Enropy 03, 5, 700-70; doi:0.3390/e500700 Aricle OPEN ACCESS enropy ISSN 099-4300 www.mdpi.com/journal/enropy Sudy on he Sabiliy of an Arificial Sock Opion Marke Based on Bidirecional Conducion Hai-Jun Yang * and Gui-Ping Sun School of Economics and Managemen, Beijing Universiy of Aeronauics & Asronauics, No. 37Xueyuan Road, Haidian Disric, Beijing 009, China; E-Mail: sgping@63.com * Auhor o whom correspondence should be addressed; E-Mail: navy@buaa.edu.cn; Tel./Fax: +86-0-833-807. Received: 9 November 0; in revised form: 8 February 03 / Acceped: 8 February 03 / Published: February 03 Absrac: Alhough sock opion markes have grown dramaically over he pas several decades, he relaion beween an opion and is underlying asse, especially bidirecional conducion, is no paricularly clear. So far, here have been many debaes abou his opic. We ry o invesigae his problem from a novel angle: an arificial sock marke including a sock opion is consruced in his paper. The model includes wo pars, one is a sock rade module based on he Sana Fe Insiue Arificial Sock Marke (SFI-ASM), and he oher is an opion rade module. In he laer module, hree ypes of opion raders are employed. The resuls show ha he model is effecive, and experimens illusrae ha opion markes have a remarkable effec on sock markes. Furhermore, by appending opions, he model replicaes some sylized properies, such as volailiy clusering and GARCH effec, which can be observed in real financial markes. Keywords: arificial financial marke; bidirecional conducion; sabiliy; informaion diffusion; opion. Inroducion In he pas few decades, here has been an explosive growh on derivaive securiies in financial markes. The impac of his growh on he World economy is immense and can influence he sabiliy of financial markes. I is clear ha here has been a massive increase in boh he ypes of financial derivaives available for rade and he size of ransacions involving derivaives. Financial innovaions
Enropy 03, 5 70 have caused more and more concerns, especially afer he 007 subprime crisis. Many people aribue he cause of he financial crisis o financial innovaions which were perceived as being ou of conrol. However, ohers ake he opposie view; hey hink ha financial derivaives can sabilize financial markes. This disagreemen emanaes from he unclear mechanism of acion beween financial derivaives and heir underlying asses. Opions are an imporan componen in a wide variey of financial derivaives. Thus, he opion will be aken as research subjec in his paper. We focus on he informaion flow in financial markes and handle i as a coninuous diffusion process among marke paricipans beween sock marke and opion marke, where he informaion diffusion is modeled by price and volume. Furhermore, he wo financial markes can be considered as a simple nework wih bidirecional informaion diffusion. The original Black and Scholes model [] was proposed o value European opions. In 973, he Chicago Board Opions Exchange (CBOE) was founded and sandardized call opions were inroduced. The impac of he opion marke on he underlying sock marke is very complicaed and no clearly undersood. Some scholars have focused on his issue, bu heir conclusions were no unanimous. Some researchers found ha he inroducion of opions had a posiive effec on he underlying sock markes. For insance, Hakansson [] and Ross [3] found ha he inroducion of opions could improve financial marke compleeness because hey expanded he range of choices available o invesors. Those auhors also hough ha opions rading could reduce he volailiy of he underlying sock. John e al. [4] inroduced a model where agens were informed agens and found ha an incremenal public informaion made he underlying marke more efficien. Similarly, Kumar e al. [5] poined ou ha he opions markes creaed higher liquidiy and greaer pricing efficiency for he underlying socks. Many oher researchers held he similar viewpoins [6 8]. On he oher hand, some researchers hough ha opions could desabilize he underlying marke and end o increase sock price volailiy. Heer e al. [9] showed ha afer he inroducion of opions he variance of he sock reurn increased. Wei e al. [0] showed ha opions increased he volailiy of OTC socks. Besides hese wo viewpoins, many oher researchers claimed ha he inroducion of opions did no direcly and significanly affec he underlying marke. For example, Bollen [] affirmed ha he inroducion of opions did no significanly affec sock reurn variance. Kabir [] also sudied he impac of opion inroducion. He poined ou ha he inroducion of opions resuled in a significan decline in sock price, bu had no significan effec on volailiy. All resuls were based on radiional mehods. In recen years, a new approach-an agen based model-for sudying financial markes has appeared. Agen-based compuaional economic and financial modelling is compleely differen from convenional economic modelling. For an agen-based model, he basic premise changes from a classical represenaive, raional agens o behavioral, boundedly raional and heerogeneous agens who use experience o forecas he economic siuaion. Currenly he Sana Fe Insiue Arificial Sock Marke and Genoa Arificial Sock Marke (GASM) models are wo of he mos popular agen-based models. Many researchers have used hese wo models o sudy many economic and financial issues [3 5]. However, here is very lile research using hese models o examine he effec of opion markes on sock markes. Ecca e al. [6] presened he firs sudy on he effecs of an opion marke relaed o an underlying sock marke, using an arificial financial marke based on heerogeneous agens invesigaed a realisic European opion by wo marke models. Their resuls showed ha he inroducion of opions, in he
Enropy 03, 5 70 proposed models, ended o decrease he volailiy of he underlying sock price. Moreover, he raders wealh can be srongly affeced by using opion o hedge risk. However, he above research sill has shorcomings. For example, he socks used in Ecca e al. s paper did no pay dividends, unlike many socks ha pay periodic dividends in real sock markes. Besides, he opion raders are only allowed o buy opions from he marke maker who has infinie wealh. In order o overcome he above shorcomings, we will use SFI-ASM o build an opion rade marke and inroduce some rading mechanisms ino he model o more closely replicae acual opions markes. This paper focuses on he effec of opion markes on sock markes. The res of he chapers are organized as follows: Secion is he inroducion of our model and he secion is mainly divided ino hree pars. Secion 3 repors he resuls of our experimens. Finally, our conclusions and recommendaions of fuure works are summarized in Secion 4.. The Model In his secion we focus on he new model which includes wo modules. Firs, he sock rade module will be inroduced and is limiaions will be poined ou. Then, he compound Poisson process will be employed o consruc a dividend process. Finally, he opion rade module will be presened... Sock Trade Module and Is Limiaions... The Inroducion of he Sock Trade Module This model consiss of wo pars, he firs par is a sock rade module, and he second par is a sock opion rade module. The following is he inroducion of he sock rade module which is based on SFI-ASM. In his module here are wo asses. One is a sock and he oher one is a riskless asse. In every period he riskless asse holders receive an ineres paymen and ineres rae denoed by r f = 0.. Similarly, sock holders can receive sock dividend which is generaed by a saionary sochasic process and he equaion of dividend process is denoed as Equaion (): d d ( ) d () where ~ N (0, e ), he expeced value of dividend, E( d ) d (0.0), he variance of dividend, D ( d ) e ( ) (0.074), he correlaion coefficien of dividend, d d (0.95). W The agen uiliy funcion is consan absolue risk aversion (CARA) U ( W ) e (γ is risk aversion parameer and equals o 0.5), he agen will maximize he expeced uiliy funcion in he nex period: Subjec o (here are only one sock and one riskless asse): max E [ U ( W )] () W x, ( p, d, ) ( rf )( W x, p, ) (3) Solving Equaion (), we can ge he demand of sock of each agen in he curren ime period:
Enropy 03, 5 703 where: x, [ E ( p, d, ) ( r f ) p, ] Var ( p d ) (4),, E ( p, d, ) a( p d ) b In he module every agen has 00 forecas rules wih differen a and b values, so hey can forecas he expeced value of ( p d ) by Equaion (5). When agens make predicions abou he fuure price and dividend, hey choose he proper forecas rule from he rule se according o previous sock informaion which includes he ime series of sock price and sock dividends. In he forecas rule se for he agen, rules wih bad predicion will be replaced by new forecas rules which are generaed by a Geneic Algorihm (GA) a regular ime inervals [7,8]. This ime inerval represens he learning speed and he reacion speed of agens o marke changes. In our model he ime inerval equals 50, because he major purpose of his paper is o sudy he effec of opion markes on sock markes, so he learning speeds of all he agens are same. Var ( p, d, ) is he condiional variance of he agen s forecas, in he module, i is updaed a he end of each loop: Var ( p, d, ) bvvar ( p, d, ) av( p, d, E ( p, d, )) (6) Where av and bv are weigh coefficiens. The sock price is given by a marke maker who gahers supply and demand informaion of all he agens and consanly adjuss he demand and supply o achieve equilibrium. The d has been deermined before he p is given, from he Equaions (4) and (5) he change of p can causes he x, o change. In he module i is imporan ha if agens are idenical wih parameers, a homogeneous linear raional expecaion equilibrium (REE) exis, a he momen he sock price is a linear funcion of dividend: where: p d e (5) fd e (7) f ( r ) f ( )( ) pd r f f ( f ) D( d ) (0) pd For he REE, a and b of Equaion (5) can be denoed as follows: (8) (9) a () b ( )( f ) d e ()... The Limiaions of he Sock Trade Module and he Counermeasure This paper mainly researches he ineracion beween an opions marke and is underlying asse marke. In he model he mechanism hrough which he sock marke impacs he opion marke is obvious because he sock price direcly influences he opions pricing. However he way in which he
Enropy 03, 5 704 opion marke impacs he sock marke is no clear. During he consrucion of model and hrough many experimens we find he limiaions of he sock rade module. Equaion (4) shows ha he only variable which deermines he sock demand is ha sock demand is unrelaed o agen wealh p. So we presume x in he previous W and he sock holding period. Then, we will verify he assumpion by he experimens. There are wo groups of experimens. Firs, he influence of he agen s wealh will be esed by he change of he iniial cash of agen which is he same o all he agens when he experimen begins. Second, he influence of he agen s sock holding will be esed hrough he sock random swap among he agens. Resuls of our experimens are given in Table. Table. The saisics of sock price under differen iniial cash scenarios. Iniial cash 0 00,000 0,000 Average 98.47 98.4 98.7 98.7 Variance 3.34 3.48 3.45 3.45 In his paper he ineracion effecs beween he sock marke and he opion marke are mainly researched hrough he changes in he ime series of he sock price. The lengh of daa used in Table is 50,000 periods and he daa is adoped afer he model sars o sabilize. Table shows ha he change of iniial cash does no obviously cause a change in sock price. Nex, we performed he second experimen for he sock holdings of agens redisribued randomly a regular inervals. Our purpose is o observe he changes of sock price. In he experimen, he ime inerval equals 00 periods. The ime of model running is more han 00,000 period and experimen akes he 00,000h period as a ransiion poin; before 00,000h period he sock holdings are no arificially redisribued, and afer 00,000h period he redisribuion is carried ou. The resuls of experimens are given in Figure and Table. Figure. The price ime series of sock (,000h is ransiion poin). 0 05 Price 00 real risk neural 95 90 50 00 50 00 Time Table. The saisics of sock price before and afer he sock holdings are redisribued. Before Afer Average 98.7 98.4 Variance 3.45 3.54
Enropy 03, 5 705 Figure shows ha wheher an agen sock holding is redisribued or no, he basic sae of he REE does no change. Table confirms his resul and shows ha he saisics of sock price change are very lile. Therefore, we conclude ha he redisribuion of sock holdings will no change sock price. The above resuls confirmed our hypohesis ha sock price is unrelaed o iniial cash and disribuion of sock holdings. Therefore, we can make he wo variables consans in laer experimens. The proposed model in his paper includes wo pars, a sock rade module and an opion rade module. They are no synchronized and he sock is raded before he sock opion is raded in every period. When opions are exercised, his resuls in he redisribuion of wealh and sock holdings among all agens, bu according o he above analysis he wo variables canno influence he sock pricing in he nex period. The effec of he opion marke on he sock marke is a very small unidirecional ransmission, so we need o consruc heir double-way relaionship from anoher perspecive. We will research his problem from an informaion angle because he informaion plays a significan role for asse pricing. This idea can be suppored by some empirical resuls [9 ]. As menioned above, during he decision-making process agens use informaion including boh sock price and sock dividend. The price and volume informaion of opion marke will be used o influence he agen s decision-making and consruc he effec of opion marke on he sock marke. This is accomplished by he addiion of opion informaion ino Equaion (5): E ( p, d, ) ( a )( p d ) b (3) where synhesizes he informaion of he opion marke, which includes opion prices and opion rade volumes in muli-periods. In he opion rade module wo opions are inroduced, call opions and pu opions for he same underlying asse: (... s s ) (4) where is a weigh coefficien and greaer han zero, which represens he sock marke impac of opion marke, he bigger he value of, he bigger he effec is.,,..., are also weigh s coefficiens of informaion in differen periods and... s, he subscrip S is he lengh of informaion used in he model: 0.5 -, 0.5, 0.5,3 0. 5,4 (5) where -, and -, denoe he price informaion and he volume informaion of call opions, -,3 and -,4 denoe he price informaion and he volume informaion of pu opions, respecively: x, i x, i and x, i x 3, i -, i x, i x, i and x, i x 3, i i,,3,4 (6) 0 ohers where x, i ( i,,3,4 ) denoes call opion price, call opion volume, pu opion price and pu opion volume, respecively. The imporan role of he opion marke is price discovery. The above definiion implies ha if call opion price or volume has a sequenial wo-period increase, i will creae a posiive expecaion for he sock marke, which can herald an increase of sock price and vice versa. The pu opion marke is
Enropy 03, 5 706 similar o i, bu he effec is he opposie. Tha is o say, Equaions (3) (6) describe an informaion diffusion process from an opion o is underlying asse... The Compound Poisson Process In he real sock marke, sock prices will rapidly flucuae due o a variey of facors. The jumped process is inroduced in he dividend process of sock o make i closer o real financial markes. According o he Equaion (7), he change of dividend will ineviably be refleced in he sock price afer he arrival of he REE. On he basis of Equaion (), he new dividend process is generaed by a compound Poisson process. If he jump range equals o, a Poisson process occurs, i is denoed as N (), and is srengh N ( ) equals o 0. 0. So he compound Poisson process can be consruced as Q ( ) Y i, 0, where N () is he Poisson process and is srengh equals o, Y Y,... are i.i.d. normal random, Y3 variables. In his paper heir expeced values equal o 0 and variances equal o 0.75. The experimenal resuls show ha he model could reach a seady sae REE. The daa in Table 3 and Figure is run from 0,000 periods afer he sysem sars o sabilize. From he above daa we can find ha he change of average price is small, bu variances of he dividend and sock price vary widely, and he change of sock price is bigger han he dividend process. The experimens address ha afer he compound Poisson process is inroduced, muageniciy of sock price appears. Moreover, he corresponding parameers also need o be recompued in he sock rade module because he variance of dividend has changed. In our model, he sock rade module also adds he compound Poisson process. Table 3. The saisics of dividend and sock price. Old dividend process New dividend process Old sock price New sock price Average 0.0 0.0 97.85 96.59 Variance 0.074 0.463 5.05 5.3 Figure. The ime series of sock dividend (lef: old sock price, righ: new sock price). i 0 0 9 9 8 0 00 00 300 400 500 600 700 800 900 000 8 0 00 00 300 400 500 600 700 800 900 000.3. The Opion Trade Module In order o research he effec of sock opions on he sock price, we will append sock opions ino he model, and hen i will be inroduced in deail from several aspecs.
Enropy 03, 5 707.3.. The Seing of Opion In he paper he European opion is used, and according o he acual siuaion, he opion s lifeime equals hree monhs, because every monh has in pracice abou weny days in which socks can be raded. Therefore, he horizons equals o 60 days and he opion will be exercised afer 60 days. Here one day is defined as he ime lengh in which model runs once. In our model he pu opions and call opions are allowed o rade among he agens. The opion premium is compued by B-S Equaion wih sock dividend. Volailiy of he sock price is compued hrough he hisorical sock price series, he lengh of daa used equals o lifeime of opion, and volailiy will be recompued as he model moves forward. The srike price of he opion is fixed a he beginning of he opion issue. I equals o he sock price added or subraced o a value which is decided by he sock price. During he lifeime of he opion he srike price will no be changed. The srike price is se by X { p, p }.For a call opion X p and for a pu opion X p. The value of depends on he curren price p and equals o.5 in he paper..3.. The Types of Opion Traders The opion rade agens are divided ino hree ypes in he model. They are random opion rader, speculaion opion rader and hedge opion rader, respecively. Random opion rader represens he noise rader exising in he real marke. A noise rader makes irraional and erraic decisions o buy, sell, or hold opions. The presence of noise raders in financial markes can cause prices and risk levels o diverge from expeced levels even if all oher raders are raional []. In our model, random opion raders decide randomly o buy or sell opions and he ypes of opions and he size of opion conrac are also decided randomly. Hedge opion raders are agens who wan o cover sock opions hrough holding opion conracs. In he model hey can buy pu opions or sell call opions o lock marke risks. The hedge opion rade coninually adjuss heir holding opions o make heir sock posiion equal o he sum of all opion conracs unless he agens do no have sufficien cash o pay he opion premium and opion margins, so he sock risk can be covered. Bu here is an implici assumpion which is ha sock price remains invarian during he ime he opion is held. The following Equaion deermines he opion demand for he hedge opion rader a he curren ime: d s xi, (7) where d denoes he curren opion demand, s denoes he curren sock posiion, x i,- is he quaniy of differen ypes opion in he previous ime ( ) and if call opion was bough or pu opion was sold by agen he iem is negaive, if call opion was sold or pu opion was bough he iem is posiive, all iems are summed o obain he opion demand in he curren ime. Speculaion opion raders are enirely he opposie of hedge opion rader. These agens will hold opion conracs he same wih he holding sock posiion, so hey will earn addiional profi; neverheless, hey will also face more serious risk when he sock price flucuaes. The following Equaion is he opion demands: xi, d s (8)
Enropy 03, 5 708 The variables represen he same values as Equaion (7)..3.3. The Opion Marke Maker When he opion marke maker is inroduced o he model, i collecs he opion demands of all agens and accomplishes he opion rade. I also execues mark o marke a opion validiy period, and exercises opions a he expiring day. In our model he maker does no buy and sell opions o paricipae he opion rade, insead i only acs as a medium. () Opion rade mechanism The call opion and pu opion are simulaneously raded in he marke. Before he opions are raded, every ype of agens firs decided he rade demands of opion; hen hey se he price hey were willing o pay for he rade. As we know, he opion price can be compued by B-S Equaion in which he only uncerain variable is volailiy. Therefore, agens forecas sock volailiy o ge he opion price. The following equaion shows how each ype of opion raders forecas volailiy. h, c, ( c, h, 3 ) speculaion opion rader hedgeopion rader randomopion rader where is he sock volailiy used o compue he opion price a he curren ime, c, is he volailiy which is compued hrough pas sock price series and h, is he implied volailiy which is i compued by he las opion price. ~ N(0,). So as o embody he characerisics of differen kind of opion raders, is se. Equaion (9) reveals ha during he volailiy forecas he speculaion opion raders focus on long-erm consideraions and he hedge opion raders are more concerned wih he shor-erm changes of opion price, which is analogous o raders in real financial markes. Afer every opion rader ges he demand or supply and price of he opion, he marker couns up he demand and supply of every opion respecively, o ge he supply and demand curve of each opion. The opion price and volume are decided by inersecion of wo curves. This can be called Muli-Agen machmaking mechanism. Figure 3 is he schemaic of supply and demand curve. Figure 3 shows ha he rade volume of opion is abou.5 and he opion price is abou 3. () Mark o marke In accordance wih he real opion marke, a daily mark o marke sysem is incorporaed ino he model. Before he opions sar o rade, mark o marke is execued. When agens sell an opion hey will deposi margins in heir margin accoun which is in he marker. If agens buy he opion he margin is no necessary, hey only pay he opion premium. Here naked opions are used. (9)
Enropy 03, 5 709 Figure 3. The supply and demand curve of call opion. 0 8 Price 6 4 0 0 0.5.5.5 Volume If he cash in he agen s margin accoun is less han he required curren margin, he agen will be forced o deposi money in heir margin accoun. If agen does no have enough cash, is opion conrac will be execued in advance. If he cash of an agen margin accoun is more han he required curren margin, he agen is allowed o wihdraw exra money from agen margin accoun. The raio coefficien leverage raio used in margin compuaion is defined as a variable, so we can adjus he variable o sudy is effecs on he opion marke. (3) Exercise opion Marke makers exercise opions ha are in-he-money a expiraion. Unlike real opion markes, in our model he underlying share of opion is no exchanged upon expiraion. Insead, he opion is exercised hrough an exchange of cash. Afer he opions are exercised, some agens ge a profi and some agens ge a loss. The cash ransfer will lead o he redisribuion of wealh among agens. Thus, he nex period sock rade will be affeced and a change in feaures of sock marke will also occur. Similarly, he sock rade also affecs he opion rade hrough changes in he wealh and posiion of agens. There is a close wo-way relaionship beween sock marke and opion marke. Nex i will be esed by experimens. Figure 4 is he ime line of a marke day. In Secion, we have consruced a simple social nework which includes wo subneworks. One is sock marke which has agens wih differen wealh. Anoher one is opion marke which has hree differen kinds of invesors, speculaion opion rade, hedge opion rade and random opion rade. In he former subnework, an agen can be considered as a node of social nework, and he marke maker is a special node in he nework which guaranees oher nodes can receive and send informaion (price and volume of sock). Similarly, in he laer subnework, opion informaion is ransmied beween he differen ypes of raders by he marker maker. Furhermore, opion and sock informaion can be ransmied beween he wo subneworks. 3. Experimens In his secion we focus on he new model which includes wo modules. Firs, he sock rade module will be inroduced and is limiaions will be poined ou. Second, he compound Poisson process will be employed o describe he dividend process. Finally, he opion rade module will be presened.
Enropy 03, 5 70 Figure 4. The ime line of a marke day (he X-axis represens ime). According o every agen s opion conracs and he curren sock price he marking-o-marke process is execued by he opion marke maker. The opion marke maker uses ransacion mach mode o accomplish he opion rades. Sock rade is accomplished and he curren sock price is go, every agen sock posiion is obained. A he expiring day he opion will be exercise and he new parameer of opion will be se and urn o nex ime sock rade. The volailiy of sock price and he opion premium are compued. Every agen obains he demand of opion conrac according o respecive rading sraegy and submis he rade order o he opion marke maker. + The sock rade module The opion rade module 3.. Experimenal Parameers This secion mainly inroduces he various parameers used in he experimens. 00 agens were used in he model. In he sock rade module, he size of agen s forecas rules se is 00. Oher deailed main parameer values are in Table 4. Table 4. Parameer values of sock rade module. Parameer Parameer value Parameer Parameer value d 0.0 a 0.95 0.95 b 4.5053 D d ) 0.463 a range 0.7. ( f 6.3333 b range 0.945 9.7053 e 6.773 The main parameers used in he opion rade module are se ou in Table 5. In addiion o he parameers menioned above, oher parameers will be briefly inroduced. Opion sar ime is momen a which opion rading begins; he purpose of his parameer is o sar o run he opion module afer he sock price has become sable. Because when he model sars o run, he price of sock flucuaes wildly. Socks per conrac is he number of sock conained in per opion conrac.
Enropy 03, 5 7 Table 5. Parameer values of opion rade module. Parameer Parameer value Parameer Parameer value lifeime 60 Leverage Raio 0. Opion sar ime 0000 0.0 Socks per conrac.0 3.. The Sabiliy of he Model Before comparing in deail he differences beween he sock marke and he opion marke, we firs analyze he sabiliy of model wih opions. In he experimen afer he running period of he sock rade module equals 5,000 he opion rade module sars o run. The model is execued more han 00,000 periods wihou any inerrupion. Figure 5 is price ime series of sock from he 0h period o he 0,000h period. In Figure 5 real price represens he price obained hrough sock rade, risk neural price represens he price r obained by equaion ( p d / rf ). Figure 6 is he forecas error ime series of sock price of he firs agen from he 0h o he 0,000h period, he forecas errors of sock price of he oher agens have he same change endency. Figure 5. The price ime series of sock. 40 0 00 Price 80 60 40 0 0 5000 0000 5000 0000 Time risk neural real Figure 6. The forecas error ime series of sock price of he firs agen. 60 50 40 Error 30 0 0 0 5000 0000 5000 0000 Time
Enropy 03, 5 7 From Figures 5 and 6 we can see ha he sysem remains sable from he 5,000h period. When he sock opion sars o rade a he 0,000h one, he ime series has no obviously changed and he sysem coninues o remain sable. The forecas error becomes very small as he sock price ends o sabilize and afer 50,000h period he forecas error of every agen is less han.0. 3.3. The Effec of Opion Marke on Sock Price and Volume Nex we will analyze he effec of he opion marke on he sock marke. So as o compare direcly, he 00,000h period is aken as a ransiion poin. Before he 00,000h period, he opion rade is no included and afer 00,000h period he opion rade sars o run. Figure 7 illusraes he sock price ime series which are chosen from a long sable price sequence. From Figure 7, we can find he obvious difference beween he wo cases. Afer he opion is inroduced, he price volailiy becomes higher and he spread of he real price and he risk neural price becomes larger. The resul implicaes ha opion informaion (price and volume) impacs on he real price of sock. Tha is o say, he opion marke affecs he sock marke by informaion diffusion hrough he nework. Figure 7. The price ime series of sock. (lef: before he opion is inroduced; righ: afer he opion is inroduced). 0 0 0 0 Price 00 90 80 real risk neural Price 00 90 80 real risk neural 70 0 00 400 600 800 000 Time 70 0 00 400 600 800 000 Time From he saisical daa of Table 6, we can find he obvious difference beween he wo saes, especially in variances. From he sock price, we can find ha he average of sock price decreases afer he sock opion sars o rade. However, variance decreases more, which fis he fac ha he higher he income, he greaer he risk. A sock s yield is he dividend per share divided by is curren price per share. Before he opion is inroduced in he model, he sock average yield equals he average dividend divided by sock price. The value equals o.3% (0.0/89.8). Afer he opion is inroduced, he sock average yield is.48% (0.0/80.). Table 6. The saisics of dividend and sock price. Sock price wihou opion Sock price wih opion Trade volume wihou opion Trade volume wih opion Average 89.8 80. 4.8.96 Variance 0.40 0.3 30. 5.98
Enropy 03, 5 73 When average rading volume decreases, he variance also decreases, unlike he change in price. The increase of risk in owning he sock and he relaively unsable sock price causes he agens o end o hold cash. This decreases boh he variance of he rade volume and he average of rade volume. The preceding analysis illusraes ha he inroducion of sock opions canno sabilize marke price, so he variance of price increases. Therefore, he risk in owning sock becomes large, which leads o a decrease in he price of he sock. However, i should be poined ou ha our conclusion is drawn based only on our model. Before he opion is inroduced, he sock rade module has reached he REE, hough he compound Poisson process is inroduced. 3.4. The Effec of Opion Marke on Sock Reurn and Volailiy This secion mainly sudies he changes of he feaures of sock reurn series marke when he opion marke is inroduced. The feaures of sock reurn include volailiy persisence and fa ails or excess kurosis. A frequencies of less han one monh, he uncondiional reurns of financial series are no normally disribued in he real marke. They usually display a disribuion wih oo few in he mid-range, oo many observaions near he mean, and again, oo many in he exreme lef and righ ails. This feaure discovered by Mandelbro [3] has puzzled financial economiss. Recenly, reurn disribuion has obained more aenion in risk managemen because he compuaion of value a risk of sock needs relaive correc reurn disribuions. Sock reurns are normally calculaed by he following expression, r ln( p / p ), p is he curren sock price. Bu in our model he compuaion of sock reurn needs o be adjused because a dividend is paid each period, which is no common in he real sock marke. The sock dividend is also paid, bu ime inervals are far longer. In our model we use r ln(( p d ) / p ) o calculae he sock reurns; d denoes he curren sock dividend. Figure 8 is sock price reurn ime series which are chosen in a long sable price sequence. The sample size in Table 7 is 50,000. Figure 8. The ime series of sock logarihmic reurns. (lef: before he opion is inroduced; righ: afer he opion is inroduced). 0.5 0.5 0. 0. Sock reurn 0.5 0. Sock reurn 0.5 0. 0.05 0.05 0 0 00 400 600 800 000 Time 0 0 00 400 600 800 000 Time
Enropy 03, 5 74 Table 7. The saisics of dividend and sock logarihmic reurn. Sock reurn wihou opion Sock reurn wih opion Average 0.05579 0.0366 Variance 0.000346 0.000937 Kurosis.443 4.499 From Figure 8 and Table 7 we can find ha he difference beween he wo sock reurns obviously. Before he opion is inroduced, he sock reurns change genly as a whole, because of he inroducion of he compound Poisson process. There are some sudden changes in he sock reurn, which makes he kurosis become very large. Afer he opion is inroduced, he average sock reurn and volailiy increase, bu he kurosis also decreases. The shape of sock reurn is closer o he real sock reurn. Sock reurn disribuions show obvious fa ails or excess kurosis (kurosis > 3) wheher or no he sock opion is inroduced. The resuls show ha he excess kurosis and fa ail has nohing o do wih he inroducion of opions. In addiion, opion rading really has an impac on he reurn of sock. Conversely, he flucuaion of sock price can coninually affec he opion marke. Nex we will analyze he volailiy of he sock which is an imporan parameer in he sock marke. Volailiy canno be observed direcly, bu i has some imporan feaures such as he persisence of volailiy ha lacks a widely acceped explanaion. In order o analyze he volailiy, we use he condiional volailiy models which seem o be appropriae. The GARCH model was proposed iniially by Bollerslev [4]. The GARCH model largely consiss of wo pars, he mean equaion and condiional variance covariance equaions. The mean equaion used in he paper is ARMA model. The condiional variance covariance equaions used is he GARCH(,) model which has been proven o be an adequae represenaion for mos financial ime series by Lamoreux and Lasrpes [5]: ARMA( p, q) model: r r... pr p a a... qaq (0) GARCH(,) model: a, c a () where follows N (0, ), a follows N(0, ), c 0,,. We firsly build he ARMA model by using EVIEWS sofware measuremen model parameer esimaion and esing. The parameers of he ARMA model are as shown in Tables 8 and 9. Table 8. The parameers of ARMA model before opion is inroduced. Variable Coefficien Sd. Error -Saisic Prob. C 0.05579 5.8E 05 000.599 0.0000 AR() 0.38857 0.037.476 0.0000 MA() 0.359 0.049 9.4488 0.0000
Enropy 03, 5 75 Table 9. The parameers of ARMA model afer opion is inroduced. Variable Coefficien Sd. Error -Saisic Prob. C 0.986 0.000857 39.744 0.0000 AR().78534 0.086 5.778 0.0000 AR() 0.5085 0.03445.789 0.0000 AR(3) 0.696 0.068 36.865 0.0000 AR(4) 0.40955 0.004987 8.90 0.0000 MA().5385 0.056 5.50 0.0000 MA() 0.398 0.0303 7.078 0.0000 MA(3) 0.674 0.0836 3.6645 0.0000 In Figures 9 and 0, he sock reurn residual refers o he residual afer he ARMA is buil. From Figure 9 we can see ha he Serial correlaion of he residual doesn exis afer he ARMA is buil. Conversely, in Figure 0 he Serial correlaion of residual squares does exis, which shows ha he ARCH effec exis in he sock reurn series. To analyze his observaion objecively, we carry ou he saisical es of he ARCH effec. Like he resuls displayed in Figures 9 and 0, he saisical es resuls show ha here is an ARCH effec in he logarihmic reurn series and he resuls passed he significance es a he % level. Figure 9. The auocorrelaion of he sock reurn residual series. (lef: before he opion is inroduced; righ: afer he opion is inroduced)... 0.8 0.8 ACF 0.6 0.4 ACF 0.6 0.4 0. 0. 0-0. 0 5 0 5 0 Lag 0-0. 0 5 0 5 0 Lag Figure 0. The auocorrelaion of he sock reurn residual squared series. (lef: before he opion is inroduced; righ: afer he opion is inroduced)... 0.8 0.8 ACF 0.6 0.4 0. 0-0. 0 5 0 5 0 Lag ACF 0.6 0.4 0. 0 0 5 0 5 0 Lag
Enropy 03, 5 76 The specific parameers of he model are in Table 0 and all parameers pass he % significance level es. Table 0. The parameers of GARCH (,) model. Parameers c Before 0.00068 0.49756 0.00509 Afer.6E 05 0.058384 0.94057 From Table 0, we can see ha he sock s volailiy persisence is clearly demonsraed hrough he ARMA-GARCH(,) model, wheher he opion is inroduced or no. Ye he resuls also show ha afer he sock opion is inroduced, he parameers of ARMA-GARCH(,) model have changed. Because he purpose of he inroducion of GARCH model is o analyze reurn volailiy, we mainly focus on condiional variance covariance equaion. The sizes of he α and β coefficiens represen he srengh of volailiy persisence. Afer he opion is inroduced he α value decreases, bu he β increases more, so in general, reurn volailiy persisence becomes more obvious following he inroducion of he opion. Furhermore, more informaion has been ransmied from opion marke o sock marke which is he reason of increases of volailiy in sock marke. 3.5. The Effec of Various Informaion Lengh From Equaion (4) we can see ha he lengh of previous opion marke informaion is a variable. In his secion we will change his variable o sudy is effec on he sock marke. We carried ou a series of six experimens and he deailed parameers used in he experimens are as shown in Table. In he parameer seing for he laes informaion (-) of he opion marke he greaer weighing coefficien was given and he oher imes he weighing coefficiens are equal. The coefficien of he laes informaion coninues o decrease wih informaion accumulaion. Table. The weighing coefficiens of informaion a differen imes in he model. Weighing coefficiens - - -3-4 -5-6.0 / / / / / 0.6 0.4 / / / / 3 0.5 0.5 0.5 / / / 4 0.4 0. 0. 0. / / 5 0.3 0.75 0.75 0.75 0.75 / 6 0. 0.6 0.6 0.6 0.6 0.6 In Equaion (4) γ is also a parameer which represens he exen of he sock marke s impac on he opion marke. The larger he value of γ, he bigger he effec, bu he model can become more volaile. If he value of γ is small, he effec is no obvious; so he selecion of appropriae value is very imporan. The parameer will be deermined hrough a series of experimens whose resuls are conained in Table. Table shows ha when he informaion of he opion marke is inroduced, he sock price varies widely wih γ increase. When γ equals o 0.03 he average decreases sharply and he variance increases dramaically, he model becomes unsable. Thus, he value of γ should be less han 0.03. If γ equals 0 he opion marke informaion is no inroduced. Considering hese wo
Enropy 03, 5 77 condiions, he value of γ equals o 0.0 in he model. Therefore, a his poin, he model is sable and he effec of opion marke on sock marke is appropriae. Table. The saisics of sock price under differen γ. γ 0.0 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Average 89.59 89.3 89. 65.55 56.8 45.67 37.7 30.34 3.89.3 Variance 9.979.89.05 54.85 64.93 87.67 98.07 9.57 09.09 97.0 Afer γ is deermined we will sudy he effec of various informaion lenghs on he sock marke. Table 3 is he saisics of sock price under differen informaion lengh scenarios. We can see ha he average price increases along wih an increase of informaion, bu he variance is decreasing. This shows ha he informaion of he opion marke is a key facor in he model. The increase of informaion for raders can decrease flucuaions of he sock price. For example, when informaion lengh increases from o, he mean of sock price increases 6.8% and he variance decreases 4.80%; when informaion lengh increases from 5 o 6, he mean of sock price increases.36% and he variance decreases.44%. Afer comparing he mean and variance of price, we found ha if he variance of sock is high, he sock price is low, and he effec of informaion diffusion is degressive, which corresponds o he real sock marke. Table 3. The saisics of sock price under differen informaion lenghs. Informaion lengh 3 4 5 6 Average 6.48 7.65 79.5 84.3 87.46 88.65 Variance 30.8 3.7 4.78 4.9. 6.67 Moreover, in he experimens we also find ha when he informaion abou he sock opion marke is inroduced and afer running a longer period of ime, he model sabilizes. There are sill big flucuaions in price in some siuaions. Figure is he price ime series of sock whose running ime is more han 50,000. From Figure we can find ha here is a downward change in he sock price. A his momen he sock dividend has no changed much. This phenomenon ells us ha even hough he inroducion of he opion marke can sabilize he sock markes, here is sill much informaion and complexiy ha can lead o dramaic changes in he sock marke. Figure. The rade volume ime series of sock. Price 0 00 90 80 70 60 50 40 50 00 50 00 Time real risk neural
Enropy 03, 5 78 3.6. The Effec of Various Proporions of Opion Traders In his subsecion we focus on he effec of he proporions of opion raders. Parameer γ equals 0.0, he lengh of informaion is 6 and he agen number is 00. The oher parameers are he same as hose in Table 4. In real financial markes here are differen kinds of opion raders whose purposes are differen and whose sraegies are relaive complicaed. So as o more closely represen a real marke, here are hree ypes of opion rader in his model. Their opion rade sraegies are differen and relaively simplified. In Table 4 he expression 0.5-0.5-0.35-0.35 represens he proporion of four ypes of opion raders which are non-opion raders, random opion raders, hedge opion raders and speculae opion raders, respecively. In Table 4 he oher expressions have similar meanings. In he experimenal design, we pay much more aenion o he speculaive opion rader and he hedge opion rader han o he non-opion rader and he random opion rader, so many experimens will be done abou he proporion changes of wo ypes of agens changes. The sample size is 50,000. Table 4. The saisics of sock price under differen proporion of opion raders. Proporion Average Variance Proporion Average Variance 0.5-0.5-0.5-0.5 87.97 63.40 0-0-0.6-0.4 9.50 3.66 0.5-0.5-0.35-0.35 88.038 48.76 0-0-0.5-0.5 89.389 6.70 0.05-0.05-0.45-0.45 89.737 6.555 0-0-0.4-0.6 89.394 4.506 0-0-0.9-0. 98.43.975 0-0-0.3-0.7 89.5 4.06 0-0-0.8-0. 97.487.78 0-0-0.-0.8 89.37 4.80 0-0-0.7-0.3 96.46 3.06 0-0-0.-0.9 89.359 6.088 Table 4 shows ha he changing proporion of raders does influence he sock marke. Decreasing he non-opion raders and random opion raders can sabilize he sock marke. The sysem becomes sable when he non-opion and random opion raders are less han 5%, respecively. Under hese condiions, he variance rapidly decreases and he average price increases. This shows ha as more people paricipae in sock opion rading, wih he excepion of random raders, he sock marke becomes more sable. The random opion rader is a negaive facor o sabilizaion of he sock marke. Increasing proporions of his ype of rader can make sock marke more volaile. We can draw a conclusion ha random opion rader is a significan facor of he sysem sable because i ransmi noise beween opion marke and sock marke. The noise no only is diffused among agens in he opion marke, bu also ransmied in he sock marke. From he proporion of he hedge and speculae opion raders he resul shows ha he increase of hedge opion raders can sabilize he sock marke. Conversely, he increase of speculae opion raders can make he sock marke more volaile, which fis he fac exising in he real financial marke, so he correcness of he consruced models is proved. 4. Conclusions and Fuure Work In order o sudy he effec of opion markes on sock markes, we presened a model of a European opion marke in his paper. This new model overcomes some shorcomings and beer replicaes some
Enropy 03, 5 79 feaures of real opion markes. To consruc a wo-sided impac mechanism, we inroduced he opion marke informaion ino he sock marke. Here, he wo markes form a simple nework wih informaion diffusion which is a key facor in he process of price and volume in boh markes. Mos imporanly, he compound Poisson process is inroduced o he dividend process which is similar o he real dividend process. Moreover, he resuls show ha muageniciy of sock price appears which can also be observed in real financial markes. In his model, hree ypes of opion raders were proposed. In he pricing mechanism, he Muli-Agen machmaking radeoff model is firsly inroduced o more closely mimic o he real opion marke. The experimens show ha he simulaion sysem is reliable afer operaing for longer ime periods. The experimens show ha he inroducion of opions could make he average of sock price decreases and sock price variance increases. For sock reurns, we found ha boh he average and he variance increase despie a decrease in kurosis. Sock reurn disribuions show obvious fa ails or excess kurosis. Through he esimaion of GARCH afer he opion is inroduced, he persisence of volailiy becomes obvious. We also find ha he average price is increasing wih an increase of informaion, bu he variance is decreasing. This proved he imporance of informaion provided by he opion marke in he model. The increase in informaion for raders can decrease flucuaions of he sock price. However, oher informaion and he complexiy of marke can sill lead o dramaic changes in he marke. Using differen proporions of hedge and speculae opion raders, he resul shows ha he increase of hedge opion raders can sabilize he sock marke. On he oher hand, an increase in he proporion of speculaive opion raders can make he sock marke more volaile which is consisen wih resuls from real financial markes, so he correcness of consruced models is proved. I should be poined ou ha our conclusion is based on our model. Before he opions are inroduced, he sock rade module has reached REE, hough he compound Poisson process is inroduced. The nex sage of he research is o employ heerogeneous agens o capure some oher sylized facs. Acknowledgemens This work was suppored in par by he Naional Naural Science Foundaion of China under Gran 7700 and he Naional Science Foundaion of China for Key Program under Gran 708300. References. Black, F.; Scholes, M. The pricing of opions and corporae liabiliies. J. Poil. Econ. 973, 8, 637 654.. Hakansson, N.H. Changes in he financial marke: Welfare and price effecs and he basic heorems of value conservaion. J. Financ. 98, 37, 977 004. 3. Ross, S. Opions and efficiency. Q. J. Econ. 976, 90, 75 89. 4. John, K.; Koicha, A.; Subrahmanyam, M. The micro-srucure of opions markes: Informed rading, liquidiy, volailiy and efficiency. Working paper, New York Universiy, New York, NY, USA, 994. 5. Kumar, R.; Sarin, A.; Shasri, K. The impac of opions rading on he marke qualiy of he underlying secuiy: An empirical analysis. J. Financ. 998, 53, 77 73.
Enropy 03, 5 70 6. Skinner, D.J. Opions markes and sock reurn volailiy. J. Financ. Econ. 989, 3, 6 78. 7. Conrad, J. The price effec of opion inroducion. J. Financ. 989, 44, 478 498. 8. Sorescu, S.M. The effec of opions on sock prices:973 o 995. J. Financ. 000, 55, 487 54. 9. Heer, B.; Trede, M.; Wahrenburg, M. The effec of opion rading a he db on he underlying socks reurn variance. Empir. Econ. 997,, 33 45. 0. Wei, P.; Poon P.S.; Zee, S. The effec of opion lising on bid-ask spreads, price volailiy, and rading aciviy of he underlying OTC socks. Rev. Quan. Financ. Accoun. 997, 9, 65 80.. Bollen, N.P.B. A noe on he impac of opions on sock reurn volailiy. J. Bank. Financ. 998,, 8 9.. Kabir, R. The price and volailiy effecs of sock opion inroducions: A reexaminaion. Discussion paper, Cener for Economic Research, Tilburg Universiy, Tilburg, he Neherlands, 997. 3. LeBaron, B.; Arhur, W.B.; Palmer, R. Time series properies of an arificial sock marke. J. Econ. Dyn. Conrol. 999, 3, 487 56. 4. Chen, S.H.; Chang, C.L.; Du, Y.R. Agen-based economic models and economerics. Knowl. Eng. Rev. 0, 7, 87 9. 5. Hommes, C.H. Handbook of Compuaional Economics; Norh-Holland Press: Oxford, UK, 006; pp. 09 33. 6. Ecca, S.; Marchesi, M.; Sezu, A. Modeling and simulaion of an arificial sock opion marke. Compu. Econ. 008, 3, 37 53. 7. Arhur, W.B.; Holland, J.H.; Lebaron, B.; Palmer, R.G.; Tayler, P. Asse pricing under endogenous expecaions in an arificial sock marke. In The Economy as an Evolving Complex Sysem II; Addison-Wesley: Boson, MA, USA, 997; pp. 5 44. 8. Su, C.H.; Chen, T.L.; Cheng, C.H.; Chen, Y.C. Forecasing he sock marke wih linguisic rules generaed from he minimize enropy principle and he cumulaive probabiliy disribuion approaches. Enropy 00,, 397 47. 9. Chen, C.R.; Lung, P.P; Tay, N.S.P. Informaion flow beween he sock and opion markes: Where do informed raders rade? Rev. Financ. Econ. 005, 4, 3. 0. Pan, J.; Poeshman, A.M. The informaion in opion volume for fuure sock prices. Rev. Financ. Sud. 006, 9, 87 908.. Hibber, A.M.; Daigler, R.T.; Dupoye, B. A behavioral explanaion for he negaive asymmeric reurn-volailiy relaion. J. Bank. Financ. 008, 3, 54 66.. DeLong, B.J.; Shleifer, A.; Summers, L.; Waldmann, R.J. Noise rader risk in financial markes. J. Poli. Econ. 990, 98, 703 738. 3. Mandelbro, B.B. The variaion of cerain speculaive prices. J. Bus. 963, 36, 394 49. 4. Bollerslev, T. Generalized auoregressive condiional heeroskedasiciy. J. Economerics 986, 3, 307 37. 5. Lamoreux, C.G.; Lasrapes, W.D. Persisence in variance, srucural change, and he GARCH model. J. Bus. Econ. Sa. 990, 8, 5 34. 03 by he auhors; licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/).