Presentation about Statistical Arbitrage (Stat-Arb), using

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


PLANPLAN

IntroductionPart I: Mathematical FrameworkPart II: Description of the proposed

strategy and first resultsConclusion


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.


Part I: Mathematical FrameworkPart I: Mathematical Framework

Description of theframework of our strategyStatistical Analysis ofmodels that are Integrated oforder 1 (ie I(1))


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


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



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



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



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



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



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



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



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



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



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 ''



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 ''


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.



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



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



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



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



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


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



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


Analyse statistique des modèles I(1) - (7)Analyse statistique des modèles I(1) - (7)

Remarks:


),...,(

),...,(

)(

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



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

' )(



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



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


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


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



)()(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




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



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



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


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



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




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.



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.



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 .


•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.



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.


First results (3)First results (3)


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


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.

Date post:	12-Sep-2021
Category:	Documents
Upload:	others
View:	8 times
Download:	0 times

Presentation about Statistical Arbitrage (Stat-Arb), using

Documents