Game-theoretic dynamic investment. information: futures contracts

Transcription

1 Game-heorec dynamc nvesmen model wh ncomplee nformaon: fuures conracs Oleg Malafeyev Shulga Andrey 2 San-Peersburg Sae Unversy Russa Absrac Over he pas few years he fuures marke has been successfully developng n he Norh-Wes regon Fuures markes are one of he mos effecve and lqud-vsble radng mechansms A large number of buyers are forced o compee wh each oher and rase her prces A large number of sellers make hem reduce prces hus he gap beween he prces of offers of buyers and sellers s reduced due o hgh compeon and hs s a good creron for he lqudy of he marke hs hgh degree of lqudy conrbued o he fac ha fuures radng ook such an mporan role n commerce and fnance A mul-sep non-cooperave n persons game s formalzed and suded Keywords: Fuures conracs goods he nal margn dynamc programmng marke sock exchange buyer seller oblgaon rsk broker Overvew Fuures are raded on specalzed fuures exchanges where far prcng s provded due o he hgh concenraon of supply and demand he fuure prces for gran meal gasolne he currency ha he fuures exchange generaes s a sure gude for any enerprse plannng s acvy n he marke condons he sysem of fnancal guaranees operang on he fuures exchange ensures uncondonal fulfllmen of oblgaons relably proecs he fuure prce of he goods malafeyevoa@malru headprvae@gmalcom

2 ha s why all over he world banks nvesmen funds wholesalers ndusral and agrculural enerprses come o he fuures exchange whch gves hem he opporuny no only o reduce commercal rsk bu also o make a prof when radng n fuures Fuures exchanges perform specfc funcons for he economy: ransfer rsk denfy prces ncrease lqudy and effcency ncrease he flow of nformaon Fuures allow us o agree oday on he prce a whch he purchase or sale wll be made n he fuure he bass s he ably o se oday he prce a whch he fuure wll be a purchase or sale A fuures conrac - s an agreemen beween wo pares o delver goods of a ceran quany and qualy n a ceran place and a a ceran me n he fuure a he prce agreed upon oday concluded accordng o he rules of he exchange he prce a whch a fuures conrac s concluded s deermned by free compeon among radng parcpans n he exchange's operang floor Each fuures conrac has wo sdes: he buyer and he seller he buyer of a fuures conrac s called a long-sakes pary and he seller s a pary wh a shor poson he buyer underakes o make a purchase a a predeermned me he seller underakes o sell a a predeermned me hese oblgaons are deermned by he name of he asse he sze of he asse he erm of he fuures and he prce agreed oday Only a whole number of fuures can parcpae n radng Durng he erm of he conrac s prce depends on he sae of he conuncure for he relevan produc Buyers benef from hgher prces as hey wll be able o receve he goods a a prce lower han he curren one Sellers benef from fallng prces as hey conraced a a prce hgher han he curren one Smulaneously wh he prce flucuaon he value of he conrac also changes For he holder of a long poson profs arse when prces rse whch ncreases he value of hs conrac he fall n prces and accordngly he decrease n he value of he conrac gves a prof o he holder of a shor poson he dfference n he value of a conrac for a long or shor poson s defned as he dfference beween he ransacon prce and he curren quoe mulpled by he conrac un Each holder of he shor and long posons of he conrac s oblgaed o provde hs broker wh a ceran amoun of money as a guaranee of he performance of he conrac hs depos s called a margn Is purpose s o proec he seller from non-fulfllmen of he conrac by he buyer f he prces have fallen and he buyer f prces have ncreased he broker uses margn o cover he losses of he clen f hs conrac suffers losses 2

3 In fuures radng here are wo ypes of margn: he nal margn - s he depos ha s made when openng he fuures poson; Varaon Margn - he ransfer of funds ensurng ha he cos of securng he new value of he conrac follows he change n prces he nal margn se n value erms s usually 2-0% of he value of he conrac In perods of hgh prce volaly and hgh rsk he exchange can se a margn a he upper lm of 0% and even hgher he margn s no he value of a fuures radng operaon he money ha he clen ransfers o a specal accoun s hs propery unl as a resul of some unsuccessful operaon he does no lose A he me of he expraon of he fuures conrac he margn currenly avalable n he accoun wh he broker s pad o he cusomer back Afer he openng of he poson and he nroducon of he nal margn change of he fuures conrac prce wll lead o a correspondng reducon or ncrease n he value of he clen's poson Afer deermnng he selemen prce of he day for each of he conracs and each poson he change n he value of each exchange conrac s calculaed based on he dfference beween he selemen prce of he gven day and he prevous day or he prce a whch he ransacon was concluded If he suaon on he fuures marke changes for some day n a favorable for he clen sde he amoun of money n he margn accoun ncreases by he sze of he poenal wn If hese prce changes are unfavorable for he clen's poson hen s nal margn decreases As soon as he amoun of he cusomer's nal margn has been reduced o he level of varaon margn addonal funds may be requred from he clen hs requremen s caused by unfavorable changes n fuures conrac prces hus he margn shows he profs and losses of he cusomer for he day hose funds ha exceed he requred amoun can be whdrawn by he clen bu more ofen hey reman on he accoun wh he broker as a reserve Real goods can be sold and bough n wo separae bu relaed markes - cash and fuures he prces of he cash and fuures markes are very closely relaed Fuures markes have a sold foundaon n realy he cash marke - s he place where he goods change he owner for a ceran prce Realzaon of ransacons n he cash marke s he purchase and sale of cash a curren prces he delvery of whch s carred ou mmedaely or whn a few days afer he concluson of he ransacon Cash prces - s he prces for whch he goods are sold n dfferen pars of he marke A each momen here are many cash prces de- 3

4 pendng on he qualy of he goods he place of delvery he processng sage of he goods and oher facors In he condons of he fuures marke s a queson of he fuure delvery of he goods n ceran erms In conras o he cash marke he fuures marke has only one se of prces he fuures prce s he curren vew of he marke on how much he commody of a ceran qualy wll cos wh ceran delvery condons a some pon n he fuure me and expecaons of marke parcpans - hese are he wo facors ha deermne he dfference n prces n he cash and fuures markes Boh markes exs n parallel bu a he expraon of each perod hey seem o merge desroyng he exsng prce dfferences A he me of delvery n he fuures and cash markes asse prces wll be he same because of quoaons for he mmedae delvery of he asse n boh markes hs concdence a one pon s called convergence he parallel movemen of hese markes s due o he fac ha facors leadng o hgher prces affec fuures prces n he same drecon For he dfference beween cash and fuures prces here s he concep of bass he bass s he dfference beween he prce of he goods n he cash marke n a parcular place and a he same prce as he commody n he fuures marke he bass s calculaed by subracng he prce n he fuures marke from he cash prce whch usually means he closes fuures monh (perod): Cash prce - Fuures prce Bass he bass s posve negave and zero If he bass s posve hen he prce of he cash marke exceeds he fuures prce or comes wh a premum o he fuures prce If he bass s negave he cash prce s lower han he fuures prce or comes a a dscoun o he fuures prce If he bass s zero he cash prce s equal o he fuures prce here can be many bases for one produc a one me because f here s only one prce of a fuures conrac for a ceran produc hen here are a lo of cash prces for hs produc dependng on he qualy and he place of delvery he bass s no a consan value Snce he prces of he cash and fuures markes are consanly changng he bass becomes wder or narrower he facors ha nfluence he sze of he bass are que a lo: hey are he demand and supply for a ceran perod he volume of he sock of producon he forecas for he producon of goods n he curren year he supply and demand for smlar producs he expor and mpor of goods he avalably of warehouses for sorage moves and a number of ohers 4

5 here are wo ypes of dynamcs of he bass: Srenghenng he bass (narrowng) s a change n he bass a whch he cash prce rses whn a ceran perod of me relave o he fuures prce Weakenng of he bass (expanson) - occurs when he cash prce s lowered relave o he fuures prce for a ceran perod of me and becomes less sable han he fuures prce he ype of dynamcs of he bass mos favorable for a parcular marke parcpan depends on wheher he s a seller or he buyer he seller wns wh a srong base and he buyer wns wh a weak base Fuures can be used n varous suaons: o avod rsk or o generae hgh reurns wh hgh rsk Fuures radng can be boh very rsky and very profable Fuures radng nvolves hedgers speculaors and arbrageurs he man goal of a hedger s o reduce he percenage of rsk he speculaor s lookng for hgh reurns due o hgh rsk he purpose of he arbrage s ncome whou rsk due o marke nconssences On fuures exchanges all ransacons are prmarly speculave n naure or are made for he purpose of nsurance agans prce rsks Speculave ransacons are made on he sock exchange n order o oban prof from he purchase and sale of exchange conracs as a resul of he dfference beween he prce of he exchange conrac on he day of mprsonmen and he prce on he day of s execuon wh a favorable prce change for one of he pares (he seller and he buyer) Moreover exchange speculaon s a mechansm for denfyng and sablzng prces for goods Fuures conracs are also used n hedgng hs s an operaon o nsure prce rsk by radng fuures conracs he prncple of nsurance here s based on he fac ha f n a ransacon one sde loses as a seller of a real commody hen wns as a fuures buyer for he same quany of goods and vce versa he mechansm of hedgng s based on he fac ha he change n marke prces for fuures are he same n her sze and drecon here are also arbrage ransacons ha are made for he purpose of makng a prof due o he dfference n quoaons on sock exchanges n dfferen counres Havng sold (bough) he fuures conrac he parcpan opens a poson n he fuures marke Close hs poson he can eher by execung a conrac or by buyng (sellng) he same conrac he sale and purchase of a conrac of he same ype compensae each oher 5

6 and are no aken no accoun n he calculaons beween he parcpans If he fuures conrac sold on he exchange s no compensaed by he purchase of he same conrac hen wh he onse of he monh of execuon mus delver a sandard produc he quany and qualy of he goods suppled he amoun of paymen me place and oher condons of delvery are srcly regulaed by he rules Every day cusomers' accouns reflec changes ha have occurred n he value of conracs ha hey have opened If he cusomer has a long poson and he prces have ncreased hs funds wll ncrease as he resul for open posons ncreases he gan wll be he dfference n he value of hs open conrac Excess funds can be ransferred by he clen or used by hm for openng new posons In he fuures marke he ransacon parcpan conrols s nvesmen of capal wh less money han n oher markes On he las day of he erm of he fuures conrac he amoun of coss s zero 2 Problem saemen Mahemacal descrpon of he model In hs paper a compeve model of decson-makng n he fuures conrac marke s consdered Agens who own he nal capal come o he fuures conracs marke and conclude conracs n order o oban maxmum prof he acons of agens are carred ou a specfc mes I s requred o choose agens conrol so ha he effec of nvesng n fuures conracs s maxmzed As a mehod of soluon we wll use he mehod of compromse soluon Le here be n agens on he marke n hey carry ou her acves hrough brokerage offces payng hem a commsson fee for her servces (we wll assume ha he commssons are precsely defned for each conrac) Le here be s conracs n crculaon s acng on he me nerval [0] Consder he paron of he nerval 0 < 2 < < f A me k k f he marke sae s descrbed by he vecor of fuures prces x ( x x x ) s 6

7 where k x f s he fuures prce h conrac a he me We wll assume ha a every momen k k agens have complee nformaon abou he hsory of he process A he me he agen Conrol eners no a se of fuures conracs u ( u u u 2 u s ) n [0] s called agen Conrol a me he se of agen conrols U ( u u n k k k f ) s called he conrol of he marke a me k k he prces of fuures conracs change a each me k k f dependng on he acons of agens on he suaon prevalng n he marke and from oher possble conrol momens he sysem goes from one sae o anoher dependng on he prevous quoes and from he conrols chosen by he agens accordng o he rule: ( ) f + x where s he ranson operaor whch s consruced on he bass of sascs colleced over a suffcenly long perod of me For smplcy we assume ha he erm of he fuures conrac whch was concluded a he me expres a he nex momen of me + A he end of each perod of me all fuures conracs are seled new fuures conracs prces are deermned and he revenue (losses) ha each agen receves from her posons are calculaed he dfference n he value of a conrac for a long or shor poson s defned as he dfference beween he ransacon prce and he curren quoaon mulpled by he conrac un ha s he change n he value of he conrac s equal o he change n he prce mulpled by he value of he conrac If has a posve value hen gves a prof o he sde ownng a long poson n he conrac and f negave hen he prof o he pary ownng he shor poson n he conrac If he agen has several conracs he oal resul s deermned by mulplyng by he number of conracs x U Le's descrbe n deal he decson-makng process by he agen a he me k f : k We use he followng noaon: x hs s he fuures prce of he -h conrac a he me k 7

8 k K f ; free - h agen capal by he me ; y he cash prce of he commody raded n he -h conrac a he me k k f ; q he quany of he goods esablshed n he -h conrac; m he margn esablshed for he -h conrac; p commsson fee esablshed for he -h conrac; s r he number of fuures conracs concluded a he r-h sep W s r s r ; ncome of he -h agen a me A me he agen k k f owns he sarng capal decdes on he concluson of fuures conracs by number makng coss n he form of commsson p and margn I s assumed ha he resrcon s fulflled: 0 < ( m + p ) By he end of he frs perod hs ncome s 2 A me he amoun of free capal changes and becomes equal o 2 K W 2 W s K - ( m + ) p s m K He s whle K s Fuures ha have been concluded earler 2 already expre and brng n revenue ( m + ( x x ) q ) afer whch he agen decdes o ener new fuures conracs n he amoun s s2 s 2 whle makng he coss of her concluson end of he second perod hs ncome s where 2 W 2 K s + s2 2 ( + ( x x ) q ) q - s he value of he conrac ( m + p m ( m + p ) purchase or sale In hs case he followng resrcon s assumed: ) By he { -} characerzes he 2 W 0 be equal o p A me p K wll expre and yeld revenue p he amoun of free capal wll change and wll p W Conracs ha were concluded a me p s p p ( m + ( x x ) q ) hen he p 8

9 agen decdes o ener new fuures conracs wh a number whle generang coss n he form of commssons and margns Hs ncome by he end of perod p wll be equal o s p p W p K s p s p p p ( + ( x x ) q ) + m ( m + p ) In hs case he followng resrcon s assumed: And so on p W 0 A he las momen of me he amoun of free capal wll change and become equal o K W Conracs ha were concluded a he me s - expre and brng n revenue - ( m + ( x x ) q ) A he me new conracs are no concluded so he agen's ncome by he end of he las perod wll amoun o W K s + ( m + ( x x ) q ) I s hs magnude ha should maxmze each agen Noe: A he begnnng of each perod each agen owns a ceran amoun of free capal Under free capal we wll assume he cash avalable a he momen a he agen's dsposal and he real goods ha he agen owns for hs perod expressed n money erms n accordance wh he prces of he cash marke a a gven me 3 Descrpon of he problem n erms of dynamc programmng Deermnsc case As a mehod of opmzaon we wll use he mehod of dynamc programmng Le's consder he case when here s no uncerany ha s when all he daa s deermned exacly and he ranson of he sysem from one sae o anoher s carred ou wh probably equal o one Agens make decsons a dscree nsans of me k k f herefore we can break hs process no f a sage Le us ake as he nal sae of he sysem he free capal of he -h agen a he me: 0: K n he free capal of agens a he 0 9

10 end of each perod K K K W wll be reaed as sysem saes a dscree momens of me 2 n k k f he effcency of he process wll be characerzed by ncome W p p receved by each agen s p s p p p K + ( + ( x x ) q ) for p frs perods of me Effcency s expressed by a funcon S U ) W We wll maxmze hs funcon m ( m + p ) n 0 ( ( U ) K 2 2 We nroduce he funcon ( k r S ( U )) denong he ncome receved by he agen for k frs seps under he opmal polcy 2 he funcons ( k r k S ( U )) k sasfy he funconal equaons of he form max k f k r 2 3 ( k S ( U )) { ( k Where k k k S S S u k U k k 2 k k S ( U )) + S } an ncrease n he ncome of he agen over a perod of me [ k k ] and he se s he se of admssble conrols a hs sep whch s deermned by he followng resrcons S ( U ) 0 U k he equaons gven above are he basc funconal equaons of dynamc programmng Le a he nal nsan of me he oal ncome of he agen be W 0 0 S 2 consequenly he funcons ( k r k S ( U )) for k f assume he form k r 2 k ( S ( U )) W k max u k U k As a resul of applyng he dynamc programmng mehod knowng he values of he nal capal we oban a sequence of 2 funcons: { ( k r S ( U )) } he funcon of he maxmum ncome k n k f and he correspondng opmal conrols u ( u u 2 u ) n [0] s 0

11 4 Sochasc case Descrpon of he game A mul-sep non-cooperave game of n persons s beng consruced { } > < I { n} n { n U } W where a se of sraeges of he agen : U - s he agen's payoff funcon U W I { n}- s he se of agens U R o solve he problem we wll use he prncple of a compromse soluon Defnon: Le be X : X U W U he se of profles n a game Г R M n max n he payoff funcon of agens W ( x) x X hen he compromse se s defned as follows: : C x X : ( M W ( x)) ( M H { max max W ( x)) x X} he algorhm for fndng a compromse se: ) Calculae he wnnng funcons for all players from each game profle 2) Fnd he maxmum value of he wn of each player for all game profles: M max W ( x) x X 3) For each profle x X calculae he devaon of he payoff funcon of he -h agen W (x) from he maxmum 4) For each profle x from X we fnd he maxmum devaon of he dfference M W (x) ha s we compue ( M W ( x)) max 5) On he se of profles X we fnd a pon x ha delvers a mnmum o he expresson ( M W ( x)) ha s we fnd he profle x : max mn max ( x X M n W ( x)) M n M W ( x) he profle n whch he mnmum s reached and wll be a compromse pon for all players n

12 Snce all compromse profles are equvalen n he sense of hs prncple of opmaly ha s each of hem gves he guaraneed wn of he leas sasfed player hen we wll choose he sraegy ha corresponds o he larges of he wnnng funcons U hus provded ha all agens have chosen specfc conrols he guaraneed ncome of he agen a he las sep s equal o he value s mn max x X K + ( ( m + ( x ( U ) x Usng he mehod of dynamc programmng he guaraneed ncome a he fnal pon n me wh fxed member conrol s calculaed Agen U conrol a each sep s seleced n accordance wh he prncple of a compromse soluon hen a each sep a sngle conrol s seleced by he mehod of backward procedures resulng n a sngle process raecory ( U U 2 )) q )) 5 Acknowledgemens he work s parly suppored by work RFBR No References [] Malafeyev OA GV Alferov AS Malseva Game-heorec model of nspecon by an-corrupon group AIP Conference Proceedngs (205) 648 hp://dxdoorg/0063/ [2] Malafeyev OA VN Kolokolsov Mean-Feld-Game Model of Corrupon Dynamc Games and Applcaons (205) hp://dxdoorg/0007/s x [3] Malafeyev OA Naural merc and pons of equlbrum n noncooperave games Vesnk Lenngradskogo Unversea Serya Maemaka Mekhanka Asronomya 4 (978) [4] Malafeyev OA Dynamc games wh dependen moons Doklady akadem nauk SSSR 23 (973)

13 [5] Avrachenkov K Negla G Elas J Margnon F Perosyan L A Nash Barganng Soluon for Cooperave Nework Formaon Games // Lecure noes n compuer scence 20 p [6] Gubko MV Conrol of organzaonal sysems wh nework agens neracon // Avomaka elemekhanka 2004 ¹8 p 5 23 ¹ 9 p 3 48 [7] Jackson MO Socal and Economc Neworks Prnceon Unversy Press 2008 [8] Jackson MO Wolnsky A A Sraegc Model of Socal and Economcs Neworks //J Econom heory 996 ¹ 7 P44 74 [9] Malafeev OA Conrolled Sysems of Conflc // Sank-Peerburg Izdaelsvo SPbGU pp [0] Perosyan LA Sedakov AA One-Way Flow wo-sage Nework Games // Vesnk Sank-Peerburgskogo unversea Serya 0 Prkladnaya maemaka nformaka prosessy upravlenya No pp 72 8 [] OA Malafeyev Dynamcal processes of conflc S Peersburg Sae Unversy San-Peersburg [2] F LChernousko NNBolonk VGGradesky Manpulaon robos: dynamcs conrol opmzaon Nauka (n Russan) [3] OA Malafeyev ND Rednskkh GV Alferov Elecrc crcus analoges n economcs modelng: Corrupon neworks Proceedngs of 2nd Inernaonal Conference on Emsson Elecroncs (204) hp://dxdoorg/009/emsson [4] OA Malafeyev LA Perosyan Dfferenal search games - Dynamcgames wh complee nformaon Vesnk Lenngradskogo Unversea Serya Maemaka Mekhanka Asronomya 2 (983) [5] OA Malafeev Exsence of equlbrum pons n dfferenal noncooperave many-person games Vesnk Lenngradskogo Unversea Serya Maemaka Mekhanka Asronomya 3 (982) [6] OA Malafeev Essenal non-cooperave n-person games Vesnk Lenngradskogo Unversea Serya Maemaka Mekhanka Asronomya (978)

14 [7] X Grgoreva O Malafeev A compeve many-perod posman problem wh varyng parameers Appled Mahemacal Scences 204 vol 8 No p [8] Malafeev OA Kolokolsov VN Undersandng game heory World Scenfc Publshng Co New Jersey p [9] Ia Arel ransfer Implemenaon n Congeson Games Dscusson Paper No 9-4 Ocober 204 pp7 [20] Malafeyev OA Kolokolsov VN Mean feld game model of corrupon Dynamc Games and Applcaons S 34 [2] Drozdov GD Malafeyev OA Nemnyugn SA Mulcomponen dynamcs of compeeve sngle-secor developmen In proc: 205 Inernaonal Conference "Sably and Conrol Processes" n Memory of VI Zubov (SCP) 205 S [22] Malafeyev OA Rednskkh ND Sohasc analyss of he dynamcs of corrup hybrd neworks In proc: 206 Inernaonal Conference Sably and Oscllaons of Nonlnear Conrol Sysems (Pyansky's Conference 206) 206 S [23] Pchugn YA Malafeyev OA Sascal esmaon of corrupon ndcaors n he frm Appled Mahemacal Scences 206 V p [24] Parfenov A Malafeyev O 2007 Equlbrum and Compromse Managemen n Nework Models of Mul-Agen Ineracon Problems of mechancs and conrol: Nonlnear dynamcal sysems 39 p [25] Rose-Ackerman S 978 Corrupon a Sudy n Polcal Economy Academc Press [26] Rose-Ackerman S 999 Corrupon and governmen Causes Consequences and Reform Cambrdge Unversy Press p 356 [27] Malafeyev O Sosnna V 2007 Model of Managng he Process of Cooperave hreeagen Ineracon Problems of mechancs and conrol: Nonlnear dynamcal sysems 39 p 3-44 [28] Bernhem B Douglas and MD Whnson 990 Mulmarke Conac and Colluskve Behavor RAND Journal of Economcs 2 () 4