Heterogeneity and Option Pricing

Transcription

1 Heterogenety and Opton Prcng Smon Bennnga Tel-Avv Unversty and Unversty of Gronngen Joram Mayshar Hebrew Unversty, Jerusalem, Israel SOM-theme E: Fnancal markets and nsttutons ABSTRACT An economy wth agents havng constant yet heterogeneous degrees of relatve rsk averson prces assets as though there were a sngle decreasng relatve rsk averson prcng representatve agent. The prcng kernel has fat tals and opton prces do not conform to the Black-Scholes formula. Impled volatlty exhbts a smle. Heterogeneous belefs about dstrbuton parameters also mples non-lognormal prcng kernels wth fatter tals and overprcng of out-of-the-money optons. Heterogenety as the source of non-statonary prcng fts Rubnsten s (994) nterpretaton of the over-prcng as an ndcaton of crash-ophoba. Rubnsten s term suggests that those who hold out-of-the-money put optons have relatvely hgh rsk averson or relatvely hgh subjectve probablty assessments of low market outcomes. The essence of ths explanaton s heterogenety n nvestor atttudes towards rsks and probablty belefs. We have benefted from dscussons wth Antono Bernardo (our AFA dscussant), Yaacov Bergman, Guenter Franke, Bjarne Astrup Jensen, Shmuel Kandel, Jonathan Paul, and Mart Subrahmanyam, and Zv Wener. We also acknowledge the helpful comments of an anonymous referee and René Stultz. Bennnga s research was fnanced n part by a grant from the Krueger Center for Fnance at the Hebrew Unversty.

2 Heterogenety and Opton Prcng I. Introducton In ths paper we nvestgate the prcng of assets n an economy n whch there are multple agents wth heterogeneous tastes. The Arrow-Debreu model of general equlbrum under uncertanty does not restrct the heterogenety of ether the probablty belefs or the preferences of nvestors. In contrast, n the theores of asset prcng that followed Lucas (978) there typcally exsts a representatve consumernvestor whose preferences and probablty assessments of the economy s stochastc endowment prce all assets. Ths representatve nvestor s almost always assumed to have tme addtve preferences wth constant relatve rsk averson. It s now wellknown that the equlbrum framework necessary to derve the Black-Scholes formula for optons, gven a proportonal Brownan dffuson of the underlyng payout, requres the exstence of such a representatve consumer wth constant relatve rsk averson (see Rubnsten (976), Breeden and Ltzenberger (978),Brennan (979), Stapleton and Subrahmanyam (984), Bck (987, 990), and He and Leland (993) ). The assumpton that all nvestors have dentcal homothetc tastes and dentcal expectatons seems partcularly unreasonable. It s well known that ths assumpton mples that all nvestors have dentcal wealth composton. The emprcal evdence seems to contradct ths assumpton: Mankw and Zeldes (99), for example, report that famles that do not own any stock account for 62% of dsposable ncome. Another a recent study fnds that n 989 the top one percent of wealth holders held 36.2% of the total non-human worth of Unted States households and 62.5% of the busness assets and corporate stock held by households (Kennckell and Woodburn 992). In addton, whle a representatve-agent framework may prce all assets, t does not explan why there exsts open nterest n assets wth zero net supply, such as the optons, wth nvestors on both sdes (short or long) of the market. Some heterogenety among nvestors, n ether endowments, tastes or For a recent crtcsm of the practce of postulatng a sngle Αrepresentatve agent see Krman (992). A number of recent studes have consdered the case of heterogenety of a dfferent knd: agents who are ex-ante dentcal end up heterogeneous ex-post, due to the exstence of dosyncratc endowment shocks and due to market mperfectons that mpede nsurance aganst such shocks (see Mankw (986) and the many studes surveyed by Heaton and Lucas (995) ). We should further note that there are artcles wthn the representatveagent framework where the preferences of the representatve agent are not of the constant elastcty type.

3 opnons seems necessary to explan why such assets wll exst at all. 2 The formal analyss of the equlbrum underpnnngs of the Black-Scholes opton prcng formula (for example Bck (987, 990) and He and Leland (993)) retans the presumpton of a representatve agent. Yet heterogenety among nvestors s embedded n most nformal dscussons of optons markets. For example, Cox and Rubnsten (985, p. 54) gve the use of certan knds of specal knowledge as one reason for the exstence of trade n optons. Accordng to such popular vews, agents wth bullsh expectatons (perhaps based on specal knowledge ) wll be attracted to out-of-the-money calls (wrtten, presumably, by others wth more bearsh expectatons). On the other hand, out-of-the-money put optons are consdered to be bought by bearsh nvestors or by nvestors who are partcularly concerned about down-sde rsk. 3 We beleve that heterogenety among agents may also be the key for resolvng the emprcal non-congruence of the Black-Scholes formula whch has attracted a szeable lterature n recent years. In an essentally a-theoretcal framework, Rubnsten (994) and others have attempted to derve a pattern of Arrow-Debreu prcng that s mpled by observed opton prces (and s nconsstent wth the Black-Scholes framework). Franke et. al. (996) attempt to reconcle ths observed pattern wthn an equlbrum framework where the representatve agent dsplays declnng relatve rsk averson. In ths paper we seek, nstead, to examne the case of the prcng of assets and optons when all agents have the standard constant-elastcty tastes, but when agents tastes, and possbly also ther probablty assessments, dffer. One may wonder however whether consumer heterogenety per se could matter. The early formulators of the CAPM were concerned about the effects of the assumpton of homogenety of opnon. As Sharpe (970) wrote (p. 04): Even the most casual emprcsm suggests that ths [homogeneous opnons] s not the case. People often hold passonately to belefs that are far from unversal. Hs concluson, however, was that heterogenety of opnon s by and large rrelevant snce (p. 29) n a somewhat superfcal sense, the equlbrum relatonshps derved for a world of 2 As summarzed by Hrshlefer and Rley (979), wth regard to futures tradng: ΑAmong the possble determnants of speculatve actvty, John Maynard Keynes and John Hcks... have emphaszed dfferental rsk averson.... In contrast to these vews, Holbrook Workng has dened that there s any systematc dfference as to rsk-tolerance between those conventonally called speculators and hedgers. Workng emphaszes, nstead, dfferences of belefs (optmsm or pessmsm) as motvatng futures tradng. 3 Ths pont was forcefully made by Leland (980) n hs dscusson of portfolo nsurance. 2

4 complete agreement can be sad to apply to a world n whch there s dsagreement, f certan values are consdered to be averages. In a smlar ven, Constantndes (982) establshed that the asset prces that arse n an economy wth heterogeneous agents could be ratonalzed as f orgnatng from the preferences of a sngle prcngrepresentatve agent. 4 In Mayshar (977), one of us argued for the prcng relevance of heterogenety of opnon, clamng that when some nvestors are n a corner soluton (e.g. all those potental nvestors n the world who do not hold any partcular asset), the sources of the heterogenety that explan the corner solutons are relevant for the determnaton of what are the relevant averages and thus also for the prcng of assets. Corner solutons, and the dentty of actual versus potental nvestors, are relevant also for the case of heterogenety n tastes. In ths paper we advance a dfferent argument aganst the practce of assumng a representatve nvestor. As s common n the related lterature, we assume that markets are Arrow-Debreu complete, and that all the heterogeneous consumer-nvestors have reasonable tme-separable, constant elastcty utlty functons wth constant tme-dscount factors, so that no corner solutons exst. We demonstrate that wthn ths framework, when consumers dffer wth respect to ther rsk averson, the nduced preferences of Constantndes s prcng-representatve agent are consderably more complcated than those of the actual agents n the economy, and n partcular do not belong to the same class of reasonable preferences as thers. In partcular, we show that when consumers have dfferent constant rsk aversons, the prcng-representatve agent s preferences exhbt declnng relatve rsk averson. The prcng representatve consumer s preferences are thus not of the same class as those of the consumers he represents. We demonstrate the sgnfcance of ths result, and of other sources of nvestor heterogenety, for the prcng of optons. We show that nvestor heterogenety provdes a smple and ntutve explanaton for the emprcal puzzle concernng the non-congruty of the Black-Scholes formula for opton prcng wth realty. These results cast consderable doubt on the standard practce n the lterature of endowng the representatve agent wth reasonable (.e., constant relatve rsk averson) 4 Constantndes s representatve agent s representatve only for a gven set of endowments and wll not prce assets correctly f there s a change n the stochastc endowment. We thus dentfy the Constantndes representatve agent as prcng-representatve. As Rubnsten (974) has shown, condtons under whch there exsts a consumer who s unversally representatve are extremely restrctve (see also the survey by Shafer and Sonnenschen (982)). 3

5 preferences. The structure of the paper s as follows: In the followng secton we set out the equlbrum model, n whch all heterogeneous consumers lve for two dates. In Secton III we characterze the preferences of the prcng-representatve agent n the model. In Sectons IV to VI we llustrate the sgnfcance of our fndngs for the prcng of optons n ths economy. II. A two-perod Arrow-Debreu economy wth heterogeneous agents We assume a one-good, two-date exchange economy. The aggregate consumpton at date ( tomorrow ) s uncertan. We normalze the scale of consumpton so that total consumpton at date 0 ( today ), Y 0, equals to unty. We denote by Y s the strctly postve total endowment of future consumpton n state s, s =,... S. Each agent s assumed to have a fxed ntal fracton of ownershp w of the economy s endowment today and n each state of nature tomorrow. We denote s ntal-perod consumpton by y 0, and denote by y s the second-perod consumpton by agent n state s. Each of the H agents has tme-separable, expected utlty preferences that take the form: S γ y () U( y) = u( y0) + β fsu( ys), where u( y) = s= γ and where f s s agent s strctly postve probablty assessment of state s, ß s her subjectve rate of tme dscount and γ her constant degree of relatve rsk averson. Each agent s thus characterzed by the subjectve parameters {ß, γ, f s } and the fracton of ownershp n the aggregate endowment at dates 0 and, w ; we note that w s also ndvdual s s fracton of total wealth. We assume ß > 0, γ > 0, and w > 0. We assume the exstence of a full ntal set of Arrow-Debreu markets, so that n equlbrum there are no potental benefts to trade. Let p s denote the Arrow- Debreu equlbrum prce of contngent consumpton n state s. Each agent selects a consumpton program whch maxmzes U (y ) n () gven her budget constrant, S S (2) y0 + psys = w Y0 + psys s= s= Let { ~ y = { y0, y,, y S}, =,..., H} be the equlbrum allocaton n the economy. As s well-known, gven the partcular preferences assumed here, there wll be no corner solutons and all ~ y wll be strctly postve. In equlbrum, all 4

6 consumers margnal rates of substtuton equal the state prces: ' γ u( ys) ys (3) ps = βfs = βfs for all, s ' u( y0) y0 The prces p s are determned n a Walrasan equlbrum so as to equate the demand and supply for all the state-contngent goods. 5 H (4) y = Y, for all s = s S Gven the partcular pattern of tastes n ths economy, t s clear that no generalty s lost by normalzng Y 0 =. Let ω = y 0 / Y0 = y 0 denote agent s date 0 share of consumpton. The equlbrum prces are, of course, a functon of the agents characterstcs {β, γ, f s, w }, and of market quanttes {Y s }. By combnng (3) and (4), we can vew p s as determned by: / H (5) ω β γ fs = Ys, for all s = ps Ths equaton determnes the equlbrum prces p s as a functon of the aggregate consumpton quanttes Y s and the agents taste parameters. However, the prces n (5) are dependent on agents endogenously-determned shares of ntal perod consumpton {ω }, nstead of ther exogenous ntal shares of total wealth {w }. Ths transformaton of varables smplfes the presentaton below. The equlbrum condtons (3) - (4) and agents budget constrants (2) establsh a one-to-one relaton between agents ntal dstrbuton of wealth {w } and the dstrbuton of ntal consumpton {ω }. Gven the latter, and gven p s as determned by (5), we can consder the ntal wealth fractons as f determned by: S / γ p f + β s s p s s (6) w = = ω S + py s s s= 5 Market clearng for frst-perod consumpton s guaranteed by Walras s Law. 5

7 III. Identfyng preferences for a prcng-representatve ndvdual In ths secton we dentfy the characterstcs of a prcng-representatve agent n the above economy. We defne a prcng-representatve agent as one whose tastes are such that f all H agents n the economy had tastes dentcal to hs, then the equlbrum state prces n the economy would reman unchanged. As noted n the prevous secton, t s not possble to fnd a representatve consumer who can mmc market prces for any possble set of aggregate endowments {Y s }; we therefore look for preferences whch can mmc the prces n the gven economy. We assume that the utlty functon for the prcng-representatve agent takes the separable form: * * * * (7) U ( Y) = u0( Y0) + β fsu ( Ys) S s= The form of the utlty functon of the prcng-representatve agent, (7), assumes tme-separablty and expected utlty maxmzaton, but does not mpose the addtonal assumpton of a constant elastcty temporal utlty functon assumed n () above for each ndvdual. What we requre from ths functon s that ts margnal rates of substtuton be equal to the equlbrum state prces p s : ( Ys ) ( Y ) *' (8) β * u f s p, for all s *' s u0 0 = Equaton (8) presents a set of condtons from whch we can dentfy propertes of the preferences of the prcng-representatve agent. Gven our normalzaton Y 0 =, we defne the probablty normalzed prces q(y), by the condton: ( Y) () *' u (9) qs ( Y) = β * *' u0. The functon q(y) s suffcent to prce all state-contngent commodtes, snce by (8): ps (0) qs( Y) =, for all s. f s We now propose to dentfy propertes of the prcng representng agent by makng two assumptons: () The set of states of nature s suffcently dense, so that every level of non-negatve future aggregate consumpton s possble. () All agents have homogeneous belefs that concde wth the objectve probabltes, so that f s = f s for all s and. In Secton IV below we 6

8 reconsder the effect of heterogeneous probablty assessments. Gven these two assumptons t follows by comparng (5) and (8) that we requre the prcng functon q(y) to satsfy the mplct condton: / γ H ω β () =, for all Y 0 = Y q( Y) The functon q(y) s mplctly defned by equaton () for every level of aggregate consumpton Y (or n fact for every rate of consumpton growth Y/Y 0 ) by the set of nvestors taste parameters, {ß, γ } and by the ntal consumpton shares {ω }, whch, as shown n equaton (6) can be taken as a proxy for the ntal endowment shares {w }. *' *' To smplfy notaton further we make the normalzaton: u () u () 0 = =. Settng Y = n condton (9) dentfes the tme-dscount factor β * of the prcngrepresentatve agent: (2) β * = q(). By () t then follows that H (3) ω β / γ * =. = β It can then easly be establshed that the representatve tme-dscount factor β * s some average of all agents tme dscount factors ß, and n partcular s between Max {b } and Mn {b }. Condtons (9) and (2) then further dentfy the temporal utlty functon u * (Y) as the soluton for the dfferental equaton: *' qy ( ) (4) u ( Y) = q() Snce equaton (4) s assumed to hold as an dentty for all non-negatve values of Y, we can, by dfferentaton, defne the temporal degree of relatve rsk averson of the prcng-representatve nvestor: ( ) (5) γ * Yq' Y ( Y) =. qy ( ) Proposton : For any Y, γ * (Y) s a harmonc weghted average of ndvduals γ s. Thus, n partcular, γ * (Y) s bounded by Max {γ } from above and by Mn {γ } from below. Proof: By equaton (), q(y) s determned as the soluton of the mplct condton: 7

9 H y 0 = Y / γ β (6) FYq (, ) q =. It follows that: / γ F ω β (7) = Y Y 2 q =. Y and γ F ω β (8) = / q Yq γ q = q where / γ ω β (9) α = α( Y). Y q( Y) Ths means that: F / Y qy ( )/ Y (20) q' ( Y) = = F / q α / γ. (2) ( ) By ts defnton n (5), * γ Y = α / γ α. γ By equaton (3), the weghts a (Y) that were defned n (9), are smply the second-perod consumpton shares of agents, when the aggregate endowment s Y. By (), the α sum to one. Proposton shows that the rsk averson of the prcng representatve consumer s not a smple average of the rsk aversons of the ndvduals n the economy. As we see n the next proposton, ths means that for our model, where all the ndvduals n the economy have constant relatve rsk averson, the prcng representatve consumer has decreasng relatve rsk averson. Proposton 2: The prcng-representatve agent dsplays decreasng relatve rsk averson. Proof: By dfferentaton of (9), and use of (20)-(2): 8

10 * α α γ (22) = Y Y γ Thus, as Y ncreases, the weght α of those nvestors wth relatvely low degree of rsk averson ncreases. From (2) and (22), (23) * γ Y = α γ Y = γ 2 α Y γ γ * 2 * ( γ ) α α ( γ ) 3 2 * 2 γ Y = α γ α 2 γ It follows that γ * / Y < 0 f and only f 2 α α (24) = * 2 2 γ < γ γ ( ) Lookng at the random varable X that obtans the value /γ wth the probablty α, we see that the rght-hand sde above s E(X 2 ) and the left-hand sde s [E(X)] 2. Our clam s now establshed, snce the varance of X, EX 2 - (EX) 2, has to be postve. As the next proposton shows, n states where the aggregate consumpton s very hgh, the prcng representatve consumer s RRA looks lke the least rskaverse consumer, and vce-versa: Proposton 3: * Mn ; () If Y, then γ { γ } () f Y 0, then γ * Max { γ }. Proof: () Assume that γ j > Mn {γ } = γ k. By (2), t s suffcent to prove that as Y, α j 0. Suppose to the contrary that α j > ε 0 > 0, even as Y. By (2) then, γ * (Y) wll be strctly greater than γ k, and there has to be ε > 0 such that [γ * (Y)/ γ k - ] > ε. Usng (22), dln ( αk )/ dln( Y) > ε. Ths dfferental nequalty mples that as Y, α j grows to nfnty, n contradcton to that t s bounded by. The proof of () s analogous. 9

11 A numercal example The above propostons demonstrate the complex nature of the preferences of the prcng-representatve agent, even n the case where all nvestor share the same probablty assessments. Propostons 2 and 3 mply that the prcng-representatve agent dsplays decreasng relatve rsk averson, wth γ * (Y) declnng from the hghest rate of relatve rsk averson to the lowest as consumpton rses. Thus, even though all nvestors dsplay constant relatve rsk averson, t s not the case that the prcngrepresentatve agent wll also dsplay constant relatve rsk averson, wth a coeffcent that s some average of those of the heterogeneous agents n the economy. As we show n the next secton, ths result has mportant mplcatons for opton prcng. A numercal example may gve some nsght nto these propostons. Consder a two-date model wth 3 possble states at date. Aggregate consumpton at date 0 s, and aggregate date consumpton n states, 2, and 3 s {0.8, 2, 3}. There are two consumers who have equal ntal shares n the consumpton at date 0 and n each state of the world at date. Each consumer has a utlty functon wth pure tme preference β = 0.99; consumer has RRA γ = and consumer 2 has RRA γ 2 = 7. An equlbrum soluton for the dvson of aggregate consumpton between the two consumers s gven n Fgure below. Although both consumers consume more n states where the aggregate consumpton s greater, the more rsk averse consumer 2 has less varablty n her consumpton than consumer. The more rsk averse consumer s more representatve of the equlbrum n state (a low consumpton state) and consumer (wth low rsk averson) s more representatve of the equlbrum n state 3, n whch there s hgh aggregate consumpton. The result s that the prcng representatve consumer s RRA n state (a low consumpton state) s hgher than the prcng representatve consumer s RRA n state 3 (a hgh consumpton state). Thus, although both consumers have constant RRA, the prcng representatve consumers RRA s decreasng. 0

12 Aggregate consumpton State prce nomenclature 0.80 p 2.00 p p 3 Consumer consumpton Consumer 2 consumpton State prces "Prcng representatve" consumer's RRA Fgure IV. The prcng of optons on aggregate consumpton: prelmnares. In ths and the next two sectons we apply the results of Secton III to the prcng of optons n a heterogeneous consumer economy. To smplfy the exposton we assume that the economy has only two compettve agents, each endowed wth constant relatve rsk averson preferences. Each agent s also assumed to beleve (correctly) that the probablty dstrbuton of aggregate consumpton at date s lognormal. To understand the ntuton behnd our clam that the prcng of optons should be partcularly senstve to heterogenety among nvestors, consder the case where the two agents dffer n ther rsk averson. As shown n the dscusson at the end of the prevous secton, the more rsk averse agent seeks to guarantee that the

13 ampltude of her future consumpton wll be small, and n partcular seeks to protect herself aganst downsde rsk. As proved formally n Proposton 3, ths agent wll thus domnate both the date consumpton n states of low aggregate consumpton and the prcng of contngent consumpton n these states. The less rsk averse agent, less concerned wth protecton aganst downsde rsk, wll correspondngly domnate n the consumpton and the prcng of consumpton n hgh states. Snce an out-of-the-money call opton (on total consumpton, n ths framework) offers the upper tal of the dstrbuton, t follows that ts prce wll be nfluenced prmarly by the atttude towards rsk of the less rsk averse nvestor. Symmetrcally, the prcng of out-of-the-money put optons wll be partcularly nfluenced by the atttude towards rsk of the more rsk averse nvestor. Ths ntuton thus suggests that the prcng of contngent commodtes by any average agent, wth constant relatve rsk averson, wll underprce the contngent consumpton n both tals of the dstrbuton, and also tend to underprce out-of-themoney optons. The ntutve dscusson above can be alternatvely consdered as an llustraton of Proposton 2, that an economy wth heterogeneous agents wth constant relatve rsk averson wll prce assets as f t conssted of a sngle nvestor wth declnng relatve rsk averson. Another way to present our case s to use Ross s (976) dea that optons can be consdered as completng the market structure, n the absence of trade n state contngent commodtes. In ths case, f there exsts a stock market, and optons can be traded for any strke prce, both nvestors n our two-agent economy wll hold a long poston n the stock. In addton, the less rsk averse agent wll purchase call optons wth hgh strke prces, wrtten by the more rsk averse agent. Complementng these transactons, the more rsk averse nvestor wll purchase the put optons wth low strke prces that the less rsk averse nvestor ssues. In effect, the two agent wll thus be able to obtan a Pareto-effcent consumpton dstrbuton by tradng call and put optons between themselves. Dfferental tastes thus provde an ntutve explanaton both for the exstence of an open nterest n optons, and for why a constant-relatve rsk averson framework s lkely to msprce out-of the-money optons. To set the stage for a more formal applcaton of ths ntutve logc, we have frst to defne the relevant assets n ths two perod economy, and then consder the reference case of asset prcng when the agent are homogeneous. Let p(y) denote the equlbrum prce at date 0 of a unt of date consumpton. The nterest rate r s determned by the condton that (+r) - s the date 0 prce of a unt of consumpton n every date contngency: 2

14 (25) ( + r) = ( ) 0 p Y dy We dentfy the future payout of the market n ths economy as consstng of the entre date endowment. The date 0 market prce S, s thus: 0 (26) S = ( ) Y p Y dy The prce of a call and a put opton on the market wth a strke prce of X s therefore: (27) ( ) = ( )( ), ( ) = ( )( ) X 0 Denote by α α( ) C X py Y X dy P X py X YdY = Y consumer s share of aggregate future consumpton when the aggregate future consumpton s Y and denote by ω consumer s equlbrum share of date 0 consumpton. By (3), the consumpton share α = α( Y ) and the equlbrum prce p = p(y) are jontly determned by the condton: γ α Y (28) ( ) ( ) ( α ) 2 Y p= βfy f Y = β 2, ω ω where the functon f (Y) s s subjectve probablty densty. Gven our assumpton that both agents beleve that the dstrbuton of aggregate future consumpton s lognormal: 2 (29) f( Y) = f( Y; µ, σ) = exp [ lny µ ]. Yσ 2π 2σ 2 As a reference for the subsequent analyss of the mplcatons of heterogenety, we now brefly summarze the well-known results for the case where the two agents are dentcal, so that the dentfyng ndex can be dropped. Proposton 4. If aggregate date consumpton Y s lognormally dstrbuted and f all nvestors share dentcal tme-addtve preferences wth constant relatve rsk averson γ, then: () The normalzed Arrow-Debreu state prces (the prcng kernel, or the rsk-neutral probabltes ), (+r)p(y), can be consdered as the probablty densty of a lognormal varable wth densty f(y; m - γs 2, s). () The prcng of call optons s accordng to the Black-Scholes formula: X γ 3

15 (30) CX ( ) = X BSXSr ( ;,,σ) SNd [ ] [ ] + r Nd σ, 2 ln ( S/ X) + ln ( + r) + σ / 2 (3) where d = σ Proof: We do not prove ths proposton n detal, snce ts elements are well known, even f not necessarly n ths smple two-perod framework (see, Rubnsten (976), Brennan (979), or Stapleton and Subrahmanyam (984) ). For ths case the state prces are gven by: γ (32) py ( ) = β Y f( Y; µ, σ) = βexp [ γµ + γ σ / 2 ] f( Y; µ γσ, σ) where the second equalty follows from standard manpulatons. Defne now the taled moment-generatng-functon of the lognormal dstrbuton n (29): [ ] 2 µ + aσ ln x m a x Y f Y µσ dy a µ aσ N x σ where N[z] s the value of the standard normal dstrbuton at z. The two parts of the proposton then follow from applyng the defntons n (25)-(27): (34) ( + r) = βm( γ, 0), S = βm( γ, 0) (35) CX ( ) = β[ m( γ, X) Xm( γ, X) ] and then performng the requred substtutons. a 2 (33) (, ) ( ;, ) = exp ( + / 2 ) V. The prcng of optons wth heterogeneous tastes Wth the case of homogenety as a reference pont, we now return to the mplcatons of heterogenety among the two agents n ths smple two-perod, two-agent economy. Gven our assumptons, the subjectve preferences and probablty assessments of each agent wll be represented by four parameters: (ß, γ, µ, σ ) for =,2, where agent beleves that Y s lognormally dstrbuted wth parameters µ and σ. To smplfy the presentaton, we consder here separately the effect of heterogenety n only one of these four parameters at a tme. Accordngly, we wll apply a standard form of notaton, where p(y; γ, γ 2 ) wll refer, for example, to the state prce for the case of the gven degrees of rsk averson, when ths s the only subjectve parameter n whch the two agents dffer. 4

16 Snce a closed form soluton for the state prces p(y) does not exst for most cases, we wll llustrate the mplcatons of several results wth a numercal smulaton. In the smulatons presented below we use a dscrete approxmaton of the lognormal dstrbuton. For our basc homogeneous case: ß = ß = ß 2 = 0.9, γ = γ 2 = 7, µ = µ = µ 2 = 0.5, σ = σ = σ 2 = 0.3. In the heterogeneous rsk averson case dscussed n Secton V.b., γ =, γ 2 = 7; for ths case we set the ntal consumpton shares of the agents so that n equlbrum they have equal wealth shares. V.a. The case of heterogenety n subjectve tme dscountng In ths case t s assumed that the only subjectve parameter n whch agents dffer s ther dscount factor ß. By solvng equaton (28) for α, t s clear that n ths case each agent s share of second-perod consumpton wll be a constant, ndependent of aggregate consumpton Y. In fact, ths economy wll generate state prces p(y) lke the ones n the case of a homogeneous economy, where the representatve agent has a dscount factor: * / γ / γ (36) β = ωβ + ( ω) β2 As a result, Proposton 4 wll apply, and a Black-Scholes formula wll prce call and put optons. V.b. The case of heterogenety n rsk averson Ths s the man case for whch the propostons of Secton III should apply. The α = α Y s here determned by the condton: consumpton share of the frst agent, ( ) (37) ( ) γ ( YY ) α( ) α ω = Y Y ω γ 2 There s no analytc soluton for the functon α(y) n ths case. However, the followng proposton s easly seen to be a corollary of Proposton 3: Proposton 5. If the two agents dffer only n ther coeffcent of relatve rsk averson and f γ < γ 2, then α(y ), the future consumpton share of the less rsk averse frst agent, wll be monotoncally ncreasng n Y, wth lm α Y = 0, lm α Y =. Y 0 ( ) ( ) Y Fgure 2 presents the shape of the consumpton share functon a(y) for the calbraton of the model as descrbed above, where n addton γ =, γ 2 = 7. As 5

17 suggested by Proposton 5, and as shown n Fgure 2 below, ths functon s ndeed monotoncally ncreasng Consumer 's Share of Consumpton aggregate consumpton Y Fgure 2 As was proved n Proposton 2, as a result of the endogenous non-constant sharng of consumpton, the prcng functon p(y) n the case of heterogeneous agents can be nterpreted as dsplayng declnng relatve rsk averson. It follows from ths mplcaton that the prcng kernel (+r)p(y) wll no longer be lognormally dstrbuted, as n the homogeneous case, and the Black-Scholes formula wll no longer apply. The relevant ssue, however, s to dentfy what specfcally wll dstngush opton prcng n ths heterogeneous economy from the opton prcng that would apply n a smlar homogeneous economy. For ths purposes we seek to compare the case of the heterogeneous economy wth another smlar, yet homogeneous, economy where both agents share some average constant coeffcent of rsk averson, γ 0. No matter how ths average γ 0 s chosen, the followng proposton apples. Proposton 6: For any γ 0, such that γ < γ 0 < γ 2, there are two postve values Y hgh and Y Row so that state prces p(y; γ, γ 2 ) > p(y; γ 0, γ 0 ) f ether Y > Y hgh or 0 < Y < Y low. As a result, () For suffcently hgh X, C(X; γ, γ 2 ) > C(X; γ 0, γ 0 ), () For X suffcently close to zero, P(X; γ, γ 2 ) > P(X; γ 0, γ 0 ). Proof: From Proposton 5 t follows that snce α(y) ncreases monotoncally towards, when Y approaches nfnty, (38) p Y; γ, γ γ γ γ 0 = β α Y Y f Y β Y f Y > β Y f Y 0 0 = p Y; γ, γ ( ) ( ) 2 [ ] ( ) [ ] ( ) [ ] ( ) ( ) Smlarly, t follows that for γ suffcently small: 6

18 (39) p Y; γ, γ γ 2 γ 2 γ 0 = β α Y Y f Y β Y f Y > β Y f Y = p Y; γ, γ ( ) ( ) [( ) ] ( ) [ ] ( ) [ ] ( ) ( ) Ths proves the frst part of the proposton and the mplcatons concernng the prcng of far out-of-the-money put and call optons now follow. Snce our concern s to examne the mpact of heterogenety on the prcng of optons, t s approprate at ths pont to llustrate the potental magntude of the mpact of Proposton 6, by use of our calbraton example. We compare the opton prce C(X; γ, γ 2 ), where γ =, γ 2 = 7, wth the opton prce C(X; γ 0, γ 0 ) n a comparable homogeneous economy where all the parameters are dentcal and where both nvestors share the same average degree of rsk averson: γ = γ 2 = γ 0. It s not obvous how to defne ths average γ 0 ; at ths pont we choose to defne t so that the homogeneous economy wll have the correct prce for an at-the-money call opton. 6 That s, γ 0 was defned to solve o o C(S( γ, γ 2); γ, γ 2)= C(S( γ, γ 2); γ, γ ). Fgure 2 compares the prces of ths average consumer (for whom we numercally obtaned that γ 0 = 2.53) wth the actual prces n the economy. In the o o graph we show the rato of these prces p (Y; γ, γ ) / p (Y; γ, γ ). Rato of Homogeneous State Prces to Actual State Prces 2 prce rato y Fgure 3 In accordance wth Propostons 3 and 7, the average consumer economy generates lower state prces than the actual heterogeneous economy, both for low and for very hgh levels of aggregate consumpton. As a drect result of ths key fndng, t s not surprsng that the homogeneous average economy wll tend to underprce 6 In Secton V.d. below we explore the mplcatons of alternatve methods for selectng γ 0. 7

19 out-of-the-money calls. 7 Ths result s depcted n Fgure 4 below, whch shows the rato of the actual call opton prce n the heterogeneous consumer economy C(X;, 7) and the Black-Scholes prce C(X; γ 0, γ 0 ), for that homogeneous economy wth the average constant relatve rsk averson γ 0 as defned above Call/BS.07 Rato of Actual Call Prce to Call Prce n Homogeneous Economy x Fgure 4 The man fndng dsplayed by Fgure 4 s that optons that are away from the money (whether n or out of the money) are more expensve n ths heterogeneous consumer economy than n the Black-Scholes case. The rato of the call prces depcted n Fgure 4 depends, of course, on how we determne the average γ 0. Were we to choose to normalze on a Black-Scholes opton wth a dfferent strke prce, we would determne a dfferent average γ 0. We return to ths topc n Secton V.d. below. However, as proved n Proposton 6, the pattern that emerges for out-of-the money optons s robust to the selecton of the average γ 0. V.c. Impled volatlty: Smles and heterogeneous consumers Gven the dffcultes n estmatng the volatlty σ, the Black-Scholes formula s often presented emprcally n terms of the stock volatlty that s mpled by the market prcng of call optons wth alternatve strke prces. Consstency wth the Black-Scholes formula should mply a horzontal curve for the mpled volatlty as a functon of the strke prce, but the emprcal pattern that researchers typcally fnd dsplays a smle. Gven the market nterest rate, r(γ, γ 2 ), and the stock value S(γ, γ 2 ) n the heterogeneous economy we now solve the Black-Scholes formula n (30) for the mpled volatlty. That s, for the functon BS(.) n (30) and for each X, we 7 Franke, Stapleton, and Subrahmanyam (996) clam that the change n sgn exhbt by ths dfference s a necessary condton for a volatlty smle. 8

20 determne σ = σ(x), such that when γ =, γ 2 = 7, the followng dentty holds: BS X ; S γ, γ, r γ, γ, σ C X ; γ, γ. ( ( 2) ( 2) ) ( 2) Fgure 5 shows the mpled volatlty of the call opton n our calbrated example (where σ = 0.3). A smle pattern s evdent: The mpled volatlty for outof-the money optons s lower than that for n-the-money optons. 8 Ths s the pattern that s presented (among others) by Rubnsten (994). As the exercse prce of the optons gets large, the mpled volatlty n ths smulated example approaches the actual, 30%, volatlty of the underlyng consumpton process from above. Ths means that for ths partcular case the mpled volatlty s everywhere larger than the volatlty of the underlyng consumpton process. 9 volatlty ImpledCallVolatlty X Fgure 5 V.d. Alternatve normalzatons In the example of the prevous secton, we chose a representatve consumer by determnng a rsk averson coeffcent γ 0 so that the prce of an at-the-money call n the homogeneous consumer economy equals that of an at-the-money call n the 0 0 heterogeneous consumer economy C(S( γ, γ 2); γ, γ 2)= C(S( γ, γ 2); γ, γ ). There are clearly several ways to choose such a normalzaton: Case : We shall refer to the normalzaton of the prevous secton as Case : 0 0 C(S( γ, γ 2); γ, γ 2)= C(S( γ, γ 2); γ, γ ). Case 2: Instead of normalzng on an at-the-money call n the heterogeneous 8 The jagged pattern n the graph s due to computatonal roundng problems. 9 Franke, Stapleton, and Subrahmanyam (996) nterpret ths as meanng that optons are too expensve. We have not succeeded n provng that ths property wll always hold. 9

21 economy, we could normalze on an at-the-money call n each economy. That s, we choose γ so that C(S( γ, γ 2); γ, γ 2)= C(S( γ, γ ); γ, γ ). Case 3: In ths case we fnd the average relatve rsk averson γ 0 whch matches the rskless nterest rates n both economes: 0 0 py, γ, γ dy = py, γ, γ dy Y ( 2) ( ) Y Case 4: In ths case we fnd the average relatve rsk averson γ 0 whch matches the market values n both economes: 0 0 p Y, γ, γ Y dy = p Y, γ, γ Y dy Y ( 2) ( ) The table below summarzes some relevant results for these four cases, and Fgure 5 shows the ratos of the actual market prce to the homogenous consumer market prce (.e., the Black-Scholes prce) for a range of exercse prces. Y 20

22 TABLE : Comparng Four Normalzatons Dfferent Methods of Fndng the Average Relatve Rsk Averson γ 0 Base Case: actual heterogenous economy Case : normalzng on at-moneycall n actual economy Case 2: normalzng on at-moneycall n average economy Case 3: normalzng on nterest rates Case 4: normalzng on market values average γ market nterest rate 9.07% 25.87% 23.97% 9.07% 25.0% market value At-the-money Call (x=s) Call (x=0.5s) Call (x=.5*s) Notes: a. The calbratons assume a lognormal aggregate consumpton process wth µ = 5%, σ = 30%. The orgnal economy has two consumers wth equal wealth shares and relatve rsk aversons γ = and γ 2 = 7; each consumer has a pure tme dscount factor β = 0.9. b. In cases 2,3,4 there s one tem n the column whch matches a correspondng tem for the base case. The excepton s case, n whch we determne the average gamma γ 0 by solvng 0 0 C(S( γ, γ 2); γ, γ 2)= C(S( γ, γ 2); γ, γ ). In case the opton prce for an at-the-money opton s calculated by C(S( γ, γ ); γ, γ )= C(S( γ, γ ); γ, γ ). 2 2 It s clear from Table and Fgure 6 that the asset prces and call prce ratos of the actual and homogeneous-equvalent economes are very senstve to the selected form of normalzaton. Dependng on the normalzaton, some opton prces may be found to be underprced relatve to the Black-Scholes case, and others to be overprced. Stll, as proved n Proposton 6, ultmately (that s, for a hgh enough strke prce), the prces of the calls n our heterogeneous consumer economy wll be larger than the Black-Scholes prce n any equvalent homogeneous economy. 2

23 Fgure 6: Four Dfferent Normalzatons For each normalzaton, we show the rato of the actual call prces to the call prce n the equvalent homogeneous-consumer economy (ths latter prce s the Black-Scholes prce). Case : Rato of Actual Call to BS Call normalzed on call whch s at-the-money n orgnal economy.7 Case 2: Rato of Actual Call to BS Call normalzed on at-the-money calls Call/BS x Call/BS x Case 3: Rato of Actual Call to BS Call normalzed on nterest rates Call/BS x Call/BS Case 4: Rato of Actual Call to BS Call normalzed on market values x 22

24 VI. Opton delta wth heterogeneous consumers In the prevous secton we proved that when there s consumer heterogenety n rsk averson, the opton prce for a suffcently out-of-the money opton s always larger n a heterogeneous consumer economy than the Black-Scholes prce. In ths secton we explore the propertes of the opton delta n a heterogeneous consumer model. We prove that n a heterogeneous consumer envronment, the opton delta s always (for suffcently hgh opton exercse prce) larger than the Black-Scholes delta. The model employed n the prevous sectons had only two dates, 0 and. In order to explore opton delta, we requre a model whch has ntermedate perods. We buld such a model n the most obvous way, assumng that the unt nterval the tme between today (date 0) and the opton s maturty (assumed to be T = ) s dvded nto n subntervals. Snce we have no analytc expresson for the opton delta, we approxmate t by showng the senstvty of the opton prce to the underlyng stock prce n some ntermedate perod. 0 The opton delta s the senstvty of the opton prce to a change n the underlyng asset prce. In the BS model, δ BS = Nd ( ). Assume that the optons are wrtten on the market and that the tme to maturty s T =. We wrte the δ n our heterogeneous model as δ HM ( S, X), where S s the market value. We prove the followng proposton: Proposton 7: For X suffcently large, δ ( S X) HM, > δ. Proof (sketch): Consder a heterogeneous consumer model n whch the consumers have relatve rsk aversons γ < γ 2 < < γ H. The Black-Scholes opton prce s essentally an opton prce for some ntermedate ( average ) rsk averson γ < γ 0 < γ H. It follows from Proposton 3 that the state prces n the heterogeneous consumer case can be regarded as derved from an average consumer whose relatve rsk averson γ A (Y) s a decreasng functon of aggregate termnal consumpton Y, where γ A (Y) mn (γ h ) as Y. It follows that the heterogeneous-model state prce for Y large enough s larger than the state prce Black-Scholes (rsk averson γ 0 ) case; furthermore, the dfference between the heterogeneous model and Black-Scholes state prces s an ncreasng functon of aggregate consumpton Y. Snce the out-of- BS 0 Suppose for example that n = 50. Then for an ntermedate perod (say j = 20), our numercal model wll calculate 2 pars of opton prces and market prces. Graphng these combnatons wll gve the senstvty of the opton prce to the underlyng market prce. 23

25 the money opton prces payoffs only n these extreme states, ths proves the proposton. It s easy to llustrate the proposton, and t s also easy to see the necessty of X beng large enough. In the example below, our two consumers are as before (rsk aversons and 7 and pure tme preference δ = 0.99). The opton s hghly outof-the-money (X = 2). As predcted by the proposton, the dfference between the HM opton prce and the BS prce s an ncreasng functon of S. dfference Actual Call Prce mnus BS Stock prce Fgure 7 It s also easy to construct counter-examples to δ ( S X) HM, > δ ; ths wll occur for lower X, as n the example below, n whch the opton exercse prce s X =. dfference Actual Call Prce mnus BS Stock prce Fgure 8 BS 24

26 The reason for ths s clear: for a lower exercse prce X the model prces can be consdered to come from both low and hgh relatve rsk aversons. State prces n the heterogeneous consumer model wll not be unformly hgher than the Black- Scholes state prces, and the resultng opton deltas wll exhbt behavor dervng from ths property. VII. Concluson People are dfferent. Some are bold and darng, whle others are overcautous. Such dversty s ndeed one of the man economc ratonales for Pareto-mprovng trade, and has been partcularly emphaszed n relaton to speculatve markets. In ths paper we consder equlbrum opton prcng n a smple two-perod economy that s characterzed by heterogenety among agents. We demonstated that an economy n whch agents have constant yet heterogeneous degrees of relatve rsk averson wll prce assets as though t had a sngle prcng representatve agent who dsplays decreasng relatve rsk averson. Ths result was shown to mply that the prcng kernel has fat tals and yelds opton prces whch do not conform to the standard Black-Scholes formula. Solvng for the mpled volatlty of ether call or put optons results n ths case n a smle pattern, typcal of those derved n practce. In addton we proved that the opton delta for suffcently out-of-the-money optons s hgher than the Black-Scholes delta. Our explanaton of heterogenety as the source for ths emprcally observed phenomenon s smple and ntutve. It seems to ft Rubnsten s (994) nterpretaton of the over-prcng of out-of-the-money put optons on the S&P 500 ndex as an ndcaton of crash-o-phoba. Rubnsten s term suggests that those who seek to hold out-of-the-money put optons as protecton aganst crashes are characterzed by relatvely hgh rsk averson or by subjectve probablty assessments wth a relatvely hgh (possbly unreasonable) weght on low market outcomes. If one were to assume that all nvestors share the same atttude towards rsk and probablty belefs wth regard to market crashes, there would be no explanaton why some nvestors hold these extreme put optons, whle others wrte them. In addton, the very complexty of the mpled bnomal tree that Rubnsten derves suggests to us that t s lkely to be the equlbrum outcome of a complex nteracton among dverse nvestors, rather than to reflect unform atttudes towards rsk shared unanmously by all nvestors. Whle t s convenent to portray the economy through the construct of a fcttous representatve nvestor, ths convenence should not blnd us to gnore the serous aggregaton problems that are nvolved by such a construct, or to regard as nnocuous the practce of endowng the fcttous representatve nvestor wth preferences and 25

27 probablty belefs that may be reasonable only for the actual nvestors n the economy. 26

28 REFERENCES Bck, Av (987). On the Consstency of the Black-Scholes Model wth a General Equlbrum Framework. Journal of Fnancal and Quanttatve Analyss 22 (September), pp Bck, Av (990). On Vable Dffuson Prce Processes of the Market Portfolo. Journal of Fnance 45 (June), pp Breeden, Douglas T. and Robert H. Ltzenberger (978). Prces of State-Contngent Clams Implct n Opton Prces. Journal of Busness 5 (October), pp Brennan, Mchael J. (979). The Prcng of Contngent Clams n Dscrete Tme Models. Journal of Fnance 34 (March), pp Constantndes, George, M. (982). Intertemporal Asset Prcng wth Heterogeneous Consumers and wthout Demand Aggregaton. Journal of Busness 55 (Aprl), pp Cox, John C. and Mark Rubnsten (985). Optons Markets. Prentce-Hall. Franke, Guenter, Rchard C. Stapleton, and Mart G. Subrahmanyam (996), Why are Optons Expensve? mmeo. Heaton, John and Deborah Lucas (995). The Importance of Investor Heterogenety and Fnancal Market Imperfectons for the Behavor of Asset Prces. Carnege-Rochester Conference Seres on Publc Polcy, Vol 42 (June), pp He, Hua and Hayne Leland (993). On Equlbrum Asset Prce Processes, Revew of Fnancal Studes, 6, pp Hrshlefer, Jack and John G. Rley (979). The Analytcs of Uncertanty and Informaton: An Expostory Survey. Journal of Economc Lterature 7 (December), pp Kennckell, Arthur B. and R. Louse Woodburn (992). Estmaton of Household Net Worth Usng Model-Based and Desgn-Based Weghts: Evdence from the 989 Survey of Consumer Fnances, Board of Governors of the Federal Reserve System, manuscrpt. Krman, Alan P. (992). Whom or What Does the Representatve Indvdual Represent? Journal of Economc Perspectves 6 (Sprng), pp

29 Leland, Hayne (980). Who Should Buy Portfolo Insurance? Journal of Fnance 35, pp Lucas, Robert E. (978). Asset Prces n an exchange economy, Econometrca 46, pp Mankw, N. Gregory (986). The Equty Premum and the Concentraton of Aggregate Shocks, Journal of Fnancal Economcs 7, pp Mankw, N. Gregory and Stephen P. Zeldes (99). The Consumpton of Stockholders and Nonstockholders, Journal of Fnancal Economcs, 29 (March), pp Mayshar, Joram (983). On Dvergence of Opnon and Imperfectons n Captal Markets. Amercan Economc Revew 73 (March), pp Ross, Stephen, (976). Optons and Effcency, Quarterly Journal of Economcs 90 (February), pp Rubnsten, Mark E. (974). An Aggregaton Theorem for Securtes Markets. Journal of Fnancal Economcs, pp Rubnsten, Mark E. (976). The Strong Case for the Generalzed Logarthmc Utlty Model as the Premer Model of Fnancal Markets. Journal of Fnance, pp Rubnsten, Mark E. (976). The Valuaton of Uncertan Income Streams and the Prcng of Optons. Bell Journal of Economcs 7 (Autumn), pp Rubnsten, Mark E. (994). Impled Bnomal Trees. Journal of Fnance 59 (July), pp Shafer, Wayne and Hugo Sonnenschen (982). Market Demand and Excess Demand Functons. Handbook of Mathematcal Economcs, Volume II, K.J. Arrow and M.D. Intrllgator (eds.), pp Sharpe, Wllam F. (970) Portfolo Theory and Captal Markets. New York: McGraw-Hll. Shefrn, Hersh (996). Behavoral Opton Prcng. mmeo. Stapleton, Rchard C. and Mart Subrahmanyam (984). The Valuaton of Optons When Asset Returns are Generated by a Bnomal Process. Journal of Fnance 39 (December), pp