)bk2

(u) du�(for k1 � k2)

�a(a � 1) p 2

(�kb)a t �a(a � 1) p 1

(�kb)a�1 t0 d b ` tj

tj�1

� a�ekbt c 1(�kb)

ta �a

(�kb)2 ta�1 �a(a � 1)(�kb)3 ta�2 � p

Iab � �tj

tj�1

ta ekbt dt

R �s1

k(e�kTBk � e�kTFk)

N � �as0

k�s1

k2 b (e�kTBk � e�kTFk) �s1

k (TBk e�kTBk � TFk e�kTFk)


14The exponential integral evaluation is available from most numerical packages, such as Ox,MATLAB, and Mathematica. An approximation formula can be found in Abramowitz and Stegun(1965).

where

and E1(z1, z2) is the exponential integral14 defined by

The mean term Equation (18) is

Evaluation of the integral yields

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ajk(2) � ajk(0) � gm(TBk � TFk)(tj � tj�1)

ajk(1) � ajk(0) � ck ek1tj � ek1tj�1

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The jump component of the volatility structure of interest rate futures markets: An international comparison