Journal of Mahemacal Fnance, 206, 6, 66-77 Publshed Onlne February 206 n ScRes. hp://www.scrp.org/journal/jmf hp://dx.do.org/0.4236/jmf.206.606 Sascal Arbrage n S&P500 Sefanos Drakos Inernaonal Cenre for Compuaonal Engneerng, Rhodes, Greece Receved 8 January 206; acceped 24 February 206; publshed 29 February 206 Copyrgh 206 by auhor and Scenfc Research Publshng Inc. Ths work s lcensed under he Creave Commons Arbuon Inernaonal Lcense (CC BY). hp://creavecommons.org/lcenses/by/4.0/ Absrac A mehodology o creae sascal arbrage n sock Index S&P500 s presened. A synhec asse based on he conegraon relaonshp of he socks wh Index was consruced. In order o capure he dynamc of he marke me adapve algorhms have been developed and dscussed. The par radng sraegy was appled n dfferen perods beween S&P500 and synhec asse and he resuls were evaluaed. Dfferen mercs have shown ha he Mulvarae Kalman Algorhm creaes sascal arbrage n ndex wh much lower Maxmum Drawdown and hgher prof. The algorhm s neural as he bea s close o zero and he Sharp Rao remans hgh n all cases. Keywords Sascal Arbrage, Mean Reverng, Par Tradng, Kalman Fler, Tradng Algorhms. Inroducon Fnancal markes are based on he general radng rule: buy wh low prce and sell wh hgh prce. The am s he developmen of sraeges wh low rsk and succeeds hs general rule. Pure Arbrage s a caegory of sraeges wh zero rsk. As an example we can refer he case of buyng and sellng a sock a he same me wh a dfferen value n wo dfferen exchanges. The prof resuls from he dfference n prces, breakng he law of one prce. Anoher caegory s he sascal arbrage whch s no rsk free a all. Sraeges of hs ype are amed a he expeced gan whch s greaer han he rsk. The prof resuls from he msprcng of he socks. To acheve hs, one needs o assess wheher he prce of a sock s overvalued or undervalued relave o he acual value whch s really hard o deermne. The fundamenals of he sock, he demand of each perod and he general economc envronmen are some of he facors ha make he far value evaluaon dffcul. Par of hs caegory s he Par Tradng and s presened n several references as n []-[0]. The Pars radng s a sraegy whch s based on he relave prcng of socks whou acually neresed n he rue value of hem. The relave prcng s based on How o ce hs paper: Drakos, S. (206) Sascal Arbrage n S&P500. Journal of Mahemacal Fnance, 6, 66-77. hp://dx.do.org/0.4236/jmf.206.606
he dea ha wo asses wh he same feaures can be prced abou he same prce as he bass of he law of one prce. When a a gven me he sock prces are dfferen hen one s overvalued and he oher undervalued relave o he acual prce. The classc radng Pars sraegy derves from he characerscs of hs ncorrec prcng (msprcng) beween he wo socks. In he curren work nsead of he wo socks he msprcng beween he ndex S&P500 and a sub-se of socks belongng o hs s consdered. Ths forms a porfolo conssng of one un of he Index n long (or shor) and he correspondng number γ (Hedge rao) of subse s socks n he oppose poson. Three mehods adoped o deermne he number γ (Hedge rao): a) he ordnary, b) he rollng ordnary leas squares (OLS) regresson and c) he Kalman fler process as s presened below. Accordng o he sraegy, he spread of he ndex S&P500 and he socks prces combnaon are compued and when hs devaes from s hsorcal average value hen he nvesor bes on he reurn o he hsorcal wh sellng and buyng respecvely he socks of he porfolo. In pracce o consruc he synhec asse s necessary o nvesgae he approprae sock exhbng long relaonshp beween hem wh he ndex. The echnque used for hs purpose s he Conegraon as ~ Y ~ I are conegraed f hs presened n [] accordng o whch wo me seres X I ( ) and ( ) ax + by ~ I ( 0) for ab, 0 and noaon I(d) means negraed order d. In [] concluded ha pars who conegrae n sample perod behave beer n he ou-of-sample perod han hose no conegrae n he sample perod. Based on he prevous, he spread of he lnear combnaon of he conegraed socks and he S&P500 ndex have o be saonary process. To analyze hs relaonshp he augmened Dckey-Fuller es (ADF) was used as presened n [2]. Accordng o he sraegy he socks are no resrced o conegrae n he ou-of-sample perod bu s an ndcaon ha hey wll presen mean-reverng behavor. Addonally he log of prces used s also nsead he prces as n [3]. The man reason s ha log-reurns are me addve. So, n order o calculae he reurn over n perods usng real reurns we need o calculae he produc of n numbers: ( + r )( + r ) ( + r ) If r defned as: And 2 n. P P P r = + = () ( r ) 0 P0 P0 log log P ( r ) + = P0 ( + r ) + ( + r ) + + ( + r ) log log log 2 P P P = + + + 2 n log log log P0 P Pn ( P) ( P0) ( P2) ( P) ( Pn) ( Pn ) ( P ) log ( P ) = log log + log log + + log log = log n Consequenly he prof of spread over a perod s equal o: 0 n (2) y A Β P P + + y = log γ log P P + A Β The classc applcaon of he radng par sraegy has publshed n many works n he pas and has been appled n pracce. In [4] he auhor wen a furher sep and presened an algorhm for he par radng of a sock baske conegraed wh he S&P500 usng consan conegrang coeffcen wh a sngle value for all socks belongng o he baske. To capure he real behavor of he markes hs work presens new Tme Adapve algorhms usng rollng ordnary leas squares (OLS) regresson and Mulvarae Kalman Fler process where he me dependen hedge rao s compued separaely for each of he socks formng he synhec asse creang hereby sascal arbrage condons n ndex S&P500 and ncreasng he sraegy performance. (3) 67
2. Perod of Mehodology The proposed mehodology and he radng algorhm desgned based on ha dvded n wo dfferen spaces. The frs refers o he n sample perod whch s used o make all he approprae es and consruc he synhec asse and he oher o he ou of sample perod (Tradng Perod) where he synhec asse radng based on he specfc rules (Fgure ). 2.. In Sample Perod The daa of he n sample perod used for he synhec asse consrucon. Workng on a daly daa doman a year of closng prces was chosen as he n sample perod for deermne he se of conegraed socks wh S&P500 and creae he synhec asse. Synhec Asse Consrucon. The paper presens dfferen algorhms n order o creae sascal arbrage n S&P500. The S&P500, based on he marke capalzaons of 500 large companes equy ndces, and many consder one of he bes represenaons of he U.S. sock marke. The nal choce of socks for conegraon es wh S&P500 s made by he S&P00 whch s a sub-se of he S&P500, and measures he performance of large cap companes n he Uned Saes. Consuens of he S&P00 are seleced for secor balance and represen abou 57% of he marke capalzaon of he S&P500 and almos 45% of he marke capalzaon of he U.S. equy markes. The socks n he S&P00 end o be he larges and mos esablshed companes n he S&P500 (Wkpeda). Usng he daa of a seleced In Sample Perod for each sock S S&P 00 he second sep of Engle and Granger approach adoped. Usng he logarhmc prce of socks and S&P500 he OLS regresson shown below s performed: log ( S ) γ log ( S ) = + ε (4) Τhe Augmened Dckey-Fuller un roo es appled for he saonary of he OLS resduals. Accordng o he successful saonary resul a subse of he S&P00 s creaed and he componens of hs se are he canddae for he synhec asse consrucon. Sll workng n he sample perod a new OLS regresson was carred ou: Or log log ( ) N S γ log ( S ) = + ε (5) N ( S ) γ log ( S ) = + ε (6) Usng agan he Augmened Dckey-Fuller he saonary of he new OLS resduals s examned. If he saonary exs hen hs an evdence of mean reverng long erm behavor of he spread log ( ) N ε = S γ log ( S ) where S S and S s he se of sock where ndvdually and as logarhmc sum conegraed wh S&P500. The dmenson of S s dm(s). 2.2. Ou of Sample Perod-Tradng Perod Accordng o he prevous perod dfferen algorhms for synhec asse radng are developed. The frs was Fgure. Perod of sudy. 68
desgned assumed ha he conegraon coeffcen s consan durng he radng perod. In ha case and usng he Equaon (3) he prof of he sraegy durng he perod + h arses from he followng equaon: 3. Tme Adapve Coeffcen γ S h N S + + h ε+ h ε= log γ log (7) S S In realy he sysem of radng s dynamc and updaed as new nformaon ge o n and he conegraon coefecen (or he hedge rao) canno say consan durng he radng perod. For ha reason me adapve algorhms are developed o capure he real condons of he markes. 3.. Rollng Ordnary Leas Squares (OLS) Regresson The frs one consderng a rollng ordnary leas squares (OLS) regresson. The frequency of regresson calculaons rased by an opmzaon procedure and he conegraon coeffcen calculaed a each sep by he regres- N son of log ( ) log. S agans he ( S ) 3.2. Kalman Fler Process The Kalman fler process can be descrbed by hree dfferen seps: he predcon he observaon and he correcon. A new approach were developed usng a Mulvarae Kalman fler process. Based on ha he hedge rao calculaed separaely for each sock owned n he synhec asse and he compued vecor of he calculaed parameers a each me sep has dmensons (N + ) where N = dm(s) whle he dmensons of he covarance marx s (N + ) (N + ). The am of hese algorhms s o calculae a each me sep he updaed hedge rao of he synhec asse. Assumng ha he hedge rao and he premum follow a random walk we have: y = y + w (8) where: y : s he curren sae of he of he parameers. y - : s he prevous sae of he of he parameers. where for he mulvarae Kalman fler process Wh: ( ) w ~ N 0, σ w T 2 n y = γ γ γ µ (9) [ ] T y = h h h µ (0) 0 0 h: s he conegraon coeffcen from n sample perod and μ 0 comng from he same perod. The vecor of logarhmc prce of socks: 2 n ( ) ( ) ( ) x = log S log S log S () And S S The process followng he seps as below: Predcon sae where he nex sysem sae s predced based on he knowledge of he prevous sae y = y + w (2) ˆ The covarance of predcon sae s gven by: Pˆ = Pˆ + V (3) w The nex sep concerns he measuremen predcon. Gven he prce of he synhec asse and he predced 69
hedge rao he measuremen predcon are gven as: The resdual of measuremen and real value a each sep calculaed as: The varance of measuremen error s equal o: z ˆ ˆ = x y (4) log ( ) ˆ ε = S z (5) S = x Pˆ x + V (6) T e The Kalman Gan s he fler, whch ells how much he predcons should be correced on me sep s gven as: The las sep of process s he updae sep where: The updaed sae s esmaed as followng: And he Updaed sae covarance s equal o T Pˆ x K = (7) S yˆ = yˆ + K ε (8) ˆ ˆ T P = P K S K (9) All he process repeaed a every me sep of ou of sample perod. The esmaon of V w and V e has been dscussed n [] [2]. 3.3. Prof of Sraeges In he case of Tme Adapve coeffcen γ of he lnear regresson case he prof of he par radng sraegy rased by he followng equaons: ε N γ+ h S S + h h ε log log + = S γ S + h N In he case of Mulvarae Kalman Fler where he hedge rao s dfferen for each sock and for each me sep n he synhec asse can been shown ha: Smlarly 2 2 n n ( S ) ( ( S ) ( S ) ( S )) ε = log γ log + γ log + + γ log N γ = log ( S ) log S N γ 2 ε 2 = log ( S2 ) log S2 = ( S h ) N γ+ h ε + h log + log S+ h And fnally he prof durng a perod + h s equal o: ε S + ε = log log S N γ+ h S h + h + h N γ S (20) (2) 70
4. The Par Tradng Sraegy The algorhm of he par radng sraegy s based on he dsance of he spread from s hsorcal mean value and s mean-reverng behavor. To measure hs dsance a normalzed varable called z-score nroduced as: z ε = σ ε [ ε] ( ) where: [ ε ] : s he mean value of he spread over a lookback perod. σ ( ε ) : s he sandard devaon of he spread over he same perod. The radng akes place when hs varable exceeds some lms based on he spread mean reverng behavor. Thus: Open-long poson f z < zlow. Open-shor poson f z > zhgh. Ex-long poson f z > z. ex-low Ex-shor poson f z < zex-hgh. When a long (shor) poson s opened we buy (sell) one un of S&P500 and sell (buy) he followng amoun of socks from synhec asse as: N γ log ( S ) n he case of consan hedge rao, N log ( S ) rollng OLS regresson s appled, and N γ log ( S ) (22) γ when he f he Kalman Fler algorhm appled. The performance of sraegy evaluaed usng he followng mercs: ) Cumulave reurn, 2) Annualzed Reurn, 3) Sharpe Rao, 4) Maxmum drawdown, 5) bea. 5. Back Tesng-Performance Evaluaon Fve dfferen me perods was suded. In each case one year daa collecon was used n order o make he enre es and consruc he synhec asse. Afer ha he algorhm sars radng wh endng day for all cases he 30/2/205. All he sample perods sared a he frs day of he year and endng a he las of he same year. The radng sared on he frs day he nex year. In Table he back esng perods are presened. In Fgure 2 and Fgure 3 he resuls of cumulave reurn of Mulvarae Kalman Fler algorhm agans S&P500 and s maxmum drawdown are shown. In Fgure 4 and Fgure 5 he cumulave reurn of rollng OLS regresson algorhm agans S&P500 and he cumulave reurn of rollng OLS regresson algorhm agans consan hedge rao are llusraed. In Table 2 he ses of synhec asses are presened whle n Table 3 he name of symbols s gven. In Tables 4-8 all he mercs of he algorhms are dsplayed. I can be shown ha he dmenson of he se and he consuens are dfferen from perod o perod. Ths s an evdence of he non consan conegraon behavor of he socks wh he ndex bu as we can see from he graphs he synhec asse of each perod connues o rade wh prof and good mercs resuls. Table. Back esng perods. 2006 2007 2008 2009 200 20 202 203 204 205 In Sample Tradng 2007 2008 2009 200 20 202 203 204 205 In Sample Tradng 2008 2009 200 20 202 203 204 205 In Sample Tradng 2009 200 20 202 203 204 205 In Sample Tradng 200 20 202 203 204 205 In Sample Tradng 7
Table 2. Ses of sock n synhec asse on each perod. Tradng Perod Sock of Synhec Asse (Se S) Dm (S) 2007-205 {BRK.B, DVN, IBM, SPG} 4 2008-205 {CAN, BAX, BMY, DELL, EMC, GE, HAL, MDLZ, NSC, T, TXN, VZ} 2 2009-205 {DVN, FCX, IBM, MON, SPG} 5 2000-205 {APA, AXP, C, CAT, COF, FCX, IBM, MMM} 8 20-205 {COST, FDX, GS, QCOM} 4 Table 3. Unon se of companes rased from conegraon es from all he perod of sudy. APA APACHE CORP FDX FEDEX CORPORATION AXP AMERICAN EXPRESS COMPANY GE GENERAL ELECTRIC CO BAX BAXTER INTERNATIONAL INC GS GOLDMAN SACHS GROUP INC BMY RISTOL MYERS SQUIBB COMPANY HAL HALLIBURTON CO (HOLDING CO) BRK.B BERKSHIRE HATHWY INC (HLDG CO) B IBM INTL BUSINESS MACHINES CORP C CITIGROUP MDLZ MONDELEZ INTERNATIONAL INC CAT CATERPILLAR INC MMM 3M COMPANY COF CAPITAL ONE FINANCIAL CORP MON MONSANTO COMPANY COST COSTCO WHOLESALE CORP NSC NORFOLK SOUTHERN CORP DELL DELL INC QCOM QUALCOMM INC DVN DEVON ENERGY CORP (NEW) SPG SIMON PROPERTIES GROUP INC EMC EMC CORPORATION T AT&T INC. COM FCX FREEPORT-MCMORAN COPPER&GOLD B TXN EXAS INSTRUMENTS INC VZ VERIZON COMMUNICATIONS Table 4. Sascal mercs performance for he radng perod 0/0/2007-30/2/205. In Sample Perod 0/0/2006-3/2/2006 Ou of Sample Perod 0/0/2007-30/2/205 Algorhm S&P500 MulVarae Kalman Fler Rollng (OLS) regresson Consan Hedge Rao Cumulave Reurn % 387.069 24.637 2.678 45.592 Annual Reurn % 9.26 2.48.337 4.268 Sharpe Rao 3.298 0.892 0.483 0.303 Bea 0.09 0.07 0.027 Maxmum Drawdown % 2.642 2.925 5.666 62.453 Duraon of Maxmum Drawdown 73.0 47.0 390.0 365.0 72
Table 5. Sascal mercs performance for he radng perod 0/0/2008-30/2/205 In Sample Perod 0/0/2007-3/2/2007 Ou of Sample Perod 0/0/2008-30/2/205 Algorhm Mul Varae Kalman Fler Rollng (OLS) regresson Consan Hedge Rao S&P500 Cumulave Reurn % 7.67 22.347 6.855 42.470 Annual Reurn % 3.32 2.556 0.833 4.528 Sharpe Rao.35 0.738 0.228 0.32 Bea 0.00 0.005 0.03 Maxmum Drawdown % 28.636 6.730 5.30 52.909 Duraon of Maxmum Drawdown 75.0 404.0 628.0 54.0 Table 6. Sascal mercs performance for he radng perod 0/0/200-30/2/205 In Sample Perod 0/0/2008-3/2/2008 Ou of Sample Perod 0/0/2009-30/2/205 Algorhm S&P500 MulVarae Kalman Fler Rollng (OLS) regresson Consan Hedge Rao Cumulave Reurn % 278.267 28.53 3.565 2.52 Annual Reurn % 20.972 3.655.837 2.054 Sharpe Rao 3.039.387 0.753 0.729 Bea 0.026 0.003 0.032 Maxmum Drawdown % 7.90 3.934 3.736 28.547 Duraon of Maxmum Drawdown 69.000 290.000 367.000 203.0 Table 7. Sascal mercs performance for he radng perod 0/0/200-30/2/205 In Sample Perod 0/0/2009-3/2/2009 Ou of Sample Perod 0/0/200-30/2/205 Algorhm Mul Varae Kalman Fler Rollng (OLS) regresson Consan Hedge Rao S&P500 Cumulave Reurn %.49 8.375 9.23 8.733 Annual Reurn % 3.294 2.857.469 0.49 Sharpe Rao 2.33.54 0.67 0.709 Bea 0.007 0.05 0.060 Maxmum Drawdown % 0.025 5.959 5.45 23.420 Duraon of Maxmum Drawdown 223.0 393.0 770.0 203.0 73
Table 8. Sascal mercs performance for he radng perod 0/0/20-30/2/205 In Sample Perod 0/0/200-3/2/200 Ou of Sample Perod 0/0/20-30/2/205 Algorhm MulVarae Kalman Fler Rollng (OLS) regresson Consan Hedge Rao S&P500 Cumulave Reurn % 8.346 4.694 5.063 62.094 Annual Reurn % 2.674 2.787 0.995 0.67 Sharpe Rao 2.643.408 0.532 0.706 Bea 0.003 0.002 0.040 Maxmum Drawdown % 8.55 2.070 3.373 20.889 Duraon of Maxmum Drawdown 25.0 43.0 255.0 203.0 Fgure 2. Cumulave reurn and drawdown dagram for mulvarae kalman fler algorhm for he radng perods sarng a 2007, 2008 and endng a 205. Comparng he mercs of each perod of sudy s clear he Mulvarae Kalman Fler algorhm gves he beer resuls among he oher and he Index S&P500. In he frs perod where he Index presens he hgher Maxmum Drawdown equal o 62% and cumulave reurns (CR) equal o 45.59% (Annualzed percenage APR = 4.26%) he Mulvarae Kalman fler algorhm (MKFA) gves he hghes cumulave rae resul = 387% (wh APR = 9.26%) wh a reasonable Maxmum Drawdown (MDD) = 2.64 and duraon = 73 days. The MKFA n oher cases shows max CR = 278% (APR = 20.9%) and Sharpe Rao = 3.039 whle n radng perod 2008-205 presen he hgher MDD = 28.63 wh duraon = 75 days. A he same perod he Index presens MDD = 52.9% wh duraon = 54 days. The prof from MKFA gven by CR = 72% (APR = 3.32) and on he oher hand S&P500 has CR = 42.47% (APR = 4.52%). In hs perod he MKFA has he lowes SR =.3 bu sll hgher han Index. The lower CR of MKFA gven n he perod 20-205 and equal o 8.34% (APR = 2.67%) wh MDD = 8.5% and duraon = 25 days. In hs radng perod he ndex has CR = 62% (APR = 0.6%) bu 74
Fgure 3. Cumulave reurn and drawdown dagram for mulvarae kalman fler algorhm for he radng perods sarng a 2009, 200, 20 and endng a 205. Fgure 4. Cumulave reurn of rollng ols regresson algorhm agans S&P500 and algorhm wh consan hedge rao, for radng perod sarng AT 2007, 2008, 2009 and endng a 205. 75
Fgure 5. Cumulave reurn of rollng OLS regresson algorhm agans S&P500 and algorhm wh consan hedge rao, for radng perod sarng a 200, 20 and endng a 205. wh hgher MDD = 23.42 and duraon = 203. The SR of MKFA s sll hgher and equal o 2.33. In he las perods he profs declned bu kep hgher prof and Sharp Rao han S&P500. The second algorhm of rollng OLS regresson gves lower cumulave prof han Index bu has almos doubled Sharp Rao han S&P500 n all cases. Fnally he evaluaon of mercs has shown ha he MKFA can bea he marke as creaes sascal arbrage condon n Index. 6. Concluson Mean-reverng algorhms wh me adapve hedge rao are presened. A mehodology of a synhec asse consrucon based on he socks of S&P500 has been dscussed. The creron of he selecon was he conegraon relaonshp of ndvdual socks as he logarhmc sum of hem wh S&P500. The resuls of back esng show ha for dfferen perod of sudy he form and he dmenson of he synhec asse are dfferen. Par radng sraegy was adoped and he evaluaon of he mercs resuls presened beer behavor of MKFA among he ohers and bea he marke. In he las perods he profs declned bu was sll hgher han S&P500 wh much hgher Sharp Rao. The algorhm defended beer s prof as he Maxmum Draw down was que lower han Index. References [] Engle, R. and Granger, C. (987) Co-Inegraon and Error Correcon: Represenaon, Esmaon, and Tesng. Economerca, 55, 25-276. hp://dx.do.org/0.2307/93236 [2] Vdyamurhy, G. (2004) Pars Tradng, Quanave Mehods and Analyss. John Wley & Sons, Hoboken. [3] Infanno, L. and Izhak, S. (200) Developng Hgh-Frequency Eques Tradng Models. Maser of Busness Admnsraon, Massachuses Insue of Technology, Cambrdge. [4] Erne, C. (203) Algorhmc Tradng: Wnnng Sraeges and Ther Raonale. John Wley & Sons, Hoboken. [5] Caldera, J. and Moura, G.V. (203) Selecon of a Porfolo of Pars Based on Conegraon: A Sascal Arbrage Sraegy. hp://dx.do.org/0.239/ssrn.29639 76
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