Volatility Clustering with Learning and Model Heterogeneity

Transcription

1 Volailiy Clusering wih Learning and Model Heerogeneiy Daniel Andrei Michael Hasler Augus 8, 11 Absrac We consider a sandard Lucas economy wih a single consumpion ree and wo agens. The agens do no observe he drif of he dividend sream, assumed o follow an Ornsein-Uhlenbeck process. Consequenly, agens filer i and updae heir beliefs under he main assumpion ha hey differenly perceive he lengh of he business cycles. We explain boh he level and he dynamics of he sock marke volailiy in close connecion wih he sae of he economy. The sock marke volailiy is significanly higher han he consumpion volailiy and srongly counercyclical a low expeced dividend growh rae riggers high sock marke volailiy. We highligh he endogenous mechanism ha generaes our resuls, namely a persisen disagreemen among agens. Indeed, using Malliavin calculus we show ha he volailiy clusering effec is exclusively implied by he laer process. The auhors are graeful for conversaions wih Bernard Dumas, Julien Cujean, Jerome Deemple, Julien Hugonnier, Arvind Krishnamurhy, Pascal S.-Amour, and for helpful commens from paricipans of he SFI finance seminar in Gerzensee 1 and he 4h Inernaional Forum on Long Term Risks in Paris 11. Swiss Finance Insiue, Universiy of Lausanne, Ecole des HEC. dandrei@unil.ch, webpage: Swiss Finance Insiue, Ecole Polyechnique Fédérale de Lausanne. michael.hasler@epfl.ch, webpage: people.epfl.ch/michael.hasler 1

2 1 Inroducion The excess-volailiy puzzle, idenified firs by Shiller 1981 and LeRoy and Porer 1981, saes ha sock reurns volailiy sysemaically exceeds ha jusified by fundamenals large changes in sock prices canno be aribued o changes in dividends or discoun raes. Even hough asse pricing models have made progress in delivering high sock marke volailiy 1, researchers recognized ha no only he level is imporan, bu he dynamics of he volailiy calls for a heoreical explanaion. Noably, he sock marke volailiy ends o ac counercyclically, being larger in bad imes han in good imes, suggesing ha i migh be driven by he volailiy of marcroeconomic fundamenals. This is no he case eiher, as shown empirically by Schwer 1989 and Mele 8. Boh sudies confirm ha he variabiliy of he macroeconomic facors is no imporan enough o explain he counercyclical marke volailiy. Thus, he non-obvious relaionship beween sock marke volailiy and macroeconomic variables calls for an endogenous mechanism relaed o he invesors maximizing behavior. Following his reasoning, i has been argued ha eiher he uiliy funcion ype or he learning mechanism migh be poenial imporan drivers of he sock marke volailiy dynamics. Campbell and Cochrane 1999, Barberis, Huang, and Sanos 1, and McQueen and Vorkink 4 assume sae-dependen uiliy funcions. In hese sudies, agens worry abou a reference consumpion level or abou pas consumpion when building heir uiliy funcions. These feaures are amenable o large changes in he sock marke volailiy. David 1997 and Veronesi 1999, show ha he filer of a coninuous-ime Markov chain is coninuous and incurs a sochasic diffusion. The ime-varying naure of he filered fundamenal s diffusion implies ineresing sock reurns properies and in paricular a sochasic volailiy. Veronesi 1999 claims ha: In coninuous-ime models, he assumpion of Gaussian-ype diffusion processes for unobservable variables has he undesirable oucome ha, if invesors priors are also assumed o be normally disribued, he condiional variance of invesors expecaions is deerminisic. Therefore he assumpion of Gaussian disribuions is ill-suied o sudy issues relaed o changing volailiy. In his paper, we show ha even hough he volailiy of he filered fundamenal is deerminisic, he disagreemen among agens implied by differen perceived business cycles lengh is persisen. Therefore, in a purely Gaussian world, he sock marke volailiy 1 See Bansal and Yaron 4, Dumas, Kurshev, and Uppal 9, and Brennan and Xia 1 among ohers. See Engle 198, Bollerslev 1986, Nelson 1991, and Mele 8 among ohers and he references herein for empirical evidence on ime-varying and counercyclical volailiy.

3 clusers and is close o be inegraed. The goal of his paper is o explain boh he level and he dynamics of he sock marke volailiy in close connecion wih he developmen of he business cycle. Our resuls are purely driven by he following assumpion: Boh agens agree o disagree on he lengh of he perceived business cycles. We show ha he persisence in he disagreemen among agens may be responsible for he high and counercyclical sock marke volailiy observed in he daa. Indeed, he model suggess ha he marke volailiy is: i, around 5 imes larger han he oupu volailiy, and ii, way more imporan in bad imes han in good imes. These resuls are novel o he difference of beliefs lieraure and are implied by he rich and ineresing dynamics of he disagreemen process. Agens heerogeneiy in beliefs has become very popular in he finance lieraure. Typically, he economy is populaed by wo agens whose filraions are generaed by he observables, namely he dividend process and poenially a macroeconomic signal. The agens do no observe he pah of he Brownian moions driving he economy, and hus have o filer ou he fundamenal he drif of he dividend process using Bayesian updaing echniques. Boh agens consider he same model for he fundamenal bu hey eiher have differen priors or hey differenly inerpre he informaiveness of he macroeconomic signal. These assumpions imply divergen forecasing dynamics and consequenly, rich asse pricing implicaions. In summary, heerogeneiy in beliefs arises because agens agree o disagree on he prior disribuion of he fundamenal or because hey miss-undersand he qualiy of he informaion source. In conras o he exising lieraure, we consider a new poenial source of diverging opinions. Tha is, we assume ha invesors differenly perceive he lengh of he business cycles he cycles incurred by he fundamenal. Since he fundamenal is non-observable, none of he agens can claim and know ha his model is he correc one. More likely, boh agens perceive business cycles lengh ha are slighly differen from he rue business cycles lengh. Our model is buil in a coninuous-ime pure exchange Lucas 1978 economy wih a single perishable consumpion good he numéraire, a single risky asse ha represens a claim o he unique oupu sream he dividend, and agens ha are unable o observe he drif of he dividend process he fundamenal. We assume ha he fundamenal follows an unobservable Ornsein-Uhlenbeck process for which agens differenly esimae is mean reversion speed. Therefore and as already menioned, boh agens perceive differen business cycles lengh. We highligh a surprising resul: a drif parameer ends up in he condiional volailiy of sock reurns and generaes high and counercyclical volailiy. Since he fundamenal is unobservable, boh invesors have o esimae i and up- 3

4 dae heir beliefs by applying he Kalman-Bucy filering mehod. Noe ha boh filers dynamics differ because boh agens consider differen probabiliy measures. One imporan and simplifying assumpion is ha agens keep heir own percepion of he world forever. In oher words, hey do no learn abou he acual business cycles lengh. We suppose ha agens exhibi differences of opinions even hough hey all possess he same se of informaion. The disagreemen among agens arises because hey do no possess complee srucural knowledge of he sochasic process underlying he economic aciviy. Insead, hey have a finie hisory of daa o esimae a reasonable and susainable model. This assumpion follows Kurz 8, who poins ou ha i is inconceivable for agens o possess perfec srucural knowledge of he economy. Following his line of reasoning, he assumpion of differen perceived business cycles lengh is relaed o he observaion ha, for coninuous-ime models wih a linear drif, sandard esimaion mehods yield biased esimaors for he mean reversion parameer. This is a raher old resul 3, bu sill of ineres in he economeric lieraure 4. More imporanly, he bias is of he order of T 1 and no of he order of n 1, where T is he daa span and n he number of observaions. Consequenly, increasing he sample size by increasing he sampling frequency canno yield a consisen esimaor. Tha is, he bias does no go away unless T goes o infiniy. As an example, le us assume ha we have o esimae he mean reversion speed of an observable Ornsein-Uhlenbeck process wih mean revering parameer worh.. Then, -years of daily daa yield a bias of magniude.1. Moreover, if he Ornsein-Uhlenbeck process is unobservable, as in our seup, we believe ha he bias should be even more severe. Using he resul of Cheng and Scaille, he augmened sae vecor of he economy is sandard affine, by consrucion. Consequenly, we can use he heory on affine processes and he resuls from Dumas, Kurshev, and Uppal 9 o solve for he price, he wealh, and all he associaed variables in closed form. Moreover, since he marke is complee in equilibrium, an applicaion of Iô s lemma on he sae-price densiy leads o a sraighforward characerizaion of he risk free rae, he marke price of risk, and hence he risk premium. Since his seup especially implies ineresing sock reurn volailiy dynamics, our main focus is he decomposiion of he volailiy. Indeed, his analysis permis o exacly disenangle he impac each sae variable has on he volailiy. We show ha even if boh agens consider business cycles lenghs ha are relaively close o each oher, he sock reurn volailiy clusers and is close o be inegraed. This is no he case in a benchmark seup where boh invesors perceive he same cycles lenghs. When he disagreemen among agens comes from a mispercepion 3 Hurwicz See Yu 9 for more deails relaed o he case where he mean revering process is observable. 4

5 of he informaion qualiy of a public signal, such as in Dumas, Kurshev, and Uppal 9, he sock reurn volailiy does no cluser eiher. We are able o show ha he origin of he clusering effec is purely due o he persisence of he disagreemen process. In such a seing, volailiy becomes counercyclical in he sense ha i is high when he fundamenal is low and vice versa. The srucure of he paper is as follows. In Secion, we provide he seup in deails, solve he filering problem using Lipser and Shiryaev 1, characerize he affine sae vecor, and solve for he equilibrium quaniies. In Secion 3, we perform an in-deph analysis of he resuls. More precisely, we show hrough Taylor expansions, simulaions, and Malliavin decomposiion ha our model always induces a GARCH-ype volailiy process. This resul is shown o be exclusively coming from he ineresing dynamics of he disagreemen process. Finally, Secion 4 provides a brief analysis of he survival and Secion 5 concludes he work. Derivaions and proofs are given in he Appendix. The Model In order o undersand he mechanisms ha ake place in a very general model, we adop a sep by sep approach in building our seup. More precisely, we sar from a very simple and sandard seing in Secion.1, a model ha we call M 1, and finish wih he broadly general model M 4 in Secion.3. Our firs seup, M 1, considers a complee informaion represenaive agen model. Tha is, he represenaive agen observes all Brownian moions and consequenly all sochasic processes driving he economy. We show ha he volailiy is large bu no counercyclical enough o imply GARCH effecs. Second, in Secion. we assume ha he fundamenal he drif of he dividend process is unobservable. Hence, he represenaive agen opimally learns abou he fundamenal by using Bayesian echniques. This is model M. Comparing M 1 and M permis o figure ou he impac of learning on he volailiy process. Third, in model M 3 we assume ha wo agens populae he economy. Agens A and B have differen beliefs regarding he unobservable fundamenal s law of moion. More precisely, Agen A believes ha he fundamenal is a consan while Agen B believes ha i follows a mean revering process, hus rying o infer consanly he sae of he business cycle. Finally, in model M 4 boh agens agree ha he fundamenal is mean revering. However, Agens A and B differ from each oher because heir esimaion of he fundamenal s mean reversion is differen. Agens are herefore in disagreemen wih respec o he lengh of business cycles. Models M 3 and M 4 are simulaneously derived in Secion.3. This sep by sep approach permis o pin down he origins of our resuls. Alhough 5

6 he mahs beyond all he compuaions are edious, we ry o make his secion as readable as possible. Therefore, and for convenience, we relegae mos of he compuaional deails o he Appendix..1 Represenaive Agen wih Observable Fundamenal: M 1 The economy is populaed by a represenaive agen who consumes he dividend sream provided by a single risky asse. The sole consumpion good s price is equal o uniy, he represenaive agen has power uiliy over consumpion wih CRRA coefficien α, and he invesmen opporuniy se consiss in one risky asse in posiive supply of one uni and one risk free asse in zero ne supply. The risky asse is defined as being a claim o he dividend process. In his basic example, we suppose ha he represenaive agen is able o observe he fundamenal; in he subsequen secions his assumpion will be relaxed. The sochasic differenial equaions characerizing he dividend sream are d = f d + σ dw df = λf f d + σ f dw f where W and W f are wo independen sandard Brownian moions. Wihou going ino he compuaional deails i can be easily shown ha he sae-price densiy saisfies α ξ = e ρ where α and ρ are respecively, he coefficien of relaive risk aversion and he subjecive discoun rae peraining o he represenaive agen s uiliy funcion. Consequenly, he sock price is given by S = α + e ρs E 1 α s ds. Since he expecaion erm can be viewed as being he momen-generaing funcion of ζ ln, we can apply he heory on affine processes from Duffie 1 o ge a closed form expression for his quaniy 5. The calibraion used is provided in Table 1. These parameers are inspired from Dumas, Kurshev, and Uppal 9 wih significanly lower values for he fundamenal s volailiy and he dividend s volailiy. The oupu process is herefore calibraed o he consumpion daa insead of he aggregae dividend daa. Ino ha respec, we are closer o he calibraion of consumpion from Brennan and Xia 1, wih wo 5 All he compuaions are sraighforward. Deails can be provided upon reques. 6

7 main differences. Firs, we consider a smaller coefficien of risk aversion he value of 15 chosen by Brennan and Xia 1 seems unrealisic. Second, we assume a posiive rae of impaience subjecive discoun facor, since our aim is o sudy second order momens of sock reurns, insead of maching he riskfree ineres rae. Parameers Symbol Relaive Risk Aversion α 3 Subjecive Discoun Facor ρ.1 Dividend s Volailiy σ.5 Fundamenal s Mean Reversion Speed λ.1 Fundamenal s Long Term Mean f.5 Fundamenal s Volailiy σ f.15 Table 1: Calibraion. Since here are only wo sae variables, he sock reurn diffusion has wo componens namely i σ, he dividend componen, and ii E f, he exposure coming from he fundamenal. Figure 1 plos he sock reurn volailiy and he diffusion componens wih respec o he fundamenal. I is obvious from Figure 1a ha he volailiy is high and monoone decreasing in he fundamenal. In oher words, volailiy has he endency o be counercyclical. However, given ha he slope of he curve is small in absolue value, he counercyclical feaure is no significan enough o induce GARCH effecs. Figure 1b shows ha he firs volailiy componen is equal o σ ; he second componen is negaive and much higher in absolue value. Figure depics he volailiy and he diffusion componens when he fundamenal is non persisen. Figures 1 and sugges ha he volailiy is i, close o be consan, and ii, higher when λ =.1 han when λ =.. Indeed, he volailiy is worh roughly 5% in he persisen case and 15% in he non persisen case while he dividend volailiy is worh only 5%. This resul arises because persisence brings uncerainy abou he long run. Thus, assuming ha he fundamenal is sochasic and slowly mean revering migh help explain he excess-volailiy puzzle 6.. Represenaive Agen wih Unobservable Fundamenal: M We now consider he same model as in he previous secion bu under he assumpion ha he represenaive agen is unable o observe he fundamenal. Thus, he only informaion available is he hisory of dividends. Because of his feaure, he represenaive agen chooses o esimae he fundamenal using he Kalman filer. This model 6 Bansal and Yaron 4. 7

8 Volailiy.3 Diffusion Componens f f a Figure 1: Volailiy and Diffusions when he Observable Fundamenal is Persisen. 1a illusraes he sock reurn volailiy and 1b he diffusion componens vs. he fundamenal. The solid line is he he dividend s componen and he dashed line he fundamenal s componen. The calibraion is given in Table 1..3 b Volailiy.3 Diffusion Componens f f a Figure : Volailiy and Diffusions when he Observable Fundamenal is non Persisen. a illusraes he sock reurn volailiy and b he diffusion componens vs. he fundamenal. The solid line is he he dividend componen and he dashed line he fundamenal s componen. λ =. and he oher parameers are provided in Table 1..3 b 8

9 is very similar wih Brennan and Xia 1. Under he assumpion ha he learning period was sufficienly large for he poserior variance o converge o is seady-sae level 7, he pos filering sysem is wrien d = f d + σ dŵ d f = λf f d + γ f dŵ σ where f = E f O is he esimaed fundamenal a ime, γ f = λσ is he poserior seady-sae variance, dŵ = 1 σ d 1 σ λ σ + σ f f d defines a sandard Brownian moion under he represenaive agen s filraion O = σ ds s : s. All he deails relaed o he Kalman filer are provided in Lipser and Shiryaev 1. Since his model is again relaively simple, all he compuaional deails are available upon reques. Inuiively, since he Kalman filering procedure is a projecion of he fundamenal on he represenaive agen s filraion, we expec ha he loading, γ f σ, ha appears in Equaion 1 should be lower han σ f 8. Consequenly, he filered fundamenal is less volaile in he model wih learning han he fundamenal in he model wihou learning and so is he sock reurn volailiy oo. Figure 3a depics he sock reurn volailiy for he cases λ =.1 and λ =.. I shows ha he sock reurn volailiy is, as in Secion.1, close o be consan. Moreover, he levels of he volailiy are respecively, 15% and %, for he wo cases considered, while hey were 5% and 15% in model M 1. Conrary o Brennan and Xia 1, we obain ha he learning decreases he sock price volailiy. While his resul seems surprising, here is a unique reason for ha: he mean-reversion speed of he fundamenal. In Brennan and Xia 1 i is calibraed o.7, which makes he fundamenal very persisen. In ha case only he change in volailiy can be reversed, and a higher sock reurn volailiy can be obained for he learning case. In general, however, he sock reurn volailiy ends o be lower in he learning case. Several insighs can be drawn a his poin from he models described in Secions.1 and.. Firs, a persisen fundamenal is likely o produce significan sock 7 This is no a srong assumpion. I is well known ha in he Kalman filer he variance follows a deerminisic process. The convergence o he seady sae is very fas. Separae compuaions show ha he variance reaches he seady sae afer roughly years of daily daa. 8 Sar wih λσ σ λ σ + σ f <. Then λσ λσ σ λ σ + σ f < or σf σ + λσ λσ σ σ λ σ + σ f λσ < σ fσ. λ σ + σ f < σf σ. I follows ha σ λ σ + σ f λσ < σf σ, or 9

10 Volailiy.3 Volailiy f f a λ =.1 b λ =. Figure 3: Volailiy when he Fundamenal is Unobservable. Sock reurn volailiy vs. he esimaed fundamenal f. If no oherwise menioned, he parameers are given in Table 1. reurn volailiy, no maer wheher he fundamenal is observable or no. Second, if he fundamenal is unobservable, by means of he learning process, he sock reurn volailiy decreases in general 9. Third, neiher he flucuaing fundamenal, nor he learning process is able o produce significan changes in he volailiy. In he following secion we consider an economy populaed by wo agens. We assume ha hese agens have slighly differen views regarding he process governing he unobservable fundamenal. We show ha his disagreemen urns ou o be a very powerful driving force of he dynamics of he volailiy..3 Two Agens wih Differen Beliefs: M 3 and M 4 In his secion we consider wo differen seups. Since he compuaional seps necessary o solve for he equilibrium are similar for boh seups, we choose o presen hem simulaneously. Separae and deailed compuaions are shown in he Appendix. In he firs seup, model M 3, Agens A and B believe ha he fundamenal is a consan and a mean revering process, respecively. We consider his inermediary sep because i permis o gain some inuiion on how and why volailiy clusers while remaining relaively simple o solve. Then, in model M 4, we assume ha boh agens agree on he ype of process ha governs he fundamenal; an Ornsein-Uhlenbeck process. Bu, Agens A and B disagree on he value of he fundamenal s mean reversion speed 1. This assumpion can be moivaed by he findings of Yu 9 who shows ha a sandard esimaion of he mean reversion speed of a coninuous- 9 Noe ha his should no necessarily be he case if he agen would observe an addiional signal correlaed wih he fundamenal. Here, he only source of informaion available is he hisory of dividends. 1 In separae compuaions we considered differences beween he oher parameers. I urns ou ha he mos imporan parameer is he mean-reversion speed. 1

11 ime process yields a significan bias. Moreover, we conjecure ha Agens A and B use cerainly ime-series ha are no of he same lengh and poenially no of he same frequency. Consequenly, heir respecive esimaed mean reversion speeds could be differen. Differen mean-reversion speeds has a clear economic meaning. I implies ha he agens do no agree on he lenghs of he business cycles. One of he agens believes ha he economy will rever faser o he long-erm mean, while he oher agen believes ha he cycles are longer. This form of disagreemen seems plausible and is ofen observed among analyss, when hey forecas he lengh of a recession or an expansion. There is one perishable good, he numeraire, wih price equal o uniy. All quaniies are expressed in unis of his good. The economy is populaed by wo agens, A and B, who opimally consume an exogenous dividend sream provided by one risky asse. Thus, he invesmen opporuniy se consiss in one sock, in posiive supply of one uni, wih price S and one risk free asse, in zero ne supply, wih price S. The ime horizon is infinie and boh agens are assumed o have power uiliy over consumpion wih RRA parameer α and subjecive discoun rae ρ. Moreover, we assume ha he observable variables are he dividend sream and he price process. Consequenly, boh agens have o filer ou he fundamenal under he assumpion ha i follows some realisic process. Since observing he price process does no add any addiional informaion, he agens filraion is defined by O { ds s : s }. On he one hand, Agen A s percepions of he processes driving he economy are d = f A d + σ dwa f A = f for he model M 3 df A = λ A f fa d + σf dw f A for he model M 4 where WA and W f A are wo independen Brownian moions under agen A s probabiliy measure P A. As previously saed, Agen A believes ha he fundamenal eiher is a consan model M 3, or is mean revering model M 4. On he oher hand, Agen B conjecures ha d = f B d + σ dwb λ B f fb d + σf dw f B for he model M 3 df B = λ B f fb d + σf dw f B for he model M 4 3 where W B and W f B are wo independen Brownian moions under Agen B s probabiliy 11

12 measure P B. Noice ha, in boh models, he long erm means of he fundamenal are equal o f. Therefore, he disagreemen among agens disappears in he long run. The sochasic differenial equaions SDE and 3 confirm ha Agens A and B have differen percepions of he process governing he fundamenal. We assume ha none of hem is righ since i would be unrealisic o assume ha one agen is lucky enough o fall exacly on he unobservable ruh. Noice ha his assumpion does no have any consequences on he equilibrium quaniies. Obviously, he equilibrium can sill be compued under eiher P A or P B irrespecive of knowing wheher one of hese wo probabiliies is he righ one..3.1 The Filering Problem We denoe by f A and f B, Agen A and Agen B s filered fundamenal, respecively. These are defined by f for he model M 3 f A E PA f A O for he model M 4 f B E PB f B O. In M 3, Agen A assumes ha he fundamenal is consan and equal o f while Agen B esimaes f using he Kalman filer. In M 4, boh agens apply he Kalman filer. Since he soluion procedure is very similar for boh models, in wha follows, we describe only he derivaions of model M 4. The compuaional deails relaed o model M 3 and M 4 can be found in Appendix A and B, respecively. Using Theorem 1.7 of Lipser and Shiryaev 1, he condiional means f A and f B have o saisfy he following SDE d f i = λ i f f γ i i d + dŵ i, for i {A, B} σ where γ i denoes he poserior variance for each of he agens. Noe ha Ŵ A and Ŵ B are Brownian moions under P A, O and P B, O, respecively. As in he previous secions and as in Scheinkman and Xiong 3 and Dumas, Kurshev, and Uppal 9, we assume ha he learning period was sufficienly large for he poserior variances o converge o heir seady-sae levels γ A and γ B. Deailed derivaions are provided in Appendix B.1. Thus, Agens A and B have he following sysem in mind d = f i d + σ dŵ i d f i = λ i f f i d + γ i σ dŵ i, i {A, B}. 1

13 In order o solve for he opimal consumpion policies as well as for he asse price, i is more convenien o work under a unique probabiliy measure, say P B. The Radon- Nikodym derivaive defining he change of measure beween P A and P B is wrien dp A dp B = e 1 O ν s ds νsdŵ Bs η for some process ν ha has o be deermined. η capures he way Agen A s probabiliy measure differs from Agen B s probabiliy measure. Dumas, Kurshev, and Uppal 9 call η he senimen variable. By applying Girsanov s Theorem we obain dη η = 1 σ ĝ dŵ B 4 dŵ A = dŵ B + 1 σ ĝ d where ĝ f B f A = σ ν represens he disagreemen among agens. Is dynamics are dĝ = λ A λ B f B f γa σ + λ A ĝ d + γ B γ A dŵ B σ. 5 The compuaional deails are provided in Appendix B.. Equaion 5 shows ha he disagreemen is mean revering, as in Dumas, Kurshev, and Uppal 9. However, in our model, ĝ mean-revers around f B ; in Dumas, Kurshev, and Uppal 9 i meanrevers around zero. More imporanly, Equaion 5 shows ha f B is muliplied by λ A λ B. Thus, if agens disagree on he value of he fundamenal s mean reversion speed, he persisence of f B parially dicaes he persisence of ĝ. In oher words, a highly persisen fundamenal f B ranslaes ino a highly persisen disagreemen among agens. We would like o emphasize ha his channel of persisence creaion in he disagreemen process is new o our knowledge. As shown laer in Secion 3, his new channel is exremely powerful in explaining he ime-varying behavior of he volailiy. Indeed, i urns ou ha he ineresing dynamics of ĝ ranslae ino a GARCH-ype sock reurn volailiy process. The parameer responsible for his new channel is exacly he one prone o esimaion errors, namely, he mean revering speed. Noe ha if he wo esimaes are equal, e.g. λ A = λ B, hen his effec compleely disappears. Since, from now on, we work under P B, O, we define he condiional expecaion operaor for noaional ease. E PB. O E. 13

14 .3. Equilibrium Pricing We assume ha boh agens have he same CRRA uiliy funcion defined by ρ c1 α U, c = e 1 α Noice ha he marke is complee in equilibrium since here is one sock and a single source or risk. Consequenly, we can solve he problem using he maringale approach of Karazas, Lehoczky, and Shreve 1987 and Cox and Huang Firs, we consider each agen s consumpion problem. Second, we wrie he firs order condiions peraining o he corresponding opimizaion problems. Third, we impose marke clearing and we obain he following characerizaion of he sae-price densiy ξ = e ρ α [ η 1/α ] 1 1/α α + 6 κ A where κ A and κ B are he Lagrange mulipliers associaed wih he budge consrains of Agens A and B, respecively. Subsiuing ξ in he agens opimal consumpion policies yields he linear consumpion sharing rule κ B c A = ω η c B = [1 ω η ] where ωη denoes Agen A s share of consumpion. The laer saisfies ω η = η 1/α κ A η 1/α 1/α. 7 κ A + 1 κ B Equaion 7 shows ha he opimal share of consumpion accouns for he senimen variable η. On he one hand, if η ends o infiniy, which means ha Agen A s measure is more likely han Agen B s measure, hen Agen A s share of consumpion converges o one. On he oher hand, if η ends o zero, hen ωη converges o zero oo. As expeced, Agen A s consumpion share ωη increases wih he likelihood of Agen A s probabiliy measure η. Equaion 6 shows ha he sae-price densiy ξ depends also on he senimen variable η. Thus, since η is driven by he disagreemen ĝ See Equaion 4, a highly persisen disagreemen implies a highly persisen sae-price densiy and consequenly a clusered sock reurn volailiy. Exended explanaions and deails will be presened in Secion We leave he echnical deails in Appendix A.1. 14

15 .3.3 Sock Price and Volailiy As in Dumas, Kurshev, and Uppal 9, we assume ha he coefficien of relaive risk aversion α is an ineger 1. This specificaion allows us o characerize he price of he single-dividend paying sock S T S T [ ] ξt = E T ξ = e ρt 1 ωη α α in he following way α α j= j 1 η j α j ωη E 1 ωη η j α T 1 α T All he derivaion s seps are provided in Appendix B.3. The sock price is hen simply he sum of he single-payoff socks. Tha is, S = S u du. Since he las expecaion in Equaion 8 is precisely he momen-generaing funcion of he vecor ζ ln, µ ln η, we can use Duffie 1 heory on affine processes o compue his quaniy. Indeed, since he sae vecor ζ, f B, ĝ, µ is affine-quadraic, we follow Cheng and Scaille and we show ha he vecor of sae variables X defined by is sandard affine. X = [ ζ f B ĝ µ ĝ ĝ f B f B Then, solving a se of Ricai equaions yields he momengeneraing funcion defined in Equaion 8. The sock price is obained by numerically inegraing he single-dividend paying socks. We again relegae all compuaional deails o Appendix B.4. Since we focus exclusively on he behavior of he sock reurn volailiy, he risk free rae, he marke price of risk, and he opimal porfolio allocaions are no repored in he paper bu are available upon reques. The ime- sock reurn volailiy saisfies ] σx σ = S X S = σx + S τ X dτ + S τ dτ where σx denoes he diffusion of he sae vecor X. 1 This ligh assumpion grealy simplifies he calculus. Under he assumpion of real coefficien of relaive risk aversion, he compuaions can sill be performed using Newon s generalized binomial heorem

16 3 Resuls Firs, we isolae he main effecs he disagreemen produces on he volailiy by considering model M 3. Then, we show ha he resuls obained in M 3 mainain for he more general model M 4. As previously saed, in model M 3 Agen B believes ha he fundamenal f is mean revering while Agen A believes i is a consan worh f. In his seup, all he economic variables are solved in closed form. For a reasonable calibraion, we perform a Taylor decomposiion of he sock price diffusion wih respec o f B a he poin f. This decomposiion allows us o show how he disagreemen affecs he dynamics of he sock reurn volailiy. Finally, we simulae he economy and we analyze he saisical properies of he sock price process. This is done in Secion 3.1. In Secion 3., we provide an explanaion on why he sock reurn volailiy dynamics are similar in models M 3 and M Model M 3 : The Resuls The parameers are adaped from Brennan and Xia 1 and Dumas, Kurshev, and Uppal 9 in order o insure finieness of he price process. The calibraion used is provided in Table. Parameers Symbol Relaive Risk Aversion α 3 Subjecive Discoun Facor ρ.1 Agen A s Iniial Consumpion Share ω.5 Dividend s Volailiy σ.5 Fundamenal s Mean Reversion Speed λ B.1 Fundamenal s Long Term Mean f.5 Fundamenal s Volailiy σ f.15 Table : Calibraion Used in Model M 3. The long erm mean has been fixed o.5 percen and he mean reversion speed used by Agen B o.1, corresponding o a half-life of approximaely 7 years. The dividend volailiy is calibraed o he volailiy of consumpion, i.e. 5 percen. This differs from Dumas, Kurshev, and Uppal 9 who assume ha σ =.13. Since in pure exchange economies markes have o clear, we believe ha equaing he dividend s volailiy σ o he empirical consumpion volailiy is a more judicious choice Volailiy Decomposiion In order o analyze how he volailiy reacs o a change in he disagreemen f B f, we perform a second order Taylor expansion of he diffusion process wih respec o f B a 16

17 he poin f. The accuracy of his approximaion is of he order of 1 3, a reasonable value. Noe ha he diffusion has hree componens namely, σ, he exposure coming from f B, and he exposure coming from he senimen variable η E E fb µ. The ime- diffusion σ is defined by σ = σ + E fb + E µ where and E E fb µ are funcions of f B and µ. Since, by definiion of he consumpion share of Agen A, here exiss a one o one mapping beween he processes ω and µ, and E E fb µ are funcions of f B and ω. The second order Taylor decomposiion of he sock reurn diffusion is hen σ C ω + C 1 ω f B f + C ω f B f. Figure 4 depics he behavior of he loadings C ω, C 1 ω and C ω wih respec o he curren share of consumpion ω. I shows ha, when he economy is populaed by a represenaive agen eiher A or B, he firs and second order loadings are worhless. In oher words, he diffusion is consan and worh eiher σ = σ =.5 or σ {.,.5,.15} depending on he value of λ and on wheher agens of ype A or ype B populae he marke. As expeced, when agens of ype A populae he marke, he sock reurn volailiy converges o he sandard Lucas volailiy σ. The lef hand side of Figure 4a suggess ha he agens learning procedure swiches he sign of he diffusion. Moreover, as λ becomes small enough, he absolue value of he diffusion becomes larger han σ. Hence, even hough agens have o filer ou he fundamenal, excess volailiy can sill be achieved. Figure 4b shows ha he disagreemen f B f has an imporan impac on he diffusion if and only if f B is very persisen. Indeed, he diffusion s impulse response o a change in he disagreemen is worh roughly 3 when λ =.1, while i is close o be worhless when λ >.1. This is no surprising given ha he sae-price densiy depends on he senimen variable which iself depends on he disagreemen. As a resul, a persisen disagreemen ranslaes ino a persisen sae-price densiy and consequenly ino a persisen volailiy. This mechanism, which represens our main conribuion, implies an almos inegraed GARCH-ype volailiy process. Noe ha he persisence belongs firs o he drif of he disagreemen see Equaion 5, hen i eners he diffusion of he senimen variable see Equaion 4, and finally i ends up in he diffusion of he sae-price densiy Iô s lemma on Equaion 17. Figure 5 depics he behavior of he sock reurn volailiy wih respec o he fundamenal when he consumpion share of Agen A is ωη =.5. Firs, when ωη =.5 he sock reurn diffusion is negaive for all values of he disagreemen. 17

18 .5 C Ω C 1 Ω Ω C Ω Ω Ω 5.15 a b c Figure 4: Taylor Decomposiion of he Sock s Diffusion. The loadings C ω, C 1 ω, and C ω are ploed agains ω for differen values of λ B. The solid line corresponds o λ B =.1, he doed line o λ B =.15, and he dashed one o λ B =.. The calibraion is provided in Table.

19 In oher words, he diffusion is minus he volailiy and i never crosses zero. This observaion can be explained by invesigaing Figure 4. Indeed, summing he Taylor componens C.5, C 1.5 f B f, and C.5 f B f yields a negaive number for all values of he fundamenal. Moreover, since he second order componen is large C.5 = 5, he curvaure of he diffusion is relaively imporan. As a resul, he diffusion remains negaive for all f B and i never crosses zero. Second, he volailiy is asymmeric. A posiive disagreemen implies a lower sock reurn volailiy han a negaive disagreemen. Figure 4 shows ha his effec is implied by boh: he slope parameer C 1.5 and he curvaure parameer C.5. Therefore, he volailiy is higher in bad imes han in good imes; a counercyclical behavior. Tha is, our model is able o reproduce he volailiy paern and volailiy smile observed in he daa. As an implicaion, we expec ha he model implied sock reurns are negaively skewed and incur fa ails. In he nex secion, we confirm hese expecaions. Volailiy f B Figure 5: Sock Reurn Volailiy vs. Fundamenal. Sock reurn volailiy as a funcion of he fundamenal f B. ωη =.5 and he calibraion is provided in Table Simulaions In order o confirm he resuls shown in Secion 3.1.1, we firs perform a 5-year simulaion of he dividend, he fundamenal, he price, and he volailiy a he daily frequency. Figure 6 illusraes one simulaed pah of hese 4 variables. The calibraion used is again provided in Table. As already explained in he previous secions, volailiy clusers over ime. Moreover, when he disagreemen f B f becomes significanly negaive, volailiy increases dramaically o a level of approximaely 35%. Figure 7 depics he corresponding reurn process. On he one hand, i confirms ha he sock reurn volailiy is very persisen and high when he disagreemen is large in absolue value. On he oher hand, i shows ha he volailiy is close o zero when he disagreemen is insignifican. 19

20 Dividend Fundamenal Time.9. Time Price Volailiy Time Figure 6: One Simulaed Pah of he Equilibrium Variables. The dividend, he fundamenal, he price, and he volailiy are simulaed over 5 years. Time Reurns Time.4.6 Figure 7: One Simulaed Pah of he Sock Reurns. Daily sock reurn behavior simulaed over 5 years. In order o make sure ha he saisical properies of sock reurns are robus, we mimic he above procedure a few imes. Then, for each simulaed ime series, we perform an ARCH1 es o check wheher our demeaned reurn series ǫ i, indeed follows a GARCH1,1 process. The Lagrange muliplier es is consruced as follows: we perform he following OLS regression ǫ i,+1 = α + βǫ i, + u where he subscrip i represens simulaion number i. defined by LM = T 1R χ 1 The lagrange muliplier is

21 where T is number of observaions considered and R he R-squared coming ou from he OLS regression. We do no repor he p-values associaed o his es because hese are equal o zero in 1% of he cases considered. Tha is, he null hypohesis of no ARCH effecs is srongly rejeced ARCH Parameer a GARCH Parameer b sa of ARCH Parameer c sa of GARCH Parameer d Figure 8: Model Implied ARCH and GARCH Parameers. Hisograms of he ARCH and GARCH parameers as well as heir corresponding -saisics. Figure 8 repors he values of he ARCH and GARCH parameers as well as heir corresponding -saisics. These values are obained hrough a GARCH1,1 fi. On he one hand, he high -sas peraining o hisograms 8c and 8d show ha he fied parameers are significan. On he oher hand, hisograms 8a and 8b confirm ha he sock reurn volailiy is close o be inegraed. Indeed, he sum of he ARCH and GARCH parameers are always very close o one. Finally, Figure 9 shows ha our model is able o replicae he negaively skewed and fa-ailed sock reurns. We do no repor he p-values associaed o a Jarque-Bera es of normaliy because hese are equal o zero for all simulaions. In oher words, 1

22 we rejec he null hypohesis of Gaussian sock reurns Reurns Skewness Reurns Kurosis Figure 9: Model Implied Sock Reurns Skewness and Kurosis. Hisograms of he skewness and kurosis of sock reurns. 3. Model M 4 : The Resuls Following exacly he same road as in Secion 3.1, we would like o know wheher he resuls of model M 3 hold in he more general model M 4 or no. Remember ha, in model M 4, boh agens believe ha he fundamenal is governed by an Ornsein- Uhlenbeck process. Agen A and Agen B conjecure ha he mean reversion speed is worh λ A =.3 and λ B =.1, respecively. Noe ha he way he mean reversion speeds differ from each oher corresponds o he bias discussed in Secion 1. The calibraion is provided in Table 3. Parameers Symbol Relaive Risk Aversion α 3 Subjecive Discoun Facor ρ.1 Agen A s Iniial Consumpion Share ω.5 Dividend s Volailiy σ.5 Fundamenal s Mean Reversion Speeds λ A.3 λ B.1 Fundamenal s Long Term Mean f.5 Fundamenal s Volailiy σ f.15 Table 3: Calibraion Used in Model M Volailiy Decomposiion hrough Malliavin Calculus In order o exacly pin down he source of he clusering effec, we go one sep furher and we decompose he sock reurn volailiy by means of Malliavin derivaive 13. We 13 Deails on Malliavin calculus can be found in Malliavin and Thalmaier 5.

23 leave all he echnical deails in Appendix C and we direcly proceed wih he soluion and is analysis. Denoe by D he Malliavin derivaive operaor a ime. Following closely he derivaions of Dumas, Kurshev, and Uppal 9, he sock reurn diffusion can be wrien σ = σ + 1 α [ ] ξ s s E s D f Bu duds S ξ + 1 [ ξ s E s ω η s ω η D ] η ds S ξ 1 S E [ ξ s ξ s ω η s σ + σ F + σ ω + σ G. 1 σ s η D ĝ u dŵ Bu + 1 σ s ĝ u D ĝ u du Equaion 9 shows ha he diffusion is given by he sandard Lucas 1978 volailiy σ added by hree erms ha represen respecively, he fuure evoluion of he fundamenal f B, he change in he fuure consumpion share ω, and he flucuaion in disagreemen ĝ. Because of he resuls obained in he previous secions, we expec he las expecaion erm o drive he enire dynamics of he volailiy. Indeed, since he sock s diffusion in he represenaive agen model wih learning M simplifies o σ = σ + 1 α [ ] ξ s s E s D f Bu duds, S ξ we know ha he erm peraining o he evoluion of he fundamenal f B does no imply any clusering effec. As a resul, he persisence has o come from he las wo erms. Since f B is persisen and since he disagreemen ĝ mean revers around he sochasic long erm mean f B, he disagreemen becomes persisen oo. Hence, given ha he disagreemen eners he diffusion of sae-price densiy hrough he senimen variable η, we srongly believe ha he clusering effec comes from he las erm of Equaion 9. In oher words, volailiy clusers because ĝ and is Malliavin derivaive Dĝ are highly persisen. Figure 1 illusraes one pah of he volailiy as well as he erms peraining o is decomposiion. Obviously, he erm ha represens he changes in ĝ and Dĝ is he sole driver of he volailiy dynamics. Indeed, he erms represening he changes in he consumpion share and he changes in he fundamenal are slighly moving over ime bu have no significan impac on he volailiy dynamics. In fac, since heir variances are small, we can almos assume ha hese wo erms are consan. Noe ha his confirms our resul of Secion.; he learning mechanism does no imply any clusering effecs. Consequenly, σ, σ ω, σ F, and σ G deermine he level of he ds ] 9 3

24 .3..1 Sock s Diffusion.1..3 σ ω σ F σ G σ σ Time Figure 1: Sock s Diffusion and Is Malliavin Componens. One simulaed pah of he sock reurn diffusion and is componens. The erms σ ω, σ F, and σ G are defined in Equaion 9. volailiy and σ G is dynamics. Figures 11a, 11b, 11c, and 11d depic he disribuions of he average level of σ ω, σ F, σ G, and σ. The laer show ha, on average, σ ω and σ G do no have a significan impac on he level of he volailiy. Indeed, Figures 3a and 11d confirm ha he average volailiy is similar in model M and M 4. To summarize, agens disagreemen on he lengh of he business cycles helps undersand he dynamics of he volailiy bu does no have a significan impac on is average level. 4 Survival The crucial assumpion of our model is ha he fundamenal is unobservable. Consequenly, i is reasonable o assume ha boh invesors have differen beliefs regarding he dynamics of he fundamenal. Addiionally, we believe ha i would be compleely arbirary and non-realisic o assume ha one of he wo agens has he correc beliefs namely, he rue probabiliy measure and he righ model. Thus, we assume ha he rue daa generaing process is d = f d + σdw 1 df = λ f f d + σ f dw f 11 where W and W f are wo independen Brownian moions under he rue probabiliyp and complee informaion filraion F. Moreover, we assume hroughou his secion ha he calibraion used by Agens A and B is given in Table 3. In oher words, boh 4

25 /T T σω,s ds a /T T σf,s ds b /T T σg,s ds c /T T σs ds Figure 11: Disribuion of he Average Diffusion Componens. Disribuions of σ ω, σ F, σ G, and σ where σ i = 1 T T σ i,sds for i {ω, F, G,.}. T is assumed o be 5 years. d agens misperceive he rue lengh of he business cycle. Table 4 provides he values of he mean reversion speeds and here corresponding half-lives. Since λ A =.3 and λ B =.1, we assume ha he rue value is λ =.. This corresponds o he bias of.1 discussed in Secion 1. To invesigae he speed a which Agen A or Agen B give up he economy, we compue he P-expecaion of he consumpion share of Agen A. Tha is, we simulae he dividend process using 1 and 11, we perform he learning exercise of Agen B o infer he perceived fundamenal f B, he disagreemen ĝ, and he senimen η, and, finally, we compue he expecaion of ωη T for T ranging from o 1 years. Figure 1 depics he P-expecaion of he consumpion share of Agen A over 1 years. I shows ha he consumpion share of Agen A decreases even hough boh agens incur a mean reversion speed s esimaion error ha is of he same magniude. This resul is even more surprising given ha Agen A s half-life is closer o he ruh han Agen B s half-life. This analysis suggess ha invesors who believe in he risk 5

26 λ Half-Life Agen A.3.31 Truh Agen B Table 4: Mean Reversion Speeds and Corresponding Business Cycle s Lengh. for he long run are expeced o gain a sligh share of consumpion. However, boh shares of consumpion remain very close o each oher and boh agens survive for way more han 1 years E P ω T Time Figure 1: Expeced Share of Consumpion of Agen A. P-expecaion of he consumpion share of Agen A, ωη T. The ime horizon T is represened on he abscissa. 5 Conclusion We build a pure exchange economy in which agens perceive differenly he lengh of he business cycles. Using he heory on affine processes, we are able o solve for he equilibrium variables in closed form. We show ha even hough he perceived business cycles lenghs are relaively close o each oher, he sock reurn volailiy clusers and is close o be inegraed. Thanks o he Clark-Ocone Theorem and o Malliavin calculus, we show ha he sole driver of he ime-varying naure of volailiy is he disagreemen process ĝ. The oher sae variables fix he level of he volailiy bu have almos no impac on is sochasic feaures. Finally, we claim ha he model implied marke volailiy is counercyclical being higher in bad imes han in good imes. 6

27 A Compuaional Deails for Model M 3 Le s consider an economy populaed by wo agens namely, A and B. Again, here is a single dividend ree whose drif is unobservable. Agen A percepions of he processes are d = f A d + σ dwa f A = f where W A is a Brownian moion under Agen A s probabiliy measure PA. In oher words, Agen A believes ha he unobservable fundamenal is consan. Agen B s percepion of he economy is d = f B d + σ dw B df B = λ B f fb d + σf dw f B where WB and W f B are wo independen Brownian moions under Agen B s probabiliy measure PB. The log-dividend ζ = log has dynamics dζ = σ + f B d + σ dwb 1 under Agen B s probabiliy measure. As usually, Agen B uses he available se of informaion o esimae he fundamenal. Using he noaion from Lipser and Shiryaev 1, we have and a = λ B f, a1 = λ B, b 1 = σ f, b = A = σ /, A 1 = 1, B 1 =, B = σ b b = b 1 b 1 + b b = σ f b B = b 1 B 1 + b B = B B = B 1 B 1 + B B = σ. Following Scheinkman and Xiong 3, we assume ha he learning period was sufficienly large for he poserior variance o converge o is seady-sae, γ B. Thus, Agen B s poserior variance, a he seady sae, solves a quadraic equaion wih a posiive and a negaive soluion. The posiive soluion is γ B = σ λ B + σ f σ λ B. Theorem 1.7 from Lipser and Shiryaev 1 implies he following fundamenal s dynamics d f B = λ B f fb d + γ [ ] B σ dζ σ + f B d. Moreover, dŵ B 1 ] [dζ σ σ + f B d represens a P B, O-Brownian moion. To summarize, Agen B s percepions are dζ = σ + f B d + σ dŵ B 13 d f B = λ B f fb d + γ B dŵ σ B, 14 7

28 while Agen A s view is simply dζ = σ + f d + σ dŵ A where Ŵ A is a Brownian moion under Agen A s probabiliy measure, PA. In order o solve for he opimal consumpion policies as well as for he asse price, i is more convenien o work under a unique probabiliy measure, say P B. Le s define he Randon-Nikodym derivaive dp A dp B = η O where O is again he se of informaion available a ime. By use of Girsanov s Theorem, we have dη = 1 fb η σ f dŵ B dŵ A = dŵ B + 1 fb σ f d. Performing he change of variable µ = log η, one obains dµ = 1 fb σ f 1 d fb σ f dŵ B. 15 Noice ha he relevan sae variables are defined by Equaions 13, 14, and 15. In order o keep he seup affine, we augmen he sae vecor wih he square of he fundamenal, f B. Is dynamics are d f γ B = B σ + fλ B fb λ B f B d + γ B fb dŵ B σ. 16 ] Consider he sae vecor X = [ζ, fb, µ, f B defined by Equaions 13, 14, 15, and 16. Following Duffie 1, X follows a mulivariae affine process of he form dx = µ X d + σ X dŵ B µ X = K + K 1 X [ σ X σ X ] = H ij + H 1ij X ij for some H ij R and H 1ij R 4. The marices K, K 1, H and H 1 are H 1 = K = γ B σ γb σ σ λ B f f σ γ B σ 1, K λ B 1 = f 1 σ σ fλ B λ B σ γ B f γ γ B fγ B H = B σ σ fγ f B f σ σ, H 11 = H 13 = [] γ B γb γ σ B σ 1 γb f, H 14 = 1 σ fγ B σ σ fγ B σ γb σ γb σ 4γ B σ. 8

29 A.1 Opimal Consumpion and Sae-Price Densiy Characerizaion We assume ha boh agens have he same CRRA uiliy funcion defined by ρ c1 α U, c = e 1 α. Noice ha he marke is complee in equilibrium since here is one sock and a single source or risk. Consequenly, we can solve he problem using he maringale approach of Karazas, Lehoczky, and Shreve 1987 and Cox and Huang Agen B s consumpion problem wries [ ] ρ c1 α B maxe e c B 1 α d [ ] s.. E ξ c B d x B where ξ denoes he sae-price densiy perceived by Agen B and x B his iniial wealh. The problem of Agen A under he measure P B is wrien [ ] ρ c1 α A maxe η e c A 1 α d [ ] s.. E ξ c A d x A. The firs order condiions are c B = κ B e ρ 1 α ξ 1 κa c A = e ρ α ξ η where κ A and κ B are he Lagrange mulipliers associaed o he budge consrains of Agen A and B, respecively. The marke clearing condiion implies ha 1 κa e ρ α ξ + κb e ρ 1 ξ α =. η This equaion gives he following characerizaion of he sae-price densiy [ 1/α ] 1/α α ξ = e ρ α η κ A κ B Subsiuing his expression in he opimal consumpion policies of boh agens yields where is Agen A s share of consumpion. ω η = c A = ω η c B = [1 ω η ] η η κ A 1/α κ A 1/α + 1/α 1 κ B 9