Table Of ContentHow does the Eurodollar Interest Rate behave?
Tiziana Di Matteo and Tomaso Aste∗
INFM-Dipartimento di Fisica, Universit`a degli Studi di Salerno,
84081 Baronissi (Salerno), Italy
e-mail: [email protected] .
∗
INFM-Dipartimento di Fisica, Universit`a di Genova
via Dodecaneso 33, 16146 Genova, Italy
AAS, Sal. Spianata Castelletto 16, 16124 Genova, Italy
e-mail: aste@fisica.unige.it .
(01/01/01)
An empirical analysis on Eurodollar interest rates daily data in the time period 1990-1996,
is performed and compared with Libor data in the time period 1984-1998. The complementary
1
cumulativedistributionsforthedailyfluctuationsatdifferentmaturitydatesandthePowerSpectral
0
Densityarecomputed. Wefindthattheprobabilitydistributionshows‘fat’tailswithnon-Gaussian
0
behaviours. Moreover, we study the correlations among Eurodollar interest rates fluctuations with
2
differentmaturitydates. Byusinganoriginalclusteringlinkage,weshowhowthecollectivemotion
n
oftheinterestratescurvecanbeanalyzedinsub-groupsofmaturitydateswithsimilarbehaviours.
a
J
I. INTRODUCTION at different maturity dates. The results concerning the
1
probability distribution and the Power Spectral Density
1 of interest rates fluctuations are presented in Section III
Thelasttwentyyearshaveseenawidedevelopmentof
v and in Section IV, respectively. In Section V, the cor-
studies on the interest rates curves. So far, most of the
9 relations are analyzed in terms of an original clustering
0 effortsinFinancehavebeendevotedtothestudyofstock
linkage. Finally, some remarks are reported in the last
0 prices, where one is dealing with a single object affected
Section.
1 bystochasticfluctuations. Oncontrast,theinterestrates
0
case deals with severalobjects that follow similar trends
1
and can be thought as a curve whose shape is varying in
0 II. EMPIRICAL DATA
/ time. This makes the subject very challenging since one
t
a isnomoredealingwiththestatisticsofsingleobjectsbut
m with the motion of a whole complex set. An Eurodollar is a dollar deposited in a bank out-
- Nowadays, these studies are becoming very attrac- side the United States by another bank. The Eurodol-
d
tive and are approached from many different perspec- lar future contract is a future contract on an interest
n
tives [1–5]. Several theoretical models have been pro- rate. These contracts are traded on an exchange. The
o
c posed[6–11]. Thesemodelsoftenassumethefluctuations three-month Eurodollar futures traded on the Chicago
: to be Gaussian, neglecting therefore the ‘fat tail’ effects Merchantile Exchange (CME) is the most popular of
v
i [12]. This is in contrast with the empirical observations the futures contract on short-term interest rates. These
X
[13]. The inadequacies of the Gaussian models in finan- contracts trade with delivery months of March, June,
r cial analysis have been reported for a long time [14,15], September,andDecemberuptotenyearsintothefuture.
a
buttheavailabilityofenormoussetsoffinancialdataand ThevariableunderlyingthecontractistheEurodollarin-
thecapabilitytotreatthem,hasrecentlyrenewedthein- terest rate applicable to a 90-days period beginning on
terest in the subject [16–24]. While the fat-tail property the third Wednesday of the delivery month. When the
of the empirical distribution of stocks price changes has delivery date is reached, and the actual interest rate for
been widely documented, the interest rates have been the 90-day period is known, the contract is settled in
less investigated [25]. Moreover, the studies of interest cash. The three-month Eurodollar interest rate is the
ratesare oftenlimited to only few maturity dates(3 and same as the three-month Libor (London Interbank Of-
6 months cash rates in [26], Bund futures in [5]). The fered Rate), that is the rate at which large international
paper [27], is an exception as the US forward rate curve banks found much of their activities. Specifically, Libor
with maturity dates up to four years is modelled. is the rate at which one large bank is willing to lend
Inthepresentpaper,weperformanempiricalanalysis money to another large bank [1,4,28,29]. All these rates
of time series based on daily prices of Eurodollar future are determined in trading between banks and change as
contracts in the time period 1990-1996. The data are economic conditions change.
described in Section II (more details can be found in the FollowingthelineofBouchaudetal.[27,30,31],wecon-
paper of Bouchaud et al. [27]). Our study is mainly fo- siderthree-monthsfuture ratesasinstantaneousforward
cused on the ‘tail’ regions of the fluctuations probability rates so obtaining daily values for the forward interest
distributions and on the correlations among time series ratecurvef(θ,t),wheretisthecurrentdateandθ isthe
1
maturity date. In the time period analyzed 1990-1996, III. PROBABILITY DISTRIBUTIONS
the variable t counts incrementally the trading days be-
tween 1 and T = 1831. On this time period we have
To investigate the nature of the stochastic process un-
16 different time series corresponding to several matu-
derlying the interest rates changes, we first analyze the
rity dates ranging from θ = 3 to 48 months with a step
probabilitydistributionofthe daily changesin the inter-
ofthreemonths. Thebehaviouroff(θ,t)asfunctionoft
est rates.
is shown in Fig.1 where, for a better visualisation of the
plot, we report only those values corresponding to the Let us start by the computation of the standard devi-
following maturity values: θ = 3,15,30,48 months. The ation of ∆f(θ,t), which is defined as:
interestratesbehavioursfortheothermaturitydatesfol-
low very similar trends in time, and stay mostly inside 1 T2
the shape traced by the two extreme maturity values, σ(θ)=v (∆f(θ,t) <∆f >)2 , (2)
namely θ =3 and θ =48 months. uutT2−T1tX=T1 −
where T1 and T2 delimit the range of t, and < ∆f > is
theaverageovertimeof∆f(θ,t)(whichtendstozerofor
10
%) 9 q=4 years 1990-1996 Tθ2h−asTa1 →min∞im).umInvtahleuewhofol0e.0p6e3rioadt θ19=903-1m99o6n,tσh(sθ,)thve.sn.
( q=30 months
q,t) 8 increases up to a maximum value of 0.087at θ = 1 year,
f( 7 once reached this maximum the value slowly decreases
es 6 until σ(θ)=0.062 at θ =4 years.
at
r 5 q=3 months
st
e 4 q=15 months
er 0.25
nt 3 Eurodollarq=3
I -250 0 250 500 750 1000 1250 1500 1750 Eurodollarq=6
0.20 Eurodollarq=15
t(days) Eurodollarq=30
Eurodollarq=48
0.15 Liborq=1
FIG. 1. Eurodollar interest rates (f(θ,t)) as function of t sq() LLiibboorrqq==36
for θ=3, 15, 30, 48 months. 0.10
Theinterestratefluctuationsareanalyzedbystudying 0.05
the changes from one day to the following day:
0.00
84 86 88 90 92 94 96 98
years
∆f(θ,t)=f(θ,t+∆t) f(θ,t) , (1)
−
FIG.3. Standard deviation σ(θ) as function of the years,
where ∆t = 1 day. In Fig.2 these fluctuations are re- for Eurodollar at θ = 3, 6, 15, 30, 48 months (time period
ported as a function of t for f(θ = 3,t). All the others 1990-1996) and Libor for θ = 1, 3, 6 months (time period
atdifferent maturity dates havesimilar behaviourschar- 1984-1998).
acterised by stochastic fluctuations around the zero.
We compare the Eurodollar standard deviation of ∆f
foreachsingleyearbetween1990-1996withtheoneofthe
Libor data corresponding to a longer time period from
1.0
0.8 q=3 months 1984 to 1998. In Fig.3 the values of σ(θ) in the different
0.6 years are plotted for several maturity dates. The stan-
0.4 darddeviationsforbothEurodollarandLiborserieswith
Dqf(,t) -000...022 tvhaeluseasmaendmsahtouwritdyedcraetaessinθg=t3reanndds6bemtwonetehns1,9h8a4vetosi1m9i9la8r.
The statistical distributions of the daily changes in
-0.4
theinterestratesarebetteranalyzedbycomputingtheir
-0.6
-0.8 1990-1996 complementarycumulativedistributionsΨ(∆f),thattell
-1.0 us the probability to find a daily change which is larger
0 250 500 750 1000 1250 1500 1750
t(days) than ∆f:
∆f
FIG. 2. Eurodollar interest rate fluctuation (∆f(θ,t)) as
Ψ(∆f)=1 p(ξ)dξ ; (3)
function of t for θ=3 months. −Z−∞
2
with p(ξ) being the probability density distributions of In Fig.5, we plot the tail region of Ψ(∆f) at θ = 3
∆f(θ,t). monthsasfunctionofbothpositiveandnegativechanges
of ∆f. We observe that the probability distributions
are slightly asymmetric with negative changes that are
100 slightly more probable than the positive ones. On the
other hand, the largevariations result more probable for
the positive changes, and the overall distribution leads
10-1 -correctly-to a zero mean value for ∆f. In the same fig-
urethetwoempiricalcurvesfortheEurodollarsarecom-
paredwiththetwocurvesfortheLiboratθ =3months.
Df)10-2 q=48 months Large fluctuations in Eurodollar and Libor data show
Y( q=30 months verysimilarbehaviourswithcomparableprobabilityval-
q=15 months
q=3 months ues. In these distributions the tails are non-Gaussian:
10-3 they are fatter than the Gaussian ones.
Instead that with the Gaussian distribution, a more
appropriate comparison should be performed with the
10-4 10-2 Df 10-1 100 generalclassofL´evy,Khinthinestable distributions[32].
A symmetric stable distribution with zero mean can be
FIG. 4. Complementary cumulative distribution Ψ(∆f) written as [2]:
v.s. ∆f forEurodollardataatdifferentmaturitydates: θ=3,
15, 30, 48 months. P (x) 1 ∞e−γqαcos(qx)dq , (4)
L ≡ π Z0
We find that our time series are highly leptokurtic
for all the maturity dates and are characterized by non- where 0 < α 2 and γ is a positive scale factor. For
Gaussian profiles for large interest rates changes. The α=2 and γ =≤σ2/2, P is a Gaussian distribution with
L
results for Ψ(∆f) are reported in Fig.4 for the tails re- zero mean and standard deviation equal to σ; whereas
gions for positive changes of ∆f with θ = 3,15,30,48 for α=1, P is the Lorentzian distribution.
L
months. These curves show, in a log-log scale, three dis- In Fig.5 we compare the Eurodollar empirical data
tinct parts characterised by different decreasing trends: with a symmetric stable distribution, given by Eq.(4),
i) a first region (0.01 < ∆f < 0.05) with slopes be- having α = 1.45. One can note that these curves have
| |
tween 0.4 0.6; ii) an intermediate region (0.05 < a tail region that seems to follow a L´evy stable distri-
− ÷−
∆f < 0.2) with a faster decreasing behaviour (slopes bution. But we may expect a faster decrease to zero
| |
1.6 2.7); iii) a final region (∆f > 0.2) which is for larger fluctuations of ∆f, as for instance the one de-
− ÷− | |
characterized by larger slopes (between -2.3 up to -10). scribedbythetruncatedL´evydistribution[33]. Adeeper
Comparableresultsareobtainedforallthetimeseriesat analysisonthisaspectwillbedoneinafuturework. Let
different maturity dates. note that alltime seriesat differentmaturity dates show
behaviours analogous to the one at θ =3 months.
100
Positive changes for Eurodollar
NPoesgiatitvivee c chhaannggeess f ofor rL Eibuorrodollar 100
Negative changes for Libor
Gaussian behaviour
Symmetric stable distribution 1984-1998
10-1
10-1
DYf() 10-2 YD(f)10-2
q=3 months
10-3
Dt=1 day
q=3 months 10-3 Dt=2 days
Dt=3 days
10-4 10-2 10-1 Df 100 DDtt==51 0d adyasys
FIG.5. Complementarycumulativedistribution(Ψ(∆f)) 10-140-2 10-1Df 100
andcumulativedistribution(1-Ψ(∆f))v.s. |∆f|,forEurodol-
lar and Libor data at θ = 3 months. The grey-line refers to FIG. 6. Complementary cumulative distribution Ψ(∆f)
the Gaussian behaviour, whereas the black-line is the L´evy v.s. ∆f for Libor data at the maturity date θ = 3 months,
stable distribution. for different values of thetime increment ∆t.
3
Fig.6reportsΨ(∆f)fortheLibornon-overlapping∆f 100
time series with θ = 3 month and ∆t values ranging be- 10-1 q=3 months
tween 1 and 10 (see Eq.(1)). Note that the Gaussian 10-2
behaviour tends to appear in the tail regions when the 10-3
∆t increases. w)(10-4
S
10-5
10-6
IV. SPECTRUM ANALYSIS 10-7 Power Spectrum÷ of Df
10-8-4 -3 -2 -1 0
10 10 10 10 10
w (1/day)
In this section we investigate the statistical properties
FIG. 8. Power spectrum S(ω) of the Eurodollar inter-
of our time series in the frequency domain. To this end
est rates fluctuations at θ = 3 months, as a function of the
wecomputethe powerspectraldensity(PSD)[34]ofour
frequency ω.
random variables by using the periodogram approach to
PSD estimation, that is currently one of the most pop-
ular and computationally efficient PSD estimator. This V. CORRELATIONS
periodogramspectralestimateisobtainedasthesquared
magnitude of the output values from an FFT performed
In order to investigate the correlations among the in-
directly on the data set. The power spectrum analysis
terestrates daily changes atdifferent maturity dates, let
for f(θ,t) indicates a stocastic signal with spectral com-
us here define the cross-correlationmatrix:
ponents which are decreasing as a power law:
<∆f ∆f > <∆f ><∆f >
i j i j
c = − , (6)
i,j
S(ω) ω−β , (5) σiσj
∼
wherei,j =1,...,16,∆f indicatestheinterestratedaily
i
withβ ∼=1.8. Fig.7showsthepowerspectrumofthespot changes for maturity θi =3i months, and σi is the stan-
interest rate r(t) = f(θ = 3,t) on a log-log scale. The dard deviation of interest rates changes ∆f . The sym-
i
slope of the linear fit is β = 1.77 0.01 and, within bol<...>denotesatimeaverageperformedoverallthe
− ±
the error, it is the same for all the other Eurodollar in- days of the investigated period. The correlation coeffi-
terestratesatdifferentmaturityvalues,anditisingood cients are computed between all the pairs of indices la-
agreement with the Libor at θ = 1, 3, and 6 months bellingourEurodollarseries. Thereforewehavea16 16
×
which have β = 1.80 0.01. symmetric matrix with c =1 on the diagonal.
i,i
− ±
The daily changes in the interest rates are highly cor-
related with coefficients in the range 0.48 c 0.98
i,j
≤ ≤
(for i=j). These strong correlations between the inter-
104 6
est rates fluctuations suggestthat one canconsider their
103 q=3 months behaviour as a collective motion of a set of quantities
102
joined together with some interaction force.
101
w)
S( 100
10-1 Correlation coefficients of Df
10-2 PLionweaerr fSitpectrum of f 1.1 Correlation coefficients of f
10-3
1.0
10-4
10-4 10-3 10-2 10-1 100 0.9
w (1/day)
0.8
Ci,j
FIG. 7. Power spectrum S(ω) of the Eurodollar interest 0.7
rates at θ=3 months,as a function of thefrequency ω. 0.6
0.5
The same analysis for ∆f(θ,t) (see Fig.8)shows a flat
spectrum,fromwhichwecaninferthattheinterestrates 0.4
i0=1 1 j=523..1064560759010512013i5=11550 1 j6=151680
changes are “pair wise uncorrelated” and the stochastic
process is approximately a white noise.
FIG.9. Correlation coefficients(ci,j,withi,j =1..16 and
i<j) for ∆f(θ,t) and f(θ,t).
4
Thecorrelationcoefficientsc for∆f =∆f(θ ,t)are max[dˆ ,dˆ ], called‘ultra-metric inequality’. Once the
i,j i i i,k k,j
shown in Fig.9 together with the correlation coefficients metric distance d between our interest rates fluctua-
i,j
for f(θ ,t) (which are obtained substituting ∆f(θ ,t) tions is known, one can introduce several ultra-metric
i i
with f(θ ,t) in Eq.(6)). The plot reads in the following distances.
i
way: the verticallines grouptogetherthe correlationco- Mantegna et. al have used the ‘subdominant ultra-
efficients ci,j for a fixed value ofi and j rangingbetween metric’, obtained by calculating the minimum spanning
i+1 to 16. Looking at the plot from left to right, i in- tree [37] connecting the stock indices [36,38]. In this pa-
creasesitsvaluefromi=1uptoi=15. Thecoefficients per we consider a different ultra-metric space that em-
ci,j have decreasing trends with j. This corresponds to phasizes the cluster-structure of the data.
the intuitive fact that interest rates with close maturity
In our case, a “cluster” is a set of elements with rela-
datesaremorecorrelatedthaninterestrateswithdistant
tive distances d which aresmallerthana giventhresh-
i,j
maturity (e.g. c1,2 > c1,4 > c1,8 > c1,16). Such decreas- old distance δ¯, whereas disjoined clusters have some ele-
ing trends support the idea that interest rates are di- mentswhichareatdistanceslargerthanδ¯. Letusdefine
rectlyinteractingbetweenneighboursonly(whichmeans the size of a cluster as the number of elements that it
thatthematurityθihasdirectinteractionswithmaturity contains, and its diameter as the maximum relative dis-
θi±1), so making a sort of ‘string’ [10,27,35]. tance between them. Therefore, an isolated element can
A closer look at Fig.9 shows that the decreasingtrend
be considered a cluster of size 1 and zero diameter.
in the correlation coefficients for ∆fi, is not perfectly We define the ultra-metric distance dˆ between two
i,j
monotonic. In particular, one can observe deviations
elements i,j belonging to two different clusters as the
from the monotonic trend for correlation coefficients in-
maximum metric distance between all the couples of el-
volving j >8 (i.e. maturity dates θ >2 years).
j ements in the two clusters. Therefore, the ultra-metric
To unde∼rstand the geometrical and∼topological struc-
distancebetweentwoisolatedelements(clustersofsize1)
tureofthe correlationcoefficients,weintroducethemet-
coincides with the metric distance. Whereas, the ultra-
ric distance d [36] between the series ∆f and ∆f
i,j i j metric distance between two elements inside the same
which is defined as:
cluster is equal to the cluster diameter.
d = 2(1 c ) . (7) The linkage procedure that we adopt is described in
i,j i,j
q − details in Appendix A. This procedure yields to a set of
By definition, ci,j is equal to zero if the interest rates clusterswhichhavediameterswhicharesmallerorequal
seriesi andj are totally uncorrelated,whereasci,j = 1 than the threshold distance δ¯, and have relative ultra-
±
inthecaseofperfectcorrelation/anti-correlation. There- metric distances dˆwhich are larger than δ¯.
fore,
d =0 if c =1
i,j i,j
d =√2 if c =0 (8)
i,j i,j
1
d =2 if c = 1 .
i,j i,j − 0.9
Note that this metric distance fulfils the three axioms of e0.8
c
a metric: 1) di,j = 0 if and only if i = j; 2) di,j = dj,i; an0.7
st
3) di,j ≤ di,k +dk,j. Indeed, di,j is the Euclidean dis- c di0.6
tance,inaT-dimensionalspace,betweenthetwovectors etri0.5
with components [∆f(θ ,1),∆f(θ ,2),....., ∆f(θ ,T)]/σ m
i i i i −0.4
a
and[∆f(θj,1),....,∆f(θj,T)]/σj. Thesevectorshaveuni- ultr0.3
tarylengths. Therefore,theyconnectthecentreofaunit
0.2
sphere, in a T-dimensionalspace, with points on its sur-
face. Similar behavioursof the interestrate changescor- 0.1
respondtotwoclosepointsonthesphere(d 1). Un- 0
i,j ≪ 39 45 33 42 48 24 27 30 36 9 12 15 18 21 3 6
correlatedbehaviourscorrespondtotwovectorspointing
toward perpendicular directions (d = √2). Whereas
i,j FIG.10. Hierarchicaltreeobtainedfromthecorrelationco-
anti-correlated behaviours are associated to two vectors efficientsofthe16Eurodollar interest ratesfluctuationstime
pointingtowardtwooppositedirections(di,j =2=sphere series in the time period 1990-1996. (On the x-axis are re-
diameter). ported thematurity dates and on they-axis theultra-metric
We study the geometrical and topological arrange- distances.)
ments of Eurodollar interest rates by introducing the
ultra-metric distance dˆ which satisfies the first two The result of this recursive linkage procedure, for the
i,j
properties of the metric distance and replaces the tri- 16 maturity dates (θ = 3,..,48) in the whole time pe-
angular inequality with the stronger condition: dˆ riod1990-1996,isreportedinthehierarchicaltreeshown
i,j
≤
5
in Fig.10. The clustering begins with maturities 39-45 Letusnowproceedlookingattheclustersarrangement
that gather together at δ¯=0.185;at a little larger value of∆f(θ,t)foreachsingleyearbetween1990to1996. We
δ¯=0.186,thecouple42-48makesanotherclusterandthe find that the cluster arrangement in each single year is
couple 30-36 merges at the δ¯= 0.206; then the maturity similartotheoneobservedinFigs.10,11wherethewhole
33 joins the first cluster at a δ¯= 0.216. The clustering timeperiodisconsidered. Firstwehavethenucleationof
processgoeson(seeFig.10)andendswhenonlyonelarge acoupleofmaturitydatesatdistancesδ¯=δ whoseval-
n
cluster is formed at δ¯=1.02. ues arebetween0.109and0.210(see Fig.12,wherethese
nucleation distances are plotted for eachyear). Then we
observe the growth of several clusters, that in the dis-
Data clustering 1990-1996 tances range between 0.3 to 0.6, makes a structure of
1.2 four well identifiable clusters. The first one is composed
1.0 by the maturity θ =3 and sometime θ = 6. The second
0.8
0.6 includes time series with maturity date values around 1
0.4
year (θ = 6 15). The third gathers maturity dates
0.2 3 6 9 12 15 18 21 24 27 30 36 33 39 45 42 48
÷
between 2 and 2.5 years (θ = 12 33) and the fourth
÷
the largest maturity dates > 2 years (θ = 27 48). Fi-
÷
nally,theseclusterscoalesceintoasingleoneatdistances
δ¯= δ , whose values (ranging between 0.852 and 1.131)
p
are plotted in Fig.12 for each year. Note that the val-
FIG.11. Grey-scale-tones visualisation of the clusters at
ues of δ and δ , calculated for the whole time period
different δ¯values, for the 16 Eurodollar interest rates fluctu- n p
(1990-1996),arewithintherangeofthevaluesplottedin
ations timeseries in thetime period 1990-1996.
Fig.12.
Thismechanismofnucleation,growth,andcoalescence
isbettervisualisedinFig.11wheretheclustersareexplic-
VI. CONCLUSIONS
itly drawn with different grey-tones for 6 distinct ranges
of δ¯ (0 0.2; 0.2 0.4; 0.4 0.6; 0.6 0.8; 0.8 1;
÷ ÷ ÷ ÷ ÷
1 1.2). ItisevidentfromFigs.10and11that,forultra-
m÷etric distances δ¯in the range between 0.4 and 0.6, the WehaveempiricallyanalyzedEurodollarinterestrates
atdifferentmaturitydates inthe time period1990-1996.
data set is gathered into 3 main clusters: Cls1 ={3,6}, We observed that the tail region has an evident non-
Cls2 = {9,12,15,18,21}, Cls3 = {24,27,....,45,48}, Gaussian behaviour - a crucial feature for the risk man-
withdiametersdˆ=0.553,0.411,0.433,respectively. The
agement analysis. Therefore, by using a Gaussian prob-
first cluster (Cls1) gathers together ∆f(θ,t) with matu- ability density function one underestimates the proba-
rity shorter than 1 year;Cls2 contains those with matu- bility of large interest rates fluctuations. These results
rity dates between 1 year and 2 years; whereas Cls3 has are confirmed by the same analysis done on Libor time
those with maturity dates which are largerthan 2 years.
seriesinthelongertimeperiodfrom1984to1999. More-
over, a power spectrum analysis has been carried out in
order to give some insights into the stochastic process
1.2 underlying the examined interest rates time series. The
powerspectrumfortheinterestratesfluctuationsshowsa
1.0 white-noise-likeflat behaviour. Whereas,the one for the
interest rates has a power law spectrum with exponent
0.8 1.77 (Eurodollar), and 1.80 (Libor) for all the different
d maturity dates.
p
0.6 d We have also analyzed the correlations between inter-
n
estratesfluctuationsatdifferentmaturitydatesbyusing
0.4 a linkage procedure that emphasizes the cluster struc-
ture. In the time period 1990-1996, we have found the
0.2 formation of three clusters joining together time series
corresponding to three maturity dates periods: shorter
0.0 than1 year,between1 to 2 years,largerthan 2 years. A
cluster structure with similar features have been found
89 90 91 92 93 94 95 96 97
years for each single year. The high correlationvalues that we
have found, confirm that the interest rates fluctuations
FIG.12. The nucleation distance δn and the distance δp behaviour at different maturity dates must be seen as a
as functions of theyears between 1990-1996. collective motion. On the other hand, the result from
6
the clustering linkage procedure indicates that the de- [5] J. P. Bouchaud, and M. Potters, “Th´eorie des Risques
viations from this collective behaviour can be gathered Financiers”, Alea (Saclay 1997).
in some maturity-groups. This might suggest that the- [6] O. A.Vasicek, Journal of Financial Economics 5 (1977)
oretical models for the interest rates behaviours should 177-188.
[7] J. C. Cox, J. E. Ingersoll and S. A. Ross, Econometrica
usefully consider different kinds of interactions in differ-
53 (1985) 368-384.
ent sub-groups of maturity dates.
[8] D. Heath, R. Jarrow, and A. Morton Econometrica 60
(1992) 77-105.
Acknowledgements
[9] J. Hull and A.White, Journal of Financial and Quanti-
Many thanks to Jean-Philippe Bouchaud, Michel Da- tative Analysis 28 (1993) 235-254.
corogna, Rosario Mantegna, for providing us the data, [10] D. Sornette, Eur. Phys. J. 3 (1998) 125-137.
[11] A. R. Pagan, A. D. Hall, and V. Martin, “Modeling the
and for many helpful discussions, comments and indica-
Term Structure”, HandBook of Statistics 14, Elsevier
tions. T. Di Matteo wishes to thank Sandro Pace for
Science B. V.(1997).
fruitful discussions andsupport. T. A. acknowledgesthe
[12] P.Embrechts,C.Kluppelberg,andT.Mikosch,“Model-
logistic support from AAS (Genova).
ing extremal eventsfor insurance and finance”, Springer
Verlag (Berlin 1997).
[13] K. C. Chan, G. A.Karolyi, Longstaff F. A.and Sanders
APPENDIX A: LINKAGE PROCEDURE A. B., Journal of Finance 47 (1992) 1209-1227.
[14] B. B. Mandelbrot, Journal of Business 36 (1963) 394-
419.
Atthe startingpointofourlinkage procedurewe have [15] B. B. Mandelbrot, “Fractals and Scaling in Finance”,
a set of N(=16) isolated elements which can be consid- Springer-Verlag (New York,1997).
ered as N clusters of size equal to 1 and diameter equal [16] D. Sornette, P. Simonetti, J. V. Andersen, Physics Re-
to zero. We define the ultra-metricdistance dˆ between ports 335 (2000) 19-92.
i,j
[17] U. A. Muller, M. M. Dacorogna, R. B. Olsen, O. V.
two elements i,j belonging to two different clusters as
Pictet,M.Schwarz,andC.Morgenegg,Journal ofBank-
the maximum metric distance between all the couples of
ing and Finance 14 (1990) 1189-1208.
elements in two clusters. Whereas, the ultra-metric dis-
[18] R. N. Mantegna, and H. E. Stanley, Phys. Rev. Lett. 73
tance between two elements inside the same cluster is
(1994) 2946-2949.
equal to the cluster diameter.
[19] S.Ghashghaie, W.Breymann,J.Peinke,P.Talknerand
Thefirststepofthelinkageprocedurefindsthe couple Y. Dodge, Nature 381 (1996) 767-770.
of elements i,j which are at the smallest distance dˆ . [20] R.N.MantegnaandH.E.Stanley,PhysicaA239(1997)
i,j
This couple is linked together in a new cluster of size 255-266.
2 and diameter dˆ . Then we seek for the pair of ele- [21] B.Piccinato,G.Ballocchi,M.Dacorogna,“Acloserlook
i,j
ments with the next-smallest ultra-metric distance, and at the Eurofutures market: intraday statistical analy-
sis”, Internal document (BPB.1997-08-25), Olsen & As-
wegenerateanewcluster,orweenlargethepreviousone
sociates (Zurich,1997).
if one of the two clusters belongs to that cluster. This
[22] G.Balocchi,M.Dacorogna,R.GencayandB.Piccinato,
procedure is recursively iterated for increasingly larger
“IntradayStatisticalPropertiesofEurofutures”,Internal
distances by joining isolated elements to clusters or by
document, Olsen & Associates, (Zurich,1997).
merging together larger clusters. The procedure termi-
[23] A.MataczandJ.-P.Bouchaud,“AnEmpiricalInvestiga-
nateswhenalltheelementsarejointedtogetherinasole tion of the Forward interest rate Term structure” cond-
cluster. This ending diameter coincides with the largest mat/9907297 (1999).
metric distance between two elements in the data set. [24] A.MataczandJ.-P.Bouchaud,“ExplainingtheForward
interest rate Term structure” cond-mat/9909396 (1999).
[25] J. Nuyts and I. Platten, “Phenomenology of the term
structure of interest rates with Pade Approximants”
cond-mat/9901096 (1999).
[26] V.A.Muller,M.M.Dacorogna,andO.V.Pictet,“Heavy
tails in high-frequency financial data”, Olsen & Asso-
[1] J. Hull, “Options, futures, and other derivatives”, Pren- ciates Working paper(Zurich, 1996).
tice Hall (New York2000). [27] J.P.Bouchaud,N.Sagna,R.Cont,N.EL-KarouiandM.
[2] R. N. Mantegna and H. E.Stanley, “An Introduction to Potters,AppliedMathematicalFinance6(1999)209-232.
Econophysics”,CambridgeUniversityPress(Cambridge, [28] F.Jamshidian,FinanceandStochastics1(1997)293-330.
2000). [29] X. Zhao and P. Glasserman, Finance and Stocastics 4
[3] P. Wilmott, S. Howison, J. Dewynne, “The mathemat- (1999) 35-68.
ics of financial derivatives”, Cambridge University Press [30] J. C. Cox, J. E. Ingersoll and S. A. Ross, Journal of Fi-
(Cambridge 1999). nancial Economics 9 (1981) 321-346.
[4] R.Rebonato, “Interest-rateoption models”, John Wiley [31] R. Rendleman, and C. Carabini, Journal of Finance 34
& Sons (NewYork 1998). (1979) 895-914.
7
[32] G. Samorodnitsky, M. S. Taqqu, “Stable non-Gaussian ley,“IdentifyingBusinessSectorsfrom StockPriceFluc-
random processes”, Chapman & Hall (NewYork,1994). tuations”, cond-mat/0011145 (2000).
[33] R. N. Mantegna and H. E. Stanley, Phys. Rev. Lett. 73 [36] R. N.Mantegna, Eur. Phys. J. B25 (1999) 193-197.
(1994) 2946-2949. [37] D. B. West, “Introduction to Graph Theory”, Prentice-
[34] S. M. Kay, S. L. Marple, Proceedings of the IEEE 69 Hall, (Englewood Cliffs NJ, 1996).
(1981) 1380-1415. [38] G. Bonanno, N. Vandewalle and R. N. Mantegna, Phys.
[35] P.Gopikrishnan,B.Rosenow,V.Plerou,andH.E.Stan- Rev. E 62 (2000).
8
