DMRG-Paris - 17/06/05
Presentation about Statistical Arbitrage (Stat-Arb), usingCointegration on the Equity Market
Presentation about Statistical Arbitrage (Stat-Arb), usingCointegration on the Equity Market
By Yoann BOURGEOIS and Marc MINKO
Derivative Models Review Group (DMRG-Paris)HSBC-CCF
By Yoann BOURGEOIS and Marc MINKO
Derivative Models Review Group (DMRG-Paris)HSBC-CCF
DMRG-Paris - 17/06/05
PLANPLAN
IntroductionPart I: Mathematical FrameworkPart II: Description of the proposed
strategy and first resultsConclusion
DMRG-Paris - 17/06/05
IntroductionIntroduction
Single stocks in the Equity Market generally are not stationary. But, their yields, in many cases are. From the econometrical point of view, they are generally told to be
Integrated of order 1. Cointegration is a mathematical theory that helps to handle the problem
generated by non-stationary data. With the help of this theory, we propose to build linear combinations of
these single stocks that are stationary. Such combinations can be traded and are called synthetic assets. Eventually, these stationary assets have the mean reversion property and
we will use this property in order to set up arbitrage strategies.
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Part I: Mathematical FrameworkPart I: Mathematical Framework
Description of theframework of our strategyStatistical Analysis ofmodels that are Integrated oforder 1 (ie I(1))
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Description of the framework of our strategy (1)Description of the framework of our strategy (1)
Reminder about Vector AutoRegressive models (VAR)In what follows, we consider a VAR process (p1), which can bewritten:
Remark: here, we suppose that the errors are i.i.d with a gaussian law, butit can easily be generalised with errors i.i.d with finite moments of order 2.
tX
),0(N law with i.i.d known X,...,X
),,1(1with
p
01k-
'2
1
t
t
tt
k
iitit
ttDTt
DXX
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Cadre dans lequel on se place (2)Cadre dans lequel on se place (2)Description of the framework of our strategy (2)Description of the framework of our strategy (2)
Definition: Let’s introduce the characteristic polynomial: with
Inversibility THEOREM for a VAR process
)(zA
k
ii
izIzA1
)(
1-
01n0
1
010t
t
A(z)C(z) ie IC(z)A(z) : radius of disc theinside and converges serie this:0/ Then,
.C(z)Let .C :1net Cwith
)()...(X
:errors theof and valuesinitial its offunction a as written becan X process VAR The
nn
nnk
jjjn
t
jjtjtj
k
sskksst
CzCI
DCXXC
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Description of the framework of our strategy (3)Description of the framework of our strategy (3)
Remark: the solution given by this theorem is valid whatever theparameters are. On the contrary, it is reminded in what follows that theparameters have to be constrained in order to define a stationary VARprocess.
Definition: a process is told strongly stationary iif
It is told weakly stationary of order 2 iif
Remark: in part I, strong stationarity is used since the errors are gaussian.
tX),...,Law(X),...,Law(X :1
11 tt htht mmXXh
constantVarXet constant t tEX
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Description of the framework of our strategy (4)Description of the framework of our strategy (4)
Remark: this fundamental hypothesis excludes explosive roots with |z|<1as well as seasonal roots (|z|=1 and z different from 1). If z=1 is a root, then theprocess is told to have a unit root.
1zor 10A(z):HYPOTHESISLFUNDAMENTA
z
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Description of the framework of our strategy (5)Description of the framework of our strategy (5)
THEOREM defining the necessary and sufficient condition for thestationarity of a VAR process
Remark: (i) when =0, one recognizes the WOLD theorem. (ii) it is checked that with gaussian errors, the strong stationarity is
recovered, whereas in the general case, the weak stationarity is obtained.
1zfor convergent is
)(C(z) :0/ where)(X:obtained is process VAR theoftion representaMA the
case, asuch In .0A(1) is stationary be toXfor conditionsufficientandnecessary a ,hypothesislfundamenta Under the
1
0n0t
t
zACzDC
EX
nn
nntntn
t
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Description of the framework of our strategy (6)Description of the framework of our strategy (6)
Basic definitions for cointegration Preliminary remark: Many economic variables are non stationary and the kind of non-
stationarity that is considered here can be removed by one or severaldifferentiations. In what follows, we will suppose that:
Definition:
),0(N law with i.i.d is p t
0 0order of
integrated is Y :/Y process a
0
0tt
ii
iitit
CC
CEY
DMRG-Paris - 17/06/05
Description of the framework of our strategy (7)Description of the framework of our strategy (7)
Remarks:
Definition:
concepts. same thedefine processes I(0) andty stationari 1, dimension In (iii)
regular. is A(1)C(1)fact in and I(0) isty theoremstationari in the defined process The (ii)
ion.cointegrat topathway a is fact thisin andsingular bemay C (i)
1-
I(0) is )( 1)d I(d),down noted (and
d"order of integrated" toldis X process a t
ttd EXX
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Description of the framework of our strategy (8)Description of the framework of our strategy (8)
Remark: the property of being integrated is connected with the stochasticpart of the process since the mean is substracted from the process in thedefinition. The concept of I(0) process is defined without consideringdeterministic terms such as the mean or the trend. Définition:
space.ion cointegrat theisrelationsion cointegratby generated is that space vectorial theLast,
t.independanlinearily arethat relationsion cointegrat ofnumber theisrank ion cointegrat The
.stationary is 0vector
ioncointegrat with theedcointegrat toldis X I(1). :XLet
'
tt
tX
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Description of the framework of our strategy (9)Description of the framework of our strategy (9)
The Vector Error Correction Model (VECM) representation We write the already used VAR model in a new way that is the VECM,
because this is the model that is used in the cointegration theory.
1
11
.
1
1t
1
1
1i
k
1i
1
11t
A(z))1(A and -A(1) : thatnote sLet'
)1(z-z)I-(1A(z) : writtenbecan X of polynomial sticcharacteri The
:used also isquantity following The
:1i and with
X : writtenbecan VAR(k)Every
k
iiz
k
ii
i
k
ii
k
ijji
tt
k
iitit
Idzd
zz
I
I
DXX
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Description of the framework of our strategy (10)Description of the framework of our strategy (10)
GRANGER Representation THEOREM (1)
of torscolumn vecr with theedcointegrat is that process I(1)an clearly is X)(C
0A : thatso is and valuesinitial on the dependsA where
A)D(L)(C)(CX
:tionrepresentaMA ith the written wbecan X case, In this
0)1(A- that is stationary be to
and Xfor condition sufficient andnecessary A : thatsor rank with r)(p ,then
pr)rank( if and 1zor 10A(z) If
t
'ortho
1ortho
'orthoortho
'
tt1
t
1iit
t
ortho'orthoortho
.'ortho
''
t
'
i
tt
t
D
XEX
XE
z
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Description of the framework of our strategy (11)Description of the framework of our strategy (11)
GRANGER Representation THEOREM (2)
Remark: clearly, from the MA writing, is stationarysince . Besides, is a representation of thedistance of from the balance position. The relationdefines underlying economic relations and supposes that all agents react to thedistance from the balance position through the adjustment coefficient andmake the variables satisfy the economic relations again.
0)( radius econvergenc as 1 with and
1z with ),(1
1)(A : thatso is )(C serie theLast, 11-
1
zCz
Czz
tt XEX
'
0Aet 0 '' C ))((1
'tt DLC
tt XEX '' tt XEX ''
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Description of the framework of our strategy (12)Description of the framework of our strategy (12)
Remark (end): it has to be noted that therelations are not asymptoticbalance relations with t+ or else relationsbetween the levels of variables in balance. Itshould be told instead that these relations arerelations between the portfolio variables that aredescribed by the statistical model and thattranslate the adjustment behaviour of the agents.
tt XEX ''
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Statistical Analysis of I(1) models - (1)Statistical Analysis of I(1) models - (1)
The existence of the cointegration vectors, which is alsoknown as the Reduced Rank hypothesis, is expressed ina parametric form, so that the Likelihood method canbe applied.
Therefore, estimators and statistical tests related to afixed number of cointegration vectors can be writtenwith closed formula.
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Statistical Analysis of I(1) models - (2)Statistical Analysis of I(1) models - (2)
Let’s consider the following general VECM model:
parameters free as),,,...,,,(and
Ω)(0,Nlaw with i.i.dε,1with
'
1
pt
11
k
tt
k
iititt
Tt
DXXX
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Statistical Analysis of I(1) models - (3)Statistical Analysis of I(1) models - (3)
As already told, an analysis of the likelihood function is done with the followingnotation:
),,...,(),,...,(
with'
1
''''12
11
0
210
k
tkttt
tt
tt
tttt
DXXZ
XZXZZZZ
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Statistical Analysis of I(1) models - (4)Statistical Analysis of I(1) models - (4)
Let’s introduce:
Remark:
With a constant, the log-likelihood can be written:
2,0with
11
'
ji
ZZT
MT
tjtitij
'jiij MM
T
ttttttt ZZZZZZ
TL
121
'0
1'21
'0 )()(
21
ln2
),,,(ln
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Statistical Analysis of I(1) models - (5)Statistical Analysis of I(1) models - (5)
First order conditions give for :
The residuals and are defined by:
(these residuals would be obtained while regressing respectively andon
)
T
ttttt ZZZZ
1
'221
'0 0)(
12212
'12202),(
MMMM
tR0 tR1
ttt
ttt
ZMMZR
ZMMZR
21
221211
21
220200
tX 1tX
ktt XX
,...,1
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Statistical Analysis of I(1) models - (6)Statistical Analysis of I(1) models - (6)
Therefore the log-likelihood can be written:
Let:
For fixed, it is easy to infer and while regressing onSo:
T
ttttt RRRRTL
11
'0
1'1
'0 )()(
21ln
2),,(ln
1,0with
12
1222
1
'
ji
MMMMRRT
S jiij
T
tjtitij
tR0 tR1'
'11
'0010
'111
'0100
111
'01
)())(()()(
)()(
SSSSSS
SS
DMRG-Paris - 17/06/05
Analyse statistique des modèles I(1) - (6)Analyse statistique des modèles I(1) - (6)Statistical Analysis of I(1) models - (7)Statistical Analysis of I(1) models - (7)
Therefore:
and the FUNDAMENTAL THEOREM of the STATISTICAL ANALYSIS ofI(1) models can be deduced:
)()()( 011
001011'
11'
002
max SSSSSS
L T
IVSVby normalized )v,...,v(V :rseigenvecto
theand 0...1 :seigenvalue with the
0S :equation following thesolving while
given is of MLE the, :)( :hypothesisUnder
11
'
p1
1
011
001011
'
p
SSS
rH
DMRG-Paris - 17/06/05
Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Statistical Analysis of I(1) models - (8)Statistical Analysis of I(1) models - (8)
p
1ri
^
i
100
T2-
max
1
)-ln(1T- :statisticfor hasr )rank(
againstr )rank( :H(r) : testlikelihood TheOLS. in the with ie above, equations in the
inserting whileobtained are parametersother the
of estimators The .)1())((L
: writtenbecan function likelihood maximized theand),...,( :by infered are relationsion Cointegrat
r
ii
r
SrH
vv
DMRG-Paris - 17/06/05
Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)
Remarks:
Statistical Analysis of I(1) models - (9)Statistical Analysis of I(1) models - (9)
),...,(
),...,(
)(
test.lstatistica JOHANSEN in the used aresmallestr -p the whilerelations,ion cointegrat the
gettingfor useful are seigenvaluebiggest r The )(
1011
00
111
01
prortho
prortho
vvSS
vvS
Sii
i
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Statistical Analysis of I(1) models - (10)Statistical Analysis of I(1) models - (10)
Models with constrained determinist termsUp to now, the coefficients of Φ were totally free.From now, we shall also consider the case when the dominant coefficient is
constrained. Therefore, we get two other models: with constrained constant:
with constrained linear trend:
t
k
iititt XXX
01
' )(
t
k
iititt XtXX
01
' )(
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Statistical Analysis of I(1) models - (11)Statistical Analysis of I(1) models - (11)
The same likelihood method can be used with the two new models. Only thenotation differ.
With a constrained constant:
With a constrained linear trend:
'''1
*2t2
''1
*1t1
*0t0
),...,( Zbecomes
)1,( Zbecomes
Zbecomes
kttt
tt
tt
XXZ
XZ
XZ
'''1
*2t2
''1
*1t1
*0t0
)1,,...,( Zbecomes
),( Zbecomes
Zbecomes
kttt
tt
tt
XXZ
tXZ
XZ
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Statistical Analysis of I(1) models - (12)Statistical Analysis of I(1) models - (12)
Conclusion: the matrix reduction problem has fordimension with
Limit lawsGenerally, the limit law of the JOHANSEN statistical test depends on the
determinist terms, constrained or not.For big samples (about 400), the asymptotic distribution of the statistic is well
known since the middle of the 90s and was tabulated by simulation. Thesestandard critical values are available in statistical tables.
For small samples, JOHANSEN proposed in 2002, a Bartlett correction whichconsists in estimating the VECM and in calculating a correction coefficientwhich is multiplied to the standard critical value.
11 pp0
1p
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Part II: Description of the proposed strategy and first resultsPart II: Description of the proposed strategy and first results
Description of an arbitragestrategy
First results
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (1)Description of an arbitrage strategy (1)
LAG choice The first problem to solve in order to work with the considered VECM is to
determine the LAG of the model: k. In an article from 1999, Lütkepohl andSaïkkonen suggest to use the AIC (Akaïke Information Criteria). After havingcollected a few pieces of information, it appeared that Hurvich and Tsai hadproposed in 1991 a corrected version of the AIC because it overestimates thereal LAG.
With a constant, this is an estimator of the expectancy of Kullback-Leibler, that is the distance between the sample and the considered VECMmodel.
The selection of k is to be done while minimizing the for differentvalues k0;…; .
cAIC
cAICmaxp
Description of an arbitrage strategy (2)Description of an arbitrage strategy (2)
For univariate time series, we have chosen the same way as FumioHayashi does in his book « Econometrics » (2000), ie:
For multivariate time series, we have decided to take:
The implementation of the with the sample has to bedone while doing the following regression:
41
max 100T12T)(p
maxp
6pmax
cAIC )X,...,X( T0
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Description of an arbitrage strategy (3)Description of an arbitrage strategy (3)
)()(1
and )(Coef :give estimators OLS The
0...X and ),0(N law with ,
Z
, )X,...,(XY with
'
'1'
'1
'0p
'
'1
'
'1
'1
'1T)(p
'1
'1
'1T)(p
'1T)(p
'T2T)(p
maxmaxmax
max
CoefZYCoefZYT
YZZZ
XCoef
D
D
X
X
XX
XX
ZCoefY
k
k
TkT
k
TT
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DMRG-Paris - 17/06/05
Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (4)Description of an arbitrage strategy (4)
In this framework:
where d is the number of determinist terms.
Estimation of the cointegration vectors: it has to be implemented exactly thesame way as it was described in Part I.
Sample size choice: different backtestings showed that the concepts ofstationarity and moreover of arbitrage are very furtive. So we decided to workon small samples, typically with a size of 50.
1))((ln
pdkpTpdkpTpTTAICc
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (5)Description of an arbitrage strategy (5)
Stationarity test In an article from March 2004 entitled « Recent Advances in Cointegration
Analysis » , Helmut Lütkepohl advises to use the ADF (Augmented DickeyFuller) test.
)dim(Dd where)()(
and
)(Coef , )N(0, law with , y
y
yy
yy Z
, ),...,y(Y with ZCoefY ie y
:Indeed model. ECM associated theof t coefficien theof statistic- ton the based is and
),,...,,(yon y of regression theuses test ADF theAR(k),an is y caseIn
T
2
2
'1'2t
'
'1
1k-T
2
1-T1-T
kk
'1k
1
11t
'111-ttt
dkkT
CoefZY
YZZZD
D
yDyay
Dyy
T
k
Ttt
k
iitit
tktt
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (6)Description of an arbitrage strategy (6)
The limit law of this statistic is non standard and depends on thedeterminist terms of the model. The major drawback of this test is its lack ofpower for small samples (this is precisely the interesting case for us). So wedecided to consider instead the ADF-GLS test, which is described in the bookof Davidson&MacKinnon « Econometric Theory and Methods ».
111
'2
1
111
'2
t
t
1
0
)(
Coef
)(
and
stationary y iestationarynon y ie
0:H0:H
:is test The
ZZZZ
Description of an arbitrage strategy (7)Description of an arbitrage strategy (7)
This ADF-GLS was proposed by Elliott, Rothenberg and Stock in an articlefrom 1996 entitled: « Efficient Tests for an Autoregressive Unit Root ».
While writing: , the idea is to infer γ (ie
the determinist coefficients) before infering β, because it appears on classicalADF tests, that the more determinist terms we have, the weaker the power ofthe test is.
So, we consider the following regression:
t
p
1jj-tj1-ttt y y D y
t)(1, D when 13.5- c)1(D when 7- cwith
1-Tc1 where v )D (D y y
t
t
t0
1-tt1-tt
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Description of an arbitrage strategy (8)Description of an arbitrage strategy (8)
Remark:
Let be the estimator of and let
The regression: helps us to calculate thet-statistic for .If then the asymptotic distribution of this statistic isIf then the asymptotic distribution was tabulated by Elliott,Rothenberg and Stock, and is close to
JOHANSEN ’s Rank Test The implementation of this test comes from an article of JOHANSEN
(2002) published in Econometrica and entitled: « A small sample correction forthe test of cointegration rank in the vector autoregressive model »
1T
^0
0
^0
tt't D y y
t
p
1j
'j-tj
'1-t
''t y y y
0'
)1(Dt nc
t)(1, Dt
c
DMRG-Paris - 17/06/05
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (9)Description of an arbitrage strategy (9)
Description of the chosen strategy The chosen strategy is simple. From a portfolio of p basic assets (p10), all the sub-portfolios of size 2,3
or 4 are extracted (meaning all pairs, triplets and quadruplets). For each sub-portfolio, we infer at instant t the vector corresponding
to the biggest eigenvalue of the associated VECM, in order to build a linearcombination of the basic assets. This combination is called a syntheticasset.
The stationarity at level 99% of the synthetic asset is checked in order tobuild an asset without trend, which is a modelling of the stochastic part ofthe synthetic asset.
Therefore, this asset without trend is stationary around zero. Consequently, a good measure of the market risk of the synthetic asset is
the standard deviation of the asset without trend.
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (10)Description of an arbitrage strategy (10)
This measure will determine the quantity of synthetic asset to trade, as well asthe conditions of opening and closing an arbitrage.Eventually, several backtestings have shown the need to use additional rules,called consistency rules, in order to get a good ratio of positive operations.The first rule is that for every proposed arbitrage, we decided to check thefollowing condition: for every moving sub-sample of a certain size (typically 20)of the asset without trend, one should have 45% of the values above zero and45% under zero. This condition is a translation of the fact that an asset that isstationary around zero is supposed to swing around zero.Last, when there are several arbitrage operations left, we decided to choose thebest one in a certain sense. Our purpose is to calculate the mean for everymoving subsample of a certain size (typically 20) and then to calculate themaximum of the absolute values of these means. The best arbitrage operation isthe one with the lowest maximum.
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)Description of an arbitrage strategy (11)Description of an arbitrage strategy (11)
Remarks: This last condition was decided in order to avoid clusters of
bad operations and to mutualize market risk over time.Moreover, this condition is synonymous with the fact that aprocess that is stationary around zero is supposed to have amean close from zero.
The spirit of the strategy is to be strict on the openingconditions of an operation because once this operation isreleased, whatever happens, the trader is charged for it.
• At instant t+1, a buying (resp. a selling) operation is released when the valueof the asset without trend is in the bracket [-2.5σ;-σ] (resp. [σ;2.5σ]).
• The quantity of synthetic asset to be traded is defined as a percentage of thevalue of the portfolio normalized by .
Description of an arbitrage strategy (12)Description of an arbitrage strategy (12)
•Conversely, once an operation is launched, it is closed only when one of thethree following conditions is satisfied: the target is completed, ie the asset without trend is positive (resp.negative) for a buying (resp. a selling) operation. the operation lasts more than 100 opened days (# 5 months). the operation generates losses bigger than 10σ per share of synthetic asset.
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Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)First results (1)First results (1)
The results we present are in no case definitive. They should be taken as anintroduction to future developments.
The following backtesting was made on the european market. Choice of the data:
The single stocks used for this backtesting were these of 15 of the biggestcapitalisations of the EuroStoxx50: ABN AMRO, Banco Bilbao VizcayaArgentaria SA, Banco Santander Central Hispano SA, BNPParibas, DeutscheBank AG, Deutsche Telekom AG, E.ON AG, ING Groep NV, Nokia OYJ,Royal Dutch Petroleum Co, Sanofi-Aventis, Siemens AG, Societe Generale,Telefonica SA et TOTAL SA.
The study period begins in 03/16/2001 and ends on 09/14/2004. The pricesused are the Last prices in Euros.
Transaction cost are worth 10bp and the daily repo rate is taken at 4% (thebiggest value from 2000 is about 3.5%)
First results (2)First results (2)
We obtained 99 operations opened and closed on the considered period. Webacktested 105 pairs, 455 triplets et 1365 quadruplets.
The Sharpe ratio is 3.67. The average P&L is 1.7, whereas the average length of an operation is 37
opened days. Last, the average annual growth rate is: 22%. The following graph describes the liquidation value of the portfolio.
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First results (3)First results (3)
DMRG-Paris - 17/06/05
EURO15
0
500000
1000000
1500000
2000000
2500000
16/03/2001
16/05/2001
16/07/2001
16/09/2001
16/11/2001
16/01/2002
16/03/2002
16/05/2002
16/07/2002
16/09/2002
16/11/2002
16/01/2003
16/03/2003
16/05/2003
16/07/2003
16/09/2003
16/11/2003
16/01/2004
16/03/2004
16/05/2004
16/07/2004
Dates
Liqu
idat
ion
Valu
e
EURO15
DMRG-Paris - 17/06/05
Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)ConclusionConclusion
The strategy set up looks like a promising one. For equity single stocks, it would be interesting to work with portfolios greater
or equal to 20 basic assets. This would generate IT problems, since thebacktesting for 15 basic assets was nearly 3 days long. But we have begun theimplementation of new libraries that should help us to divide by 2 or 3 thecalculation time.
It should be recalled that the only very important condition for such a strategyto work is the liquidity of the considered market.
Therefore, our next step will be the application of this strategy to CMS rates.On one hand, we expect interesting results since CMS rates are verycorrelated, but on the other hand the way to valuate swaps is morecomplicated than for single stocks.