Behavioral Heterogeneity in the Options Market. - Preliminary Draft -

Transcription

1 Behavioral Heerogeneiy in he Opions Marke - Preliminary Draf - March 2008 Bar Frijns Deaprmen of Finance, Auckland Universiy of Technology, New Zealand. Thorsen Lehner Limburg Insiue of Financial Economics (LIFE), Maasrich Universiy, The Neherlands. Nijmegen Cener for Economics (NiCE), Radboud Universiy Nijmegen, The Neherlands. Remco C.J. Zwinkels 1 Nijmegen Cener for Economics (NiCE), Radboud Universiy Nijmegen, The Neherlands. 1 Corresponding Auhor: Nijmegen Cener for Economics (NiCE), P.O.Box 9108, 6500HK Nijmegen, The Neherlands. T: +31 (24) , F: +31 (24) , E: r.zwinkels@fm.ru.nl (Remco Zwinkels).

2 Behavioral Heerogeneiy in he Opion Marke - Preliminary Draf - March 2008 Absrac This paper develops and ess a heerogeneous agens model for he opion marke. Our agens have differing beliefs abou he level of volailiy of he underlying sock index and rade accordingly. We consider wo ypes of agens: fundamenaliss, who are assumed o expec he condiional volailiy o reurn o he uncondiional volailiy, and chariss who respond solely o noise from he level process. Agens are able o swich beween groups according o a mulinomial logi swiching mechanism. The model simplifies o a GARCH-ype specificaion wih ime-varying parameers, which depend on he disribuion of agens across ypes. Esimaion resuls for index opions on he German DAX30 reveal ha differen ypes of raders are also acively involved in rading volailiy. We find evidence ha he observed paerns in opion prices are he resul of heerogeneiy in expecaions abou fuure volailiy. Keywords: Heerogeneous Agens, Opion Markes, Fundamenaliss, Chariss. JEL-Classificaion: G12

3 1. Inroducion Volailiy is priced and raded in he opions marke. If marke paricipans have diverging views abou fuure volailiy of e.g. a sock index, hey engage in direcional volailiy bes, for example hrough he use of opion sraegies. 2 If hey believe ha markes become more volaile, hey buy a-he-money pus and calls (a long sraddle), since he value usually increases wih a rise in volailiy. If hey believe ha volailiy is overpriced in he marke, hey shor a sraddle. Volailiy rading creaes uncerainy abou he fair value of volailiy. Differing expecaions abou fuure volailiy implies ha volailiy is no consan and ha volailiy iself becomes volaile. In recen years, sochasic volailiy models are successfully used for he purpose of opion valuaion. The volailiy of volailiy was found o be in paricularly imporan for he pricing performance of he model (Chrisoffersen and Jacobs, 2004). The evidence agains he Efficien Marke Hypohesis (EMH, see Fama, 1971) has been mouning in he previous decades. In a broad range of markes and using a similarly broad range of echniques, researchers have found evidence agains he noion of raionaliy in financial markes. One area in which he noion of raionaliy is consisenly rejeced is in sudies using observed (survey) expecaions (see Frijns e al., 2008 for experimenal evidence or MacDonald, 2000 for an overview). Mos surveys focus on he foreign exchange marke, bu similar resuls are found for bond and sock markes. A second srand of observaions ha raises doubs on he EMH is he exisence of numerous anomalies in financial markes. Phenomena like excess volailiy, small firm effecs, overshooing and he January effec canno be explained by represenaive agen raional expecaion models. Also heoreically one can cas doub on he EMH. No rade heorems, inroduced by Milgrom and Sokey (1982), hypohesize ha raional agens will never rade because expecaions are equal and all informaion is discouned in he curren marke price. Therefore, price changes occur wihou rade aking place. This is hard o combine wih he observaion of enormous volume in especially FOREX markes. 2 One alernaive for example is o buy or sell an over-he-couner volailiy conrac like a volailiy swap, where e.g. he buy receives he difference beween he realized volailiy and he fixed swap rae. Anoher is o rade fuures and opions on a volailiy index. Surprisingly, he volume in hese conracs has been disappoining. 1

4 One of he responses in he academic lieraure o he demise of he EMH is he behavioral finance lieraure (Kahneman and Tversky, 1979 or Barberis e al., 1998, iner alii). The behavioral finance lieraure seps aside from he noion of raionaliy and inroduces elemens from social psychology in economic decision making. Models of behavioral finance are usually geared owards explaining he observed marke anomalies. These models can, for example, explain he excess volailiy observed in financial markes (e.g. De Long e al., 1990) by he exisence of differen rader ypes. 3 These models, which explain he excess volailiy observed in financial markes, are based on he idea ha differen rader ypes, hrough heir acions, affec he condiional volailiy of he price process. Indeed, Avromov e al. (2006) show ha he exisence of boh ypes of raders canno only explain he differences in daily volailiy, bu can also explain he asymmery observed in daily volailiy. The lieraure on heerogeneous expecaions in volailiy so far has been relaively limied. Guo (1998) assumes ha opion invesors hold heerogeneous expecaions abou he parameers of he lognormal process of he underlying asse price. Esimaion resuls for S&P500 index call opions indicae ha here are wo groups: bulls and bears. Ziegler (2002) models wo ypes of agens who differ in heir iniial beliefs on he dividend process and invesigaes he effec on opion prices. We argue ha differen ypes of raders are also acively involved in rading volailiy. Alhough his migh no necessarily be he case in he sock marke, i is he case when one urns o he opion marke. Being he only unobserved variable in an opion pricing model, volailiy plays a pivoal role in he deerminaion of he value of an opion. Hence opion sraegies could be a direc consequence of expecaions abou fuure volailiy. If differen rader ypes have differen expecaions abou he fuure volailiy of he underlying, his may induce rade and cause volailiy o change (see e.g. Carr and Madan (2002)). To evaluae wheher differen rader ypes are presen, we make use of he heerogeneous agens lieraure, which argues ha due o he presence of agens wih bounded raionaliy expecaions abou fuure values may differ. In line wih his lieraure we assume ha wo differen rader ypes are acive in rading volailiy, where some raders (ermed 3 The exisence of differen rader ypes and heir impac on volailiy has no only been brough forward wihin he behavioral finance lieraure, bu has also been proposed wihin he field of marke microsrucure (see e.g. Kyle, 1985). 2

5 fundamenaliss) rade on he long-run mean reversion of he condiional volailiy o he uncondiional volailiy and oher rader (called chariss) rade on shor-run persisence in he volailiy process. These raders may change heir sraegy based on he performance of heir sraegy compared o he performance of he sraegy of he oher raders. Ineresingly, when combining he sraegies of boh ypes of raders we find ha our model reduces o an asymmeric GARCH model, bu wih ime-varying coefficiens, where he ime-variaion is due o changes in he proporion of fundamenaliss and chariss presen in he marke. This ime variaion inroduces anoher ineresing feaure, namely allowing he volailiy process o be locally unsable while guaraneeing global sabiliy. When charis raders dominae he marke heir persisence may cause he volailiy process o become unsable. However, when he proporion of fundamenalis raders increases, heir presence ensures ha he volailiy process remains sable in he long-run. In his manner, he process can swich beween sable and unsable phases, providing an economic inerpreaion o he noion of volailiy clusering observed in financial markes. When empirically esing our model on opion prices we find evidence ha suppor he presence of boh ypes of raders. Over ime, he fracions of fundamenalis and charis raders change and we find evidence ha our model ouperforms a sandard model wihou swiching in erms of pricing performance boh in-sample and ou-ofsample. We conribue o he lieraure on hree differen grounds. Firs, we do no assume heerogeneiy in beliefs on he price process, bu on he volailiy process; fundamenally differen ypes of agens wih differen opion sraegies are inroduced. Second, we allow agens o swich beween differen sraegies insead of assuming fixed proporions and, finally, nex o his novel heoreical seup we empirically assess he pricing performance of he model. The remainder of he paper is organized as follows. Secion 2 reviews some of he lieraure on heerogeneous agen models. In secion 3 we presen he economeric framework of our model. Secion 4 presens he daa & mehodology of he esimaion procedure. In secion 5 we show he resuls, and secion 6 concludes. 3

6 2. Heerogeneous Agen Models Divergence from he assumpion of raionaliy implies ha one can inroduce heerogeneiy in expecaions as well; here is only one way of behaving raional, while here are infinie ways of behaving irraional, or boundedly raional. Three explanaions for being heerogeneous can be discerned from he lieraure. Firs is he exisence of asymmeric informaion. Differen marke paricipans are assumed o hold differen ses of informaion, whereby par of he informaion is common for all paricipans and par is privae. The concep of asymmeric informaion was firs inroduced in he New Classical Theory of he macro economy, where agens were assumed o be unable o obain informaion ha is public in oher pars of he economy, and where agens are raional in he Muh (1961) sense in ha hey use ha informaion ha is available o hem in he bes possible way o form heir expecaions of a paricular variable. Second is he claim ha agens migh differ in he way (symmeric) informaion is inerpreed. To argue why he difference in inerpreaion occurs we can follow he raional belief heory due o Kurz (1994), which assumes ha heerogeneiy of beliefs is caused by he fac ha economic agens do no know he srucural relaions of he economy. Agens only have informaion or empirical knowledge, which is readily observable from he economy. Third and final ground for heerogeneiy in expecaions is he exisence of fundamenally differen ypes of agens. DeLong e al. (1990) formally model he noion of noise raders, which do no ge driven ou of he marke. Frankel and Froo (1986, 1990) popularize he view ha he foreign exchange marke is dominaed by wo ypes of marke paricipans ha differ in which informaion hey use for forming heir expecaions. Fundamenaliss hink of he exchange rae as an economic model, while chariss predominanly use he exchange rae s own hisory as inpu in heir expecaions formaion process. The lieraure on heerogeneous agens models, or HAMs, coninues on he line of hough ha here can be fundamenally differen ypes of agens, see Hommes (2006) and LeBaron (2006) for an overview. The lieraure on heerogeneous agens applied o financial markes aims o describe he evoluion of sock price by relaxing he assumpion of homogeneiy among invesors. By allowing for heerogeneiy among invesors, differen ypes of invesors can be classified along wih heir poenial sraegies, and one 4

7 can evaluae how likely i is ha hese raders are acive in a marke and wha he consequences of heir rading is for he price and volailiy process. Generally, raders are classified in wo caegories, being eiher fundamenaliss, who rade on he basis of fundamenals, or chariss, who rade on observed paerns in pas prices, as firs inroduced by Frankel and Froo (1986). Revoluionary in he models described in Hommes (2006) is ha agens do no only differ, bu ha hey are able o swich beween ypes, condiional on performance. This swiching inroduces a non-linear model ha mixes differen regimes, based on economic foundaions. Up ill now he majoriy of sudies on HAMs has been conduced in experimenal seings. Using eiher deerminisic or sochasic simulaion echniques, he presence of differen rader ypes in financial markes can explain some sylized facs of reurns from financial markes; see e.g. De Grauwe and Grimaldi (2005, 2006) and Lux (1998). The irregular swiching beween ypes induces volailiy clusering, heavy ails, slow mean reversion, and excess volailiy. To our bes knowledge, here is only a handful of papers ha direcly aemp o esimae a HAM wih full-fledged swiching mechanism. Boswijk e al. (2007) examine he S&P500; Weserhoff and Reiz (2005, 2007) look a commodiy markes; De Jong e al. (2007) focus on EMS exchange raes. All sudies, hough, find significan evidence of heerogeneiy among raders, and swiching beween sraegies. 3. The Economeric Framework Le S be he value of an underlying asse a ime, and D be he expeced cash dividend paymens over he lifeime of he opion. Then, in a Gaussian discree-ime economy he (log) reurn of he asse a ime (r ) is assumed o follow he following dynamics, r = S d h ln + = µ + ε S, (1) 1 ε ~ N( 0, ) under probabiliy measure P Ω 1 1 where d is he dividend yield, µ is he (condiional) mean of r, h is he condiional volailiy of he asse and ε is a sandard normal random variable. I is on he process of h ha we focus in his paper and we assume ha here are wo differen groups of raders, 5

8 so-called fundamenaliss and chariss, which have differen expecaions regarding he fuure evoluion of h. Le F h 1 + be he predicion of he condiional volailiy for he fundamenaliss. These fundamenaliss are assumed o rade on he basis of mean reversion, where hey expec he condiional volailiy mean-revering o he uncondiional volailiy. Their bes predicion for he volailiy process is F h + 1 = h + α ( h h ), (2) where h is he long-run uncondiional volailiy 4 and α measures he speed a which he fundamenaliss expec he volailiy process o mean rever. Since volailiy needs o remain posiive wih probabiliy 1, α is bounded beween [-2, 0], bu is ypically expeced o be beween [-1, 0]. When α 0 he process becomes very persisen and lile mean reversion akes place. When α -1 he process revers back o he uncondiional volailiy almos immediaely. Equaion (2) reveals ha fundamenaliss essenially follow a GARCH (1, 0), no aking ino accoun any shocks in he volailiy process. The chariss do no believe in mean reversion, bu rade on recenly observed shocks in he marke. Given he curren level of volailiy, hey use recenly observed shocks o predic he fuure level of condiional volailiy. Given ha condiional volailiy behaves differenly in he presence of posiive or negaive shock, we allow for an asymmeric impac of hese shocks (Glosen, Jagannahan and Runkle, 1993). We herefore define heir predicion of he volailiy process as h C = h + 0 ( h ε ) + β1( h ε ) β, (3) where C h 1 + is he volailiy predicion of he chariss, ε ( ε ) is he pas posiive (negaive) shock in he volailiy process and β 0 (β 1 ) measures he exen o which chariss incorporae posiive (negaive) shocks ino heir predicion. + 4 When we empirically implemen he model, we assume ha h is equal o he variance of he underlying reurn series calculaed over he previous 250 rading days wih a moving window. 6

9 Because we have defined a marke where only hese wo ypes of raders are presen, he condiional volailiy ha is observed in he marke (h +1 ) is a funcion of he predicions of chariss and fundamenaliss and he fracion a which each rader ype is represened in he marke. Since boh sraegies involve no paricular skill or informaion from raders, raders can swich o eiher sraegy a any poin in ime wihou incurring ransacion coss. Le w be he fracion of fundamenaliss presen in he marke. Then a naural choice for w is a rule ha considers he profiabiliy or pricing error of following a fundamenalis sraegy. 5 We define w as a mulinomial logi swiching rule, as firs inroduced by Brock and Hommes (1997, 1998), where he swiching depends on he absolue forecas error of fundamenaliss versus chariss. The swiching rule is given as w = 1+ e F ln( ) ln( ) C ln( ) ln( ) h h h h γ abs ln( ) abs ln( ) h h 1, (4) where γ measures he sensiiviy of marke paricipans (fundamenaliss or chariss) o heir respecive percenage forecasing errors in erms of volailiy and is expeced o be beween 0 and infiniy. This sensiiviy of choice parameer can be inerpreed as he saus quo bias of raders. Wih γ = 0 agens are disribued uniformly across ypes. As γ increases, agens become increasingly sensiive o differences in forecasing performance beween he sraegies. In he limiing case, asγ, all agens direcly swich o he more profiable rule, such ha w is eiher 0 or 1. Given his definiion, w will always be sricly bounded beween 0 and 1. Wih he given weighs and he differen rading sraegies we can now esablish he process for he condiional volailiy. Since he condiional volailiy is a consequence of he proporion of marke paricipans following each sraegy i is compued as a weighed average of he fundamenalis and he charis volailiy predicion, 5 An example for he definiion of w is he profis fundamenaliss make relaive o he charis on an opion sraegy ha involves sraddles. When e.g. fundamenaliss expec volailiy o increase hey will go long in a sraddle and vice versa. If heir sraegy works and pays off well relaive o he sraegy of he chariss, more raders may be inclined o follow his fundamenal sraegy and hence he proporion of fundamenaliss will increase. If heir sraegy does no work and performs poorly relaive o chariss, more raders may be inclined o follow a charis sraegy in he fuure. 7

10 F C h + 1 wh ( 1 w ) h + 1 =. (5) The paricular specificaion of w ensures ha more marke paricipans follow a paricular rading sraegy he beer he sraegy performed in he previous period. Therefore, if he fundamenalis predics volailiy more accuraely han he charis, w +1 increases. A beer predicion of he chariss consequenly reduces w +1. Equaion (5) defines he process for he condiional volailiy and shows ha his is a weighed average of he condiional volailiy predicions of chariss and fundamenaliss. Subsequenly, we provide an economic inerpreaion of (5). We sar by subsiuing (2) and (3) ino (5). Afer rewriing we obain h = wα h + ( 1 + wα )h + ( 1 w ) β0( h ε ) + ( 1 w ) β1( h ε ), (6) or h = h + h + β0, ( h ε ) + β1, ( h ε ) α, (7) where h = wαh, α = ( 1 + wα ), β 0, = ( 1 w ) β0, and β 1, = ( 1 w ) β1. Equaion (7) shows ha he model essenially reduces o a GJR-GARCH(1,1) model wih ime varying coefficiens. The ime variaion in hese coefficiens is driven by he profiabiliy of being a fundamenalis or a charis. Apar from his ime variaion, here are several ineresing feaures abou our model. Firsly, since our model reduces o a sandard GJR-GARCH, we can provide an economic inerpreaion of he GARCH model. Up ill now he GARCH model has mainly been moivaed by he empirical observaion of ime variaion in condiional volailiy. The model proposed provides an economic inerpreaion of he source of ime variaion in volailiy and of GARCH effecs. The model shows ha he mean reversion of he condiional volailiy is driven by he presence of fundamenaliss, and ha persisence in volailiy is driven by he presence of he chariss. When very few chariss 8

11 are presen in he marke, mean reversion would occur a a faser rae han when many chariss are presen. Also, he impac of news shocks on he condiional volailiy is solely driven by he presence of chariss, who expec recen news o be informaive abou he fuure level of volailiy. The GARCH effec and ARCH effec can herefore be explained by he presence of hese wo ypes of raders in he marke. A second ineresing feaure of he model concerns he sabiliy condiions of (7). Under normal circumsances, fundamenaliss follow a sraegy ha ensures ha he condiional volailiy remains bounded. However, he charis sraegy is an unsable sraegy when β 0 and β 1 are posiive and volailiy prediced by chariss will no remain bounded. However, he fac ha boh ypes of raders are presen and w flucuaes over ime allows he volailiy process (7) o be locally unsable, while guaraneeing sabiliy of he GARCH process in he long run. Wheher (7) is sable in he long run depends on he parameer values for α, β 0 and β 1 and is an issue ha will be addressed in he empirical secion. A hird feaure abou he model is he ime varying uncondiional volailiy. This ime variaion in uncondiional volailiy is no caused by slow-moving change in he underlying uncondiional volailiy (as suggesed by Engle and Lee, 1999), bu is also driven by he amoun of fundamenaliss or chariss presen. The model presened above represens he mos simplisic form fundamenalis and charis behavior. There are several exensions possible o he sraegies for boh ypes of raders. Firsly, we can exend he fundamenalis sraegy by allowing for dynamics in he uncondiional volailiy. Such ypes of model follow from Engle and Lee (1999) and are ofen referred o as wo-componen GARCH models. Allowing for such addiional dynamics can be done sraighforwardly, and would imply ha (4) akes he form of a GJR-GARCH(2, 2). Secondly, chariss may also consider oher sochasic variables, such as rading volumes, number of ransacions, ec. The sochasic variables can be added o he model. The model presened in secion 3 could essenially be applied o any ype of securiy in financial markes. However, in he curren paper we esimae he model on 9

12 opion prices. From Black and Scholes we know ha he only unknown variable in he opion pricing model is he volailiy. Hence rading in opions is essenially rading on he expecaions abou he fuure volailiy of he underlying. The following secion discusses he daa and mehodology followed in esimaing he model and subsequenly we summarize he empirical resuls. 4. Daa and Mehodology We use daily closing DAX 30 index opions and fuures prices for a one year period from January 2000 unil December The raw daa se is direcly obained from he EUREX, European Fuures and Opions Exchange. The marke for DAX index opions and fuures is he mos acive index opions and fuures marke in Europe. For index opions he expiraion monhs are he hree neares calendar monhs, he hree following monhs wihin he cycle March, June, Sepember and December, as well as he wo following monhs of he cycle June, December. For index fuures, he expiraion monhs are he hree neares calendar monhs wihin he cycle March, June, Sepember and December. The las rading day is he hird Friday of he expiraion monh, if ha is an exchange rading day; oherwise on he exchange-rading day immediaely prior o ha Friday. We exclude opions wih less han one week and more han 25 weeks unil mauriy and opions wih a price of less han 2 Euro o avoid liquidiy-relaed biases and because of less useful informaion on volailiies. We filer he available opion prices and include all opions ha are acively raded, inside or ouside he 10% absolue moneyness inerval. In pracice, in volaile periods deep ou-of-he money opions are highly informaive if hey are acively raded. As a resul, each day we use a minimum of 3, bu ypically 4 differen mauriies for he calibraion. Inser Table 1 Here 10

13 The DAX index calculaion is based on he assumpion ha he cash dividend paymens are reinvesed. Therefore, when calculaing opion prices, heoreically we do no have o adjus he index level for he fac ha he sock price drops on he ex-dividend dae. Bu he cash dividend paymens are axed and he reinvesmen does no fully compensae for he decrease in he sock price. Therefore, in he conversion from e.g. fuures prices o he implied spo rae, one empirically observes a differen implied dividend adjused underlying for differen mauriies. For his reason, we work wih he adjused underlying index level implied ou from fuures or opion marke prices. In a nushell, he opion pricing procedure boils down he following. Firs of all, he dividend adjused value for he underlying is deermined for a cerain day; in our case, ha is he DAX30 on January 1s Nex, a se of opions is observed wih differen imes o mauriy and differen srikes for ha same day. Using Mone Carlo simulaions, he model generaes a cerain forecass for all he differen expiraion daes. In oher words, i sars off from he observed dividend-adjused underlying of oday, and ieraes forward unil expiraion. Nex, opion prices are calculaed wih hese forecass using he sandard Black and Scholes approach, and compared wih he empirically observed opion prices. The opimisaion procedure hen consiss of minimizing he roo-meansquared pricing error of he oal se of opions per day. The equilibrium se of coefficiens is hen used as saring values for he opimizaion procedure for he nex day. This whole procedure is repeaed for each rading day in he daase. In paricular we are using he following procedure for one paricular day o price opions on he following rading day: Firs, we compue he implied ineres raes and implied dividend adjused index raes from he observed pu and call opion prices. We are using a modified pu-call pariy regression proposed by Shimko (1993). The pu-call pariy for European opions reads: c i, j pi, j = [ S PV ( D j )] i r j ( T j ) X e (8) where c i,j and p i,j are he observed call and pu closing prices, respecively, wih exercise prices X i and mauriy (T j -), PV(D j ) denoes he presen value of dividends o be paid 11

14 from ime unil he mauriy of he opions conrac a ime T j and r j is he coninuously compounded ineres rae ha maches he mauriy of he opion conrac. Therefore, we can infer a value for he implied dividend adjused index for differen mauriies, S - PV(D j ), and he coninuously compounded ineres rae for differen mauriies, r j. To ensure ha he implied dividend adjused index value is a non-increasing funcion of he mauriy of he opion, we occasionally adjus he sandard pu-call pariy regression. Therefore, we conrol and ensure ha he value for S - PV(D j ) is decreasing wih mauriy, T j. Since we are using closing prices for he esimaion, one alernaive is o use implied index levels from DAX index fuures prices assuming ha boh markes are closely inegraed. Second, we esimae he parameers of he paricular models by minimizing he loss funcion. Parameers of he model are calibraed by minimizing he roo mean squared absolue pricing error beween he marke prices and he heoreical opion prices: RMSE = 1 N min n mi ( cˆ i, j ci, j ) i= 1 j= 1 2 (9) where N is he oal number of call opions evaluaed, he subscrip i refers o he n differen mauriies and subscrip j o he m i differen srike prices in a paricular mauriy series i. Given reasonable saring values, we price European call opions wih exercise price X i and mauriy T j. Using well-known opimizaion mehods (e.g. Newon-Raphson mehod), we obain he parameer esimaes ha minimize he loss funcion. The goodness of fi measure for he opimizaion is he mean squared valuaion error crierion. Third, having esimaed he parameers in-sample, we urn o ou-of-sample valuaion performance and evaluae how well each day s esimaed models value he raded opions a he end of he following day. We filer he available opion prices according o our crieria for he in-sample calibraion. The fuures marke is he mos liquid marke and he opions and he fuures marke are closely inegraed, herefore i can also be assumed ha he fuures price is more informaive for opion pricing han jus 12

15 using he value of he index. For every observed fuures closing price we can derive he implied underlying index level and evaluae he opion. Given a fuures price F j wih ime o mauriy T j, spo fuures pariy is used o deermine S -PV(D j ) from S -r j T j PV ( D ) = F e (10) j j where PV(D j ) denoes he presen value of dividends o be paid from ime unil he mauriy of he opions conrac a ime T j and r j is he coninuously compounded ineres rae (he inerpolaed EURIBOR rae) ha maches he mauriy of he fuures conrac (or ime o expiraion of he opion). If a given opion price observaion corresponds o an opion ha expires a he ime of delivery of a fuures conrac, hen he price of he fuures conrac can be used o deermine he quaniy S -PV(D j ) direcly. The mauriies of DAX index opions do no always correspond o he delivery daes of he fuures conracs. In paricular for index opions he wo following monhs are always expiraion monhs, bu no necessarily a delivery monh for he fuures conrac. When an opion expires on a dae oher han he delivery dae of he fuures conrac, hen he quaniy S -PV(D j ) is compued from various fuures conracs. Le F 1 be he fuures price for a conrac wih he shores mauriy, T 1 and F 2 and F 3 are he fuures prices for conracs wih he second and hird closes delivery monhs, T 2 and T 3, respecively. Then he expeced fuure rae of dividend paymen d can be compued via spo-fuures pariy by: r3t3 r2t2 log (F3 / F2 ) d = (11) (T T ) 3 2 Hence, he quaniy S -PV(D) = S e dt associaed wih he opion ha expires a ime T in he fuure can be compued by 6 S e dt ( ( r1 d ) T1 dt ) = F1e. (12) 6 See e.g. he appendix in Poeshman (2001) for deails. 13

16 This mehod allows us o perfecly mach he observed opion price and he underlying dividend adjused spo rae. Given he parameer esimaes and he implied dividend adjused underlying we can calculae opion prices and compare hem o he observed opion prices of raded index opions. For he ou-of-sample par he same loss funcions for call opions are used. The predicion performance of he various models are evaluaed and compared by using he roo mean squared valuaion error crierion. We compare he prediced opion values wih he observed prices for every raded opion. We repea he whole procedure over he ou-of-sample period and conclude, which model minimizes he ou-of-sample pricing error. In order o evaluae opions, he physical process has o be ransformed o a riskneural process. We make use of he Local Risk Neural Valuaion Relaionship (LRNVR) developed in Duan (1995). Under he LRNVR he condiional variance remains unchanged, bu under he pricing measure Q he condiional expecaion of r is equal o he risk free rae r f : [ r ) ] exp( r ) Q E exp( Ω 1 = f, (13) The risk-neural Gaussian process reads: S PV ( D) r = rf σ + h ε S PV D 1 2 = ln 2, 1 1( ) ε Ω 1 ~ N(0,1) under he risk-neuralized probabiliy measure Q (14) In Equaion (14), ε is no necessarily normal, bu o include he Black-Scholes model as a special case we ypically assume ha ε is a Gaussian random variable. The locally risk-neural valuaion relaionship ensures ha under he risk neural measure Q, he volailiy process saisfies 14

17 P [ r Ω ] = Var [ r Ω 1] h Q Var = 1. (15) price equal o: A European call opion wih exercise price X and ime o mauriy T has a ime Q ( rt ) E [ max( S X,0) Ω ] = exp 1 c (16) For his kind of derivaive valuaion models wih a high degree of pah dependency, compuaionally demanding Mone Carlo simulaions are commonly used for valuing derivaive securiies. We use he recenly proposed simulaion adjusmen mehod, he empirical maringale simulaion (EMS) of Duan and Simonao (1998), which has been shown o subsanially accelerae he convergence of Mone Carlo price esimaes and o reduce he so called simulaion error. In he empirical par of he paper, we model he expecaions of condiional volailiy of fundamenaliss (and chariss) in an EGARCH seing, which is moivaed by he empirical daa fiing (see Lehner (2003)). Applying a sandard GARCH framework resuled in numerous violaions of parameers in heir permissible parameer space. The EGARCH seing resolves hese issues, as i imposes no resricions on he parameer space (see Nelson, 1991). 5. Resuls This secion presens he empirical resuls of he opion pricing applicaion of our heerogeneous agens model for he second momen. Firs, we presen one specific pah from he Mone Carlo simulaions in order o gain somewha more feeling on he behaviour of he proposed model. Second we focus on boh he esimaion resuls and he sabiliy of he esimaes hrough ime. Finally, we look a he pricing errors of our model, boh in-sample and ou-of-sample. The esimaion exercises are conduced in a seing wih and wihou swiching. This allows us o examine he direc effec of inroducing more flexibiliy in he model; in oher words, i allows us o see he advanage of our model over a sandard GARCH. 15

18 Inser Figure 1 Here Figure 1 presens a close-up of a one simulaion pah ou of in he Mone Carlo seup; i uses he (opimized) coefficiens from a random day, May 5 h in his case 7. A number of observaions can be made. As one would expec, he volailiy h lies beween he expecaions of he fundamenaliss h F and chariss h C. The disance beween he hree is governed by he weigh w. Weighs flucuae coninuously around he benchmark of one half wih a minimum of 0.14 and maximum of The naure of he wo groups is clearly illusraed by he course of he volailiy process. High spikes in volailiy always coincide wih low weighs; i.e., a relaively high volailiy is caused by he fac ha he marke is dominaed by chariss. The mos clear example of his can be seen around observaion number 40 where w reaches is minimum and h is maximum. The reverse is rue as well; when fundamenalis make up over 80% of he marke around period 70, volailiy drops owards is long-run value. Therefore, fundamenaliss are sabilizing, and chariss desabilizing. None of he groups ges driven ou of he marke, and boh groups experience periods of dominance. Inser Table 2 Here 7 The coefficiens are given by α=-0.087; β 0 =-0.365; β 1 =0.337; and γ=

19 Table 2 presens he esimaion resuls of he opion pricing applicaion of he heerogeneous agens model; he disribuional characerisics of he esimaes over ime are depiced. Overall, we observe ha all coefficiens have he sign and magniude as hypohesized by he model. Boh fundamenaliss and chariss appear o be acive on he marke; heir individual effecs on he variance process are as expeced (sabilizing and desabilizing, respecively). Also, we find significan evidence of swiching beween he wo rules. Focusing on he saic seup firs (Panel A); he mean-reversion parameer α is negaive hroughou he sample, implying ha fundamenaliss consisenly apply a sabilizing expecaion formaion rule. They herefore inroduce a mean revering dynamics ino he variance process, as hey expec he variance o reurn o he long run volailiy. The absolue magniude of he mean esimae of α indicaes ha fundamenaliss expec on average 4% of he excess volailiy o disappear in he nex period. The average esimaed local volailiy, i.e. he saring value for he volailiy dynamics, is esimaed o be equal o 27% (annualized), which is very much in line wih he ime series volailiy of he DAX index in ha period. Parameer esimaes for he charis expecaion formaion rule in he model, β 0 and β 1, have he expeced sign as well. The resuls for his asymmeric seup imply ha here is a clear leverage effec: Posiive shocks in he level resul in a reducion of he variance β 0 <0; negaive shocks in he level resul in an increase of he variance β 1 >0. Therefore, negaive shocks in he level have a desabilizing effec o he variance process because of he chariss expecaion formaion rule. The resuls for he swiching seup, in Panel B, are generally consisen wih he saic seup. The difference, he sensiiviy of choice parameer γ, is posiive hroughou he sample. This implies ha he swiching mechanism funcions as a posiive feedback rule. In oher words, he posiive sign of he coefficien indicaes ha agens swich owards he group wih he smalles forecasing error. The magniude of γ is condiional on he funcional form of he profi funcion (in our case, a loss funcion consising of he percenage forecasing error). Therefore, i is no possible o make any saemens abou he sensiiviy o profi differences of raders in he opion marke a his ime. We will, 17

20 however, be able o say somehing abou he evoluion of individual s behaviour over ime in he sensiiviy analysis below. As addiional empirical evidence for our model, we examined boh he in-sample and ou-of-sample pricing errors. The resuls for he models wih and wihou swiching are depiced in he final wo columns of Table 2. Resuls sugges ha he assumpion ha agens always swich o he more profiable forecasing rule is very much suppored by he daa. Comparing Panel A and B reveals ha our mos sophisicaed model ouperforms he benchmark wih on average 0.92 for he in-sample and 0.49 for he ou-of-sample pricing error. In oher words, nex o inroducing a more inuiive appeal o volailiy models, our heerogeneous agens seup for he second momen also proves o be more effecive in explaining and forecasing opion prices. To our bes knowledge, a heerogeneous agens model has never been applied o he opions marke. We can, however, compare our resuls wih relaed lieraure. Firs of all, he signs and magniudes of he charis expecaion formaion funcion are direcly comparable o he sandard EGARCH-resuls, due o Nelson (1991). The relaive impac of posiive versus negaive shocks corroboraes previous findings; he ypical resuls for he leverage effec indicae ha he relaive effec of negaive shocks on he variance process is larger han he posiive shocks. Second, our resuls are direcly in line wih previous findings on esimaes of heerogeneous agens models for alernaive markes. Boswijk e al. (2007) find significan evidence of he co-exisence of chariss and fundamenalis for he S&P500 from 1870 o 2006; De Jong e al. (2007) presen similar resuls for he Briish Pound during he EMS crisis. Our resuls on he swiching mechanism, however, are sronger compared o Boswijk e al. (2007) and De Jong e al. (2007); evidence for swiching is limied given heir esimae of he swiching parameer. This implies ha raders in he opions marke are more prone o change heir sraegy in response o a difference in profis compared o raders in he S&P500 or foreign exchange marke. Given ha he model is esimaed for differen mauriies and levels of moneyness for each day of he year, we can examine he sabiliy of he esimaed coefficiens during he esimaion process. By following he evoluion of he esimaed coefficiens, we will 18

21 be able o say somehing abou he condiional behaviour of heerogeneous raders. Figure 2 depics he developmen of he coefficien esimaes over ime. Inser Figure 2 Here Figure 2 displays he developmen of he wo expecaion formaion funcions, fundamenaliss and chariss, and he inensiy of choice parameer. Overall, he parameers of he fundamenalis and charis expecaion formaion funcions are relaively sable; α, β 0 and β 1 move in a relaive small band wihin he region you expec hem o be. A around wo-hirds of he sample α, β 0 and β 1 sar moving owards zero, while γ becomes larger and more volaile. This evoluion of he parameers can be direcly explained by he logic of he underlying heerogeneous agens model. Apparenly, he volailiy in he underlying is relaively consan in his middle period, which can be seen from he fac ha he coefficiens of he expecaion formaion funcions go o zero. Boh groups form heir expecaion as he mos recenly observed volailiy, plus some correcion erm; as he correcion erm goes o zero, agens expec a consan volailiy. As boh fundamenaliss and chariss expec small innovaions o he volailiy process, he profi difference beween he wo sraegies as well as he forecasing error iself will be small. As he forecasing errors are small, large shifs in γ will no induce large shifs in he disribuion of weighs over sraegies (see Equaion 4). This is exacly why he esimae of γ shows large shifs in his period. Inser Figure 3 Here 19

22 Figure 3 presens he evoluion of he esimaed local volailiy and he in-sample pricing error of our model. There is a clear posiive correlaion beween he esimaed fundamenal volailiy and he pricing error. Consisen wih previous lieraure, we find ha volailiy shows disincive periods of high and low volailiy. Ineresingly, he local volailiy esimaes exacly flucuae around he long-run volailiy level esimaed from reurn daa. 6. Conclusions In his paper we inroduce a model of behavioral volailiy rading. Being he only unobserved variable in an opion pricing model, volailiy plays a pivoal role in he deerminaion of he value of an opion. Our marke consiss of wo ypes of agens ha have differen views on volailiy and rade accordingly. Fundamenaliss are expecing he condiional volailiy o mean rever o a long-run volailiy level. Chariss on he oher hand respond solely on noise from he level process and bid up (down) volailiy if hey receive a negaive (posiive) signal from he sock marke. Depending on he profiabiliy of heir sraegy, agens are able o swich beween groups according o a mulinomial logi swiching mechanism. The model is shown o reduce o a GJR- GARCH(1,1) wih ime varying coefficiens. The difference, however, lies in he fac ha we provide a behavioral underpinning; also, ime variaion in he coefficiens is dependen on rader behavior. In an applicaion of he model o DAX index opions, using he GARCH opion pricing model, we find evidence ha differen ypes of raders are acively involved in rading volailiy. Boh fundamenaliss and chariss are shown o be acive in he opions markes, and boh groups are consisenly presen. Hence, we find evidence ha observed opion prices are he resul of heerogeneiy in expecaions abou fuure volailiy. Also, we find evidence of swiching beween he groups; ha is, a cerain poins in ime he marke is dominaed by mean-revering fundamenaliss, a oher poins by desabilizing chariss. Inroducing he possibiliy o swich gives a subsanial reducion in boh in- and 20

23 ou-of-sample pricing errors. In oher words, volailiy raders indeed change heir forecasing behavior dependen on he relaive profiabiliy. Exensions o he curren resuls are boh possible and necessary. Our curren daase only comprises of call-opions for a limied ime span. More daa will obviously yield more confidence in he resuls. The esimaion procedure as i is now esimaes he model daily; using he daa as a panel, so using boh he ime-series as he cross-secional variaion, would make i possible o consruc sandard errors around he esimaes. I would also be ineresing o experimen wih alernaive specificaions of he model. Think of alernaive profi funcions, as is common in he heerogeneous agens lieraure. Also, he expecaion formaion funcions are flexible o incorporae numerous differen specificaions, including ones wih exogenous informaion. 21

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27 Tables and Figures Table 1: Number of Observaions Trading Days o Expiraion < 21 [21,63] > 63 Moneyness Toal Daily Median Toal Daily Median Toal Daily Median Toal < [0.92,0.96) [0.96,1.00) [1.00,1.04) > Toal Noes: This Table repors he number of observaions for differen moneyness and mauriies during he period January 2000 unil December Moneyness is defined in he following way: a call opion is said o be deep in-he-money if X/F < 0.92, in-hemoney if X/F [0.92,0.96), near- or a-he-money if X/F [0.96,1.00] or X/F [1.00,1.04) and ou-of-he-money if X/F > 1.04, where X is he srike price and F is he forward price. Toal is he oal number of opions of a paricular moneyness and/or mauriy caegory. Max is he maximum number of opions priced on one rading day of he paricular moneyness and/or mauriy caegory. 25

28 Table 2: Parameer Esimaes Alpha Gamma Bea0 Bea1 Local Volailiy Pricing Error (In-sample) Pricing Error (Ou-of- Samplesample) Panel A Mean SD Min Max quarile quarile Panel B Mean SD Min Max quarile quarile Noes. The able presens he average parameers esimaes, sandard deviaion, min, max, 1. and 3. quarile of he daily esimaions of he model during he period January 2000 unil May Addiionally, we repor in-sample and ou-of sample pricing errors. Panel A shows he resuls wihou swiching and Panel B wih swiching. 26

29 Figure 1: Simulaion Pah 0,0009 0,9 0,0008 0,8 0,0007 0,7 0,0006 0,6 0,0005 0,0004 0,5 0,4 HF HC H w (righ axis) 0,0003 0,3 0,0002 0,2 0,0001 0,

30 Figure 2: Parameer Esimaes over ime 0, , ,3 0, , ,1 60-0,2 40-0,3-0,4-0,5 Bea0 Alpha Bea1 Gamma (righ scale)

31 Figure 3: Esimaed Volailiy and in-sample Pricing Error over ime 0,4 6 0,35 5 0,3 0,25 4 0,2 3 0,15 2 0,1 0,05 0 Local-/LG- VOL In-sample PE