By By Yoann BOURGEOIS and Marc MINKO

Transcription

1 Presenaon abou Sascal Arbrage (Sa-Arb, usng Conegraon on on he Equy Marke By By Yoann BOURGEOIS and Marc MINKO Dervave Models Revew Group (DMRG-Pars HSBC-CCF

2 PLAN Inroducon Par I: Mahemacal Framework Par II: Descrpon of he proposed sraegy and frs resuls Concluson

3 Inroducon Sngle socks n he Equy Marke generally are no saonary. Bu, her yelds, n many cases are. From he economercal pon of vew, hey are generally old o be Inegraed of order. Conegraon s a mahemacal heory ha helps o handle he problem generaed by non-saonary daa. Wh he help of hs heory, we propose o buld lnear combnaons of hese sngle socks ha are saonary. Such combnaons can be raded and are called synhec asses. Evenually, hese saonary asses have he mean reverson propery and we wll use hs propery n order o se up arbrage sraeges.

4 Par I: I: Mahemacal Framework Descrpon of he framework of our sraegy Sascal Analyss of models ha are Inegraed of order (e I(

5 Descrpon of of he framework of of our sraegy ( ( Remnder abou Vecor AuoRegressve models (VAR In wha follows, we consder a VAR process (p, whch can be wren: k wh D (,, -k,..., D Remark: here, we suppose ha he errors are..d wh a gaussan law, bu can easly be generalsed wh errors..d wh fne momens of order known..d wh law N p (0,

6 Descrpon Cadre of of dans he framework lequel on on se se of of place our sraegy (2 (2 (2 (2 Defnon: Le s nroduce he characersc polynomal: wh A( z Inversbly HEOREM for a VAR process I k z A(z he VAR process k s wh C 0 C s ( C(zA(z I s I e n : 0... C n e C(z A(z can be wren as a funcon of s nal values and of k - k s k n j C n j hen, 0/ : hs sere converges and nsde he dsc of radus : j0 j C. j ( j D Le C(z j n0 z n C n. he errors :

7 Descrpon of of he framework of of our sraegy (3 (3 Remark: he soluon gven by hs heorem s vald whaever he parameers are. On he conrary, s remnded n wha follows ha he parameers have o be consraned n order o defne a saonary VAR process. Defnon: a process s old srongly saonary f h : Law(,..., Law(,..., m h I s old weakly saonary of order 2 f m h E consan e Var consan Remark: n par I, srong saonary s used snce he errors are gaussan.

8 Descrpon of of he framework of of our sraegy (4 (4 FUNDAMENAL HYPOHESIS: A(z 0 z or z Remark: hs fundamenal hypohess excludes explosve roos wh z < as well as seasonal roos ( z = and z dfferen from. If z= s a roo, hen he process s old o have a un roo.

9 Descrpon of of he framework of of our sraegy (5 (5 HEOREM defnng he necessary and suffcen condon for he saonary of a VAR process Under he fundamenal for he MA represenaon of n0 E ( C s convergen for z n n o be saonary s D n hypohess, he VAR process s obaned : where a A( necessary and suffcen C(z A( z Remark: ( when =0, one recognzes he WOLD heorem. ( s checked ha wh gaussan errors, he srong saonary s recovered, whereas n he general case, he weak saonary s obaned. 0. In such a case, 0/ : n0 z n C n condon

10 Descrpon of of he framework of of our sraegy (6 (6 Basc defnons for conegraon Prelmnary remark: Many economc varables are non saonary and he knd of nonsaonary ha s consdered here can be removed by one or several dfferenaons. In wha follows, we wll suppose ha: s..d wh law Np(0, Defnon: a process Y / : Y EY 0 C s negraed of order 0 C 0 C 0

11 Descrpon of of he framework of of our sraegy (7 (7 Remarks: ( C may be sngular and n fac hs s a pahway o conegraon. ( he process defned n he saonary heorem Defnon: a s I(0 and n fac C( A( s regular. ( In dmenson, saonary and I(0 processes defne he same conceps. process (and noed down I(d, d - s old "negraed of order d" d ( E s I(0

12 Descrpon of of he framework of of our sraegy (8 (8 Remark: he propery of beng negraed s conneced wh he sochasc par of he process snce he mean s subsraced from he process n he defnon. he concep of I(0 process s defned whou consderng deermnsc erms such as he mean or he rend. Défnon: Le vecor he conegraon rank s ha are lnearly ndependan. Las, s : I(. 0 s he vecoral space ha s generaed by conegraon relaons he conegraon space. old conegraed s saonary. he number of wh he conegraon conegraon relaons

13 Descrpon of of he framework of of our sraegy (9 (9 he Vecor Error Correcon Model (VECM represenaon We wre he already used VAR model n a new way ha s he VECM, because hs s he model ha s used n he conegraon heory. Every VAR(k wh he Les k characersc noe ha : can be : he followng quany s also used : I and wren polynomal A( - and : of. A( I k j can be d dz j k A(z k wren z D : k A(z (- zi - z I k ( z z

14 Descrpon of of he framework of of our sraegy (0 GRANGER Represenaon HEOREM ( If A(z 0 z or z and f rank( r p hen, (p r wh rank r so ha : A necessary and suffcen condon for E and E o be saonary s ha - orho. A( orho orho orho 0 In hs case, where A depends on he nal values and s so ha : C C ( orho D ( can be wren wh he MA represenaon : orho C orho (L( orho D A A 0 sclearly an I( process ha s conegraed wh he r column vecors of

15 Descrpon of of he framework of of our sraegy ( GRANGER Represenaon HEOREM (2 - Las, he sere C( z s so ha : A ( z C C( z, z and wh as convergence radus ( 0 wh z Remark: clearly, from he MA wrng, s saonary snce C 0 e A 0. Besdes, C( L( s a represenaon of he D dsance of E from he balance poson. he relaon E defnes underlyng economc relaons and supposes ha all agens reac o he dsance from he balance poson hrough he adjusmen coeffcen and make he varables sasfy he economc relaons agan. E

16 Descrpon of of he framework of of our sraegy (2 Remark (end: has o be noed ha he relaons E are no asympoc balance relaons wh + or else relaons beween he levels of varables n balance. I should be old nsead ha hese relaons are relaons beween he porfolo varables ha are descrbed by he sascal model and ha ranslae he adjusmen behavour of he agens.

17 Sascal Analyss of of I( models --( ( he exsence of he conegraon vecors, whch s also known as he Reduced Rank hypohess, s expressed n a paramerc form, so ha he Lkelhood mehod can be appled. herefore, esmaors and sascal ess relaed o a fxed number of conegraon vecors can be wren wh closed formula.

18 Sascal Analyss of of I( models --(2 (2 Le s consder he followng general VECM model: wh ε and (,,..d wh law N, k,..., k D p (0,Ω,, as free parameers

19 Sascal Analyss of I( models - (3 Sascal Analyss of I( models - (3 As already old, an analyss of he lkelhood funcon s done wh he followng noaon:,,..., (,,..., ( wh k k D Z Z Z Z Z Z

20 Sascal Analyss of of I( models --(4 (4 Le s nroduce: Remark: M wh Wh a consan, he log-lkelhood can be wren: ln L(,, 2 ( Z, j 0 Z Z, M M j j 2 ln j j 0 Z Z2 ( Z0 Z Z2 2

21 Sascal Analyss of of I( models --(5 (5 Frs order condons gve for : (, R 0 0 Z Z2 Z 2 0 ( Z he resduals and are defned by: R R 0 M R Z Z 0 02 (hese resduals would be obaned whle regressng respecvely and on M M M M M 22 22,..., Z Z k M M 22

22 Sascal Analyss of I( models - (6 Sascal Analyss of I( models - (6 herefore he log-lkelhood can be wren: Le: For fxed, s easy o nfer and whle regressng on So: R R R R L 0 0 ( ( 2 ln 2,, ( ln, 0 wh j M M M M R R S j j j j R 0 R ( ( ( ( ( ( ( S S S S S S S S

23 Analyse Sascal sasque Analyss des of of I( modèles models I( -- (7 -(7 (6 (6 herefore: 2 Lmax S00 ( ( ( S S0S00S0 S and he FUNDAMENAL HEOREM of he SAISICAL ANALYSIS of I( models can be deduced: Under hypohess : H ( r : whle solvng he followng equaon :, he MLE of S S 0 S s gven 00 S 0 0 wh he egenvalues :... p 0 and he egenvecors : V (v,..., v p normalzed by V S V I

24 Analyse Sascal sasque Analyss des of of I( modèles models I( -- -(8 (7 (7 Conegraon relaons are nfered by : and he maxmzed lkelhood funcon can be wren : L 2 - max 00 he oher parameers ( H ( r n he he lkelhood rank( S es r ( equaons above, e wh : H(r : r has for sasc :. rank( - p r are obaned whle nserng ^ ln(- ( v he esmaors of r agans,..., v n he OLS. r

25 Analyse Sascal sasque Analyss des of of I( modèles models I( -- -(9 (7 (7 Remarks: ( he r bgges egenvalues are useful for geng he conegraon relaons, whle he p - r smalles are used n he JOHANSEN sascal es. ( orho orho S 0 S S 00 ( v S r 0 ( v,..., v r p,..., v p

26 Analyse sasque des modèles I( - (7 Analyse sasque des modèles I( - (7 Sascal Analyss of I( models - (0 Sascal Analyss of I( models - (0 Models wh consraned deermns erms Up o now, he coeffcens of Φ were oally free. From now, we shall also consder he case when he domnan coeffcen s consraned. herefore, we ge wo oher models: wh consraned consan: wh consraned lnear rend: k 0 ( k 0 (

27 Analyse Sascal sasque Analyss des of of I( modèles models I( --( - -(7 he same lkelhood mehod can be used wh he wo new models. Only he noaon dffer. Wh a consraned consan: Z Z Z Wh a consraned lnear rend: Z Z Z becomes Z becomes Z becomes Z becomes Z becomes Z becomes Z * 0 * * 2 * 0 * * 2 ( ( ( (,,,...,,..., k, k

28 Analyse Sascal sasque Analyss des of of I( modèles models I( --(2 - -(7 Concluson: he marx reducon problem has for dmenson wh Lm laws p 0 p p Generally, he lm law of he JOHANSEN sascal es depends on he deermns erms, consraned or no. For bg samples (abou 400, he asympoc dsrbuon of he sasc s well known snce he mddle of he 90s and was abulaed by smulaon. hese sandard crcal values are avalable n sascal ables. For small samples, JOHANSEN proposed n 2002, a Barle correcon whch consss n esmang he VECM and n calculang a correcon coeffcen whch s mulpled o he sandard crcal value.

29 Analyse sasque des modèles I( --(7 (7 Par Par II: II: Descrpon of of he he proposed sraegy and and frs frs resuls Descrpon of an arbrage sraegy Frs resuls

30 Analyse Descrpon sasque of of an an arbrage des modèles sraegy I( --( (7 ( (7 LAG choce he frs problem o solve n order o work wh he consdered VECM s o deermne he LAG of he model: k. In an arcle from 999, Lükepohl and Saïkkonen sugges o use he AIC (Akaïke Informaon Crera. Afer havng colleced a few peces of nformaon, appeared ha Hurvch and sa had proposed n 99 a correced verson of he AIC because overesmaes he real LAG. Wh a consan, hs AIC c s an esmaor of he expecancy of Kullback- Lebler, ha s he dsance beween he sample and he consdered VECM model. he selecon of k s o be done whle mnmzng he AIC c for dfferen values k{0; ; p max }.

31 Descrpon of of an an arbrage sraegy (2 (2 For unvarae me seres, we have chosen he same way as Fumo Hayash does n hs book «Economercs» (2000, e: p max ( p max For mulvarae me seres, we have decded o ake: p max 6 AIC,..., ( he mplemenaon of he c wh he sample 0 has o be done whle dong he followng regresson:

32 Descrpon of an arbrage sraegy (3 Descrpon of an arbrage sraegy (3 ( ( and ( Coef esmaors gve : he OLS 0... and (0, wh law N, Z,,..., ( wh Y 0 p ( p ( p ( p 2 ( p max max max max Z Coef Y Z Coef Y Z Y Z Z Coef D D ZCoef Y k k k k

33 Analyse Descrpon sasque of of an an arbrage des modèles sraegy I( --(4 (7 (4 (7 In hs framework: AIC c ln ( p ( kp kp d d p p where d s he number of deermns erms. Esmaon of he conegraon vecors: has o be mplemened exacly he same way as was descrbed n Par I. Sample sze choce: dfferen backesngs showed ha he conceps of saonary and moreover of arbrage are very furve. So we decded o work on small samples, ypcally wh a sze of 50.

34 Analyse Descrpon sasque of of an an arbrage des modèles sraegy I( --(5 (7 (5 (7 Saonary es In an arcle from March 2004 enled «Recen Advances n Conegraon Analyss», Helmu Lükepohl advses o use he ADF (Augmened Dckey Fuller es. In case and s based on he y y Z y and 2 y y k - s an AR(k, k y y k - a y he ADF es uses D y y Y Z Coef ( k ( k d 2 2 -k -sasc where of D D e he k, d dm(d he regresson of coeffcen Y ZCoef wh law of he wh Y ( y N(0, y 2 assocaed ECM, on (y k -, y,..., y Coef ( Z Z,..., y model. Indeed :, Z Y k, D

35 Analyse Descrpon sasque of of an an arbrage des modèles sraegy I( --(6 (7 (6 (7 he es s : and 2 H H ( Z 0 : : Z 0 e 0 Coef 2 ( Z y e y Z non saonary saonary he lm law of hs sasc s non sandard and depends on he deermns erms of he model. he major drawback of hs es s s lack of power for small samples (hs s precsely he neresng case for us. So we decded o consder nsead he ADF-GLS es, whch s descrbed n he book of Davdson&MacKnnon «Economerc heory and Mehods».

36 Descrpon of of an an arbrage sraegy (7 (7 hs ADF-GLS was proposed by Ello, Rohenberg and Sock n an arcle from 996 enled: «Effcen ess for an Auoregressve Un Roo». p Whle wrng: y D y y, he dea s o nfer γ (e - he deermns coeffcens before nferng β, because appears on classcal ADF ess, ha he more deermns erms we have, he weaker he power of he es s. So, we consder he followng regresson: y wh c y c - (D D - 7 when D -3.5 when D - j 0 ( (, j v - j where c -

37 Remark: ^ Descrpon of of an an arbrage sraegy (8 (8 Le 0 be he esmaor of and le y y 0 p he regresson: - j - j helps us o calculae he -sasc for 0. j If D ( hen he asympoc dsrbuon of hs sasc s nc If D (, hen he asympoc dsrbuon was abulaed by Ello, Rohenberg and Sock, and s close o c JOHANSEN s Rank es he mplemenaon of hs es comes from an arcle of JOHANSEN (2002 publshed n Economerca and enled: «A small sample correcon for he es of conegraon rank n he vecor auoregressve model» y y y D ^ 0

38 Analyse Descrpon sasque of of an an arbrage des modèles sraegy I( --(9 (7 (9 (7 Descrpon of he chosen sraegy he chosen sraegy s smple. From a porfolo of p basc asses (p0, all he sub-porfolos of sze 2,3 or 4 are exraced (meanng all pars, rples and quadruples. For each sub-porfolo, we nfer a nsan he vecor correspondng o he bgges egenvalue of he assocaed VECM, n order o buld a lnear combnaon of he basc asses. hs combnaon s called a synhec asse. he saonary a level 99% of he synhec asse s checked n order o buld an asse whou rend, whch s a modellng of he sochasc par of he synhec asse. herefore, hs asse whou rend s saonary around zero. Consequenly, a good measure of he marke rsk of he synhec asse s he sandard devaon of he asse whou rend.

39 Descrpon Analyse sasque of of an an arbrage des modèles sraegy I( -(0 - (7 (7 hs measure wll deermne he quany of synhec asse o rade, as well as he condons of openng and closng an arbrage. Evenually, several backesngs have shown he need o use addonal rules, called conssency rules, n order o ge a good rao of posve operaons. he frs rule s ha for every proposed arbrage, we decded o check he followng condon: for every movng sub-sample of a ceran sze (ypcally 20 of he asse whou rend, one should have 45% of he values above zero and 45% under zero. hs condon s a ranslaon of he fac ha an asse ha s saonary around zero s supposed o swng around zero. Las, when here are several arbrage operaons lef, we decded o choose he bes one n a ceran sense. Our purpose s o calculae he mean for every movng subsample of a ceran sze (ypcally 20 and hen o calculae he maxmum of he absolue values of hese means. he bes arbrage operaon s he one wh he lowes maxmum.

40 Descrpon Analyse sasque of of an an arbrage des modèles sraegy I( -( - (7 (7 Remarks: hs las condon was decded n order o avod clusers of bad operaons and o muualze marke rsk over me. Moreover, hs condon s synonymous wh he fac ha a process ha s saonary around zero s supposed o have a mean close from zero. he spr of he sraegy s o be src on he openng condons of an operaon because once hs operaon s released, whaever happens, he rader s charged for. A nsan +, a buyng (resp. a sellng operaon s released when he value of he asse whou rend s n he bracke [-2.5σ;-σ] (resp. [σ;2.5σ]. he quany of synhec asse o be raded s defned as a percenage of he value of he porfolo normalzed by.

41 Descrpon of of an an arbrage sraegy (2 Conversely, once an operaon s launched, s closed only when one of he hree followng condons s sasfed: he arge s compleed, e he asse whou rend s posve (resp. negave for a buyng (resp. a sellng operaon. he operaon lass more han 00 opened days (# 5 monhs. he operaon generaes losses bgger han 0σ per share of synhec asse.

42 Analyse sasque Frs resuls des modèles ( ( I( --(7 (7 he resuls we presen are n no case defnve. hey should be aken as an nroducon o fuure developmens. he followng backesng was made on he european marke. Choce of he daa: he sngle socks used for hs backesng were hese of 5 of he bgges capalsaons of he EuroSoxx50: ABN AMRO, Banco Blbao Vzcaya Argenara SA, Banco Sanander Cenral Hspano SA, BNPParbas, Deusche Bank AG, Deusche elekom AG, E.ON AG, ING Groep NV, Noka OYJ, Royal Duch Peroleum Co, Sanof-Avens, Semens AG, Socee Generale, elefonca SA e OAL SA. he sudy perod begns n 03/6/200 and ends on 09/4/2004. he prces used are he Las prces n Euros. ransacon cos are worh 0bp and he daly repo rae s aken a 4% (he bgges value from 2000 s abou 3.5%

43 Frs resuls (2 (2 We obaned 99 operaons opened and closed on he consdered perod. We backesed 05 pars, 455 rples e 365 quadruples. he Sharpe rao s he average P&L s.7, whereas he average lengh of an operaon s 37 opened days. Las, he average annual growh rae s: 22%. he followng graph descrbes he lqudaon value of he porfolo.

44 Frs resuls (3 Frs resuls (3 EURO5 6/03/200 6/05/200 6/07/200 6/09/200 6//200 6/0/2002 6/03/2002 6/05/2002 6/07/2002 6/09/2002 6//2002 6/0/2003 6/03/2003 6/05/2003 6/07/2003 6/09/2003 6//2003 6/0/2004 6/03/2004 6/05/2004 6/07/2004 Daes EURO5 Lqudaon Value

45 Analyse sasque Concluson des modèles I( --(7 (7 he sraegy se up looks lke a promsng one. For equy sngle socks, would be neresng o work wh porfolos greaer or equal o 20 basc asses. hs would generae I problems, snce he backesng for 5 basc asses was nearly 3 days long. Bu we have begun he mplemenaon of new lbrares ha should help us o dvde by 2 or 3 he calculaon me. I should be recalled ha he only very mporan condon for such a sraegy o work s he lqudy of he consdered marke. herefore, our nex sep wll be he applcaon of hs sraegy o CMS raes. On one hand, we expec neresng resuls snce CMS raes are very correlaed, bu on he oher hand he way o valuae swaps s more complcaed han for sngle socks.