Cres H, Markeprand T, Tvede M. Incomplete financial markets and jumps in asset prices. Economic Theory 2015.

Transcription

1 Cres H, Markeprand T, Tvede M. Incomplete fnancal markets and jumps n asset prces. Economc Theory 205. Copyrght: The fnal publcaton s avalable at Sprnger va Date deposted: 3/07/205 Embargo release date: 20 May 206 Ths work s lcensed under a Creatve Commons Attrbuton-NonCommercal-NoDervatves 4.0 Internatonal lcence Newcastle Unversty eprnts - eprnt.ncl.ac.uk

2 Incomplete fnancal markets and jumps n asset prces Hervé Crès Tobas Markeprand Mch Tvede Abstract For ncomplete fnancal markets jumps n both prces and consumpton can be unavodable. We consder pure-exchange economes wth nfnte horzon, dscrete tme, uncertanty wth a contnuum of possble shocks at every date. The evoluton of shocks follows a Markov process and fundamentals depend contnuously on shocks. It s shown: () equlbra exst; (2) for effectvely complete fnancal markets asset prces depend contnuously on shocks; and, (3) for ncomplete fnancal markets there s an open set of economes U such that for every equlbrum of every economy n U, asset prces at every date depend dscontnuously on the shock at that date. Keywords fnancal markets general equlbrum jumps n asset prces JEL-classfcaton D52 D53 E32 G G2 The authors wsh to thank an anonymous referee for constructve suggestons and M. Pascoa as well as partcpants n GEdays n York for nterestng comments. New York Unversty n Abu Dhab, PO Box 2988 Abu Dhab, Unted Arab Emrates, emal: herve.cres@nyu.edu DREAM, Amalegade 44, 256 Copenhagen K, Denmark, emal: tma@dreammodel.dk Newcastle Unversty, 5 Barrack Road, Newcastle upon Tyne, NE 4SE, Unted Kngdom; e-mal: mch.tvede@ncl.ac.uk.

3 Introducton Consder an economy wth uncertanty. Suppose fundamentals of the economy depend contnuously on the state of the world, whle asset prces depend dscontnuously on the state of the world, or n other words asset prces jump. Then markets amplfy uncertanty n the sense that on top of the uncertanty about fundamentals there s uncertanty about whether asset prces jump or not. The present paper shows that the amplfcaton of uncertanty can be unavodable by provdng a general equlbrum explanaton of jumps n asset prces. The study of jumps n asset returns has a long hstory n fnance. Indeed, the consequences of jumps have been consdered snce Merton (976). Emprcally jumps have receved some attenton n recent years as can be seen n Andersen, Benzon & Lund (2002), Chrstoffersen, Jacobs & Ornthanala (20) and Eraker, Johannes & Polson (2003). Estmates n these artcles range from -2 to 8-9 jumps per year and jumps accountng for from 8-5 to 2-5 percent of the total varance for the S&P 500. The theoretcal lterature on jumps n asset prces s lmted. A couple of contrbutons use partal equlbrum models. In Balduzz, Fores & Hat (997) consder a model wth a mx of utlty maxmzng nvestors and buy-and-hold nvestors. Buy-and-hold nvestors trade only n case prces ht lower or upper barrers, perhaps because of transacton costs. It s shown that, f all buy-and-hold nvestors have dentcal barrers, then prces jump n case prces ht the barrers. Lm, Martn & Teo (998) consder a partal dsequlbrum model wth contnuous tme. Prce movements depend on the dfference between demand and supply. Some supplers use portfolo-nsurance hedgng strateges, whch result n supply beng downward slopng for low prces. Therefore there can be multple equlbra and the movement of prces depend on the vews of agents. It s shown that n case agents swtch vews asset prces jump. A couple of contrbutons study jumps n asset prces n general equlbrum models wth nfntely lved consumers. Bansal & Shalastovch (2009) consder pure-exchange economes wth dscrete tme, uncertanty and ncomplete nformaton. There s a representatve agent, who can buy new nformaton. It s shown that prces jump n case the representatve agent buys new nformaton. Calvet & Fsher (2004) consder pure-exchange economes wth nfnte horzon, contnuous tme and uncertanty. Asset returns follow geometrc Brownan motons wth jumps n drft and volatlty parameters. For these processes asset returns depend contnuously on tme, but asset prces depend dscontnuously on tme: jumps n the drft parameter result n jumps n the expected dscounted dvdends of assets; and, jumps n the volatlty parameter result n jumps n the expected dscounted utlty of dvdends of assets so asset prces jump. An ssue related to jumps s volatlty. Prce volatlty s studed n Calvet (200) and 2

4 Ctanna & Schmedders (2005). The set of possble shocks at every date s fnte and the evoluton of shocks follows a general Markov process. The basc dea s that ncomplete fnancal markets prevent consumpton smoothng, so prces are more volatle wth ncomplete fnancal markets than wth complete fnancal markets. Consumpton volatlty for pure-exchange economes wth dscrete tme, uncertanty and ncomplete fnancal markets s consdered n Beker & Chattopadhyay (200) where dynamc propertes of equlbra are studed. For a large class of economes wth one good and two consumers t s shown that n any equlbrum ether consumpton of one of the consumers s eventually zero or consumpton of both consumers s arbtrarly close to zero at nfntely many dates. Other papers have studed ncomplete fnancal markets and asset prces n pure-exchange economes wth nfnte horzon, dscrete tme, uncertanty and ncomplete fnancal markets. In Telmer (993) and Lucas (994) endowments of consumers are ht by transtory dosyncratc shocks. In both papers t s found that consumers are able to smooth consumpton by accumulatng assets. In Constantndes & Duffe (996) endowments of consumers are ht by persstent dosyncratc shocks. It s shown that for all stochastc processes for aggregate ncome and asset prces, the process for aggregate ncome can be splt nto processes for ndvdual ncomes such that the processes for asset prces are equlbrum processes. The result s obtaned for no-trade equlbra. However dscontnutes n fundamentals are necessary for dscontnutes n assets prces at no-trade equlbra. Therefore economes for whch asset prces jump are not ncluded n the analyss. In Heaton & Lucas (996) endowments of consumers are ht by both dosyncratc and aggregate shocks and consumers face transacton costs. It s found that for large transacton costs consumers are not able to smooth ncome. In the present paper we study the exstence of jumps n assets prces n general equlbrum economes where jumps are generated by the nteracton of real markets and ncomplete fnancal markets. We consder pure-exchange economes wth nfnte horzon, dscrete tme, uncertanty wth a contnuum of possble shocks at every date and ncomplete fnancal markets. The evoluton of shocks follows a general Markov process. We show: () equlbra exst; (2) f fnancal markets are effectvely complete, so equlbrum allocatons are Pareto optmal, then prces depend contnuously on shocks, and; (3) f fnancal markets are ncomplete, then there s an open set of economes U such that for every equlbrum of every economy n U and almost all hstores of shocks, prces at every date depend dscontnuously on the shock at that date. Hence jumps n prces can be unavodable. To the best of our knowledge, the only other paper on exstence of equlbrum for economes wth nfnte horzon, dscrete tme, uncertanty wth a contnuum of possble shocks at every date and ncomplete fnancal markets s Araujo, Montero & Pascoa (996) (AMP). The major dfferences n terms of assumptons are: () n AMP the probablty ds- 3

5 trbuton on the set of possble shocks at every date s the Lebesgue measure makng the probablty dstrbuton ndependent of the hstory of shocks, here the probablty dstrbuton on the set of possble shocks at date t depends on the shock at date t ; (2) n AMP fundamentals at date t depend on the hstory of shocks up to and ncludng date t, here fundamentals at date t depend on the shock at date t; and, (3) n AMP consumpton sets are non-negatve orthants and utlty functons are contnuous, here consumpton sets are postve orthants and utlty functons are twce dfferentable functons tendng to mnus nfnty as consumpton converges to the boundary of the consumpton set. The two frst dfferences () and (2) more or less cancel out. However the thrd dfference (3) s mportant. Indeed a strong form of dscountng s needed to get exstence of equlbrum n AMP (see Theorem 4. n AMP). No such assumpton s needed n the present paper because of (3) (see Theorem below). Contnuty of prces s studed n Chchlnsky & Zhou (998) and Covarrubas (200). Pure-exchange economes wth one date and uncertanty are consdered. The set of possble shocks s a compact subset of some Eucldean space and fundamentals depend contnuously on shocks. It s shown that prces depend contnuously on shocks n Walrasan equlbra. Optmalty of equlbrum allocatons for economes wth nfnte horzon, uncertanty wth a fnte number of possble shocks and ncomplete fnancal markets s studed n Kubler & Schmedders (2003). It s shown that f an equlbrum allocaton s Pareto optmal, then equlbrum prces and allocatons at every date only depend on the shock at that date, and that genercally equlbrum allocatons are not Pareto optmal. Multplcty of equlbra s studed n Mas-Colell (99). Pure-exchange economes wth two dates and uncertanty wth a contnuum of possble shocks at the second date are consdered. It s shown that f fnancal market are ncomplete, then some economes have a contnuum of equlbra and that n these equlbra prces of goods at the second date depend dscontnuously on shocks even though fundamentals depend contnuously on shocks. The present paper s organzed as follows: n Secton 2 the set-up s ntroduced and the assumptons are stated; n Secton 3 results on exstence of equlbrum and results on economes wthout and wth jumps n prces are stated and establshed; and, Secton 4 contans some fnal remarks. 2 The model Below we ntroduce our set-up, state the consumer problem and state our assumptons. 4

6 Set-up Consder a pure exchange economy wth nfnte horzon and uncertanty. There s an nfnte number of dates t N 0 = {0,,2,...}. The set of possble shocks at every date s S = [0,] wth s t S and the shock at the frst date s s 0 S. The evoluton of shocks as tme passes s descrbed by a bounded and measurable tme ndependent transton densty π : S S R ++ wth π(s t+,s t )ds t+ = for almost all s t. Let s t = (s t,...,s ) denote the hstory of shocks up to and ncludng date t and let S t denote the set of possble hstores of shocks up to and ncludng date t. There are fnte numbers of goods l wth h {,...,l}, consumers m wth {,...,m} and short-lved real assets n wth j {,...,n}. Consumers have dentcal consumpton sets X = R l ++ and dscount factors ρ ]0,[. Consumers are descrbed by ther tme ndependent endowments ω : S X, state utlty functons u : X R and lower bounds on short sales of assets δ R n ++. Assets are descrbed by ther tme ndependent dvdends a j : S R l +\{0}. An economy s a lst of consumers and a lst of assets E = ((ω,u,δ ),(a j )). Prces of goods and assets are normalzed such that they sum to one. Let = {v R l+n + k v k = } be the set of normalzed prces. A prce system s a collecton of maps (p,q) = (p t,q t ) where (p t,q t ) : S t for all t. A consumpton bundle s a collecton of maps x = (x t ) wth xt : St X for all t. An allocaton x = (x ) s a lst of ndvdual consumpton bundles. A portfolo s a collecton of maps θ = (θ t ) wth θ t : S t {δ } + R n + for all t. A consumpton plan s a par of collectons of maps (x,θ ). Fnancal markets are ncomplete for two reasons. Frst the set of assets s fnte and the set of shocks s nfnte. Second there are bounds on short sales. Consumers At date t for a hstory of shocks s t let θ t (s t ) {δ } + R n + be the portfolo from date t and (p t (s t ),q t (s t )) the prces. Then at date t for a hstory of shocks s t the budget constrant of consumer s p t (s t ) (x t (s t ) ω (s t )) + qt j (s t )θ t j (s t ) p t (s t ) a j (s t )θ t j (s t ). j j Assume u s contnuous, x s measurable and there exsts a compact set C X such that x t (st ) C for all t and almost all s t. Then the expected utlty of consumer s U (x ) = u (x 0) + ρ u (x (s ))π(s,s 0 )ds +ρ 2 +ρ t u (x 2 (s 2 ))π(s 2,s )π(s,s 0 )ds u (x t (s t ))π(s t,s t )...π(s,s 0 )ds t

7 The problem of consumer s to choose a consumpton plan that maxmzes expected utlty subject to the budget constrants. Assumptons The consumers are assumed to satsfy the followng assumptons (A.) ω C(S,R l ++). (A.2) u C 2 (X,R) wth Du (x ) R l ++ and v T D 2 u (x )v < 0 for all x and v 0. (A.3) lm xk x u (x k ) = where x k X for all k and x X. (A.2) states that the utlty functon s twce dfferentable, smoothly strongly monotonc and smoothly strctly concave. (A.3) mples consumpton s bounded away from the boundary of the consumpton set. The assets are assumed to satsfy the followng assumpton (A.4) a j C(S,R l +\{0}). The transton densty s assumed to satsfy the followng assumpton (A.5) π C(S S,R ++ ). (A.), (A.4) and (A.5) mply fundamentals depend contnuously on shocks. In order to defne a topology on the set of economes satsfyng (A) (A.5), topologes on endowments, utlty functons, assets and transton denstes have to be defned. The sets of endowments, assets and transton denstes are endowed wth the maxmum norm topologes. The set of utlty functons s endowed wth the Whtney C 2 -topology. Fnally the set of economes E = ((ω,u,δ ),(a j )) s endowed wth the product topology. 3 Equlbra Below we state and establsh results on exstence of equlbrum and jumps n prces. Exstence of equlbrum In equlbrum consumers maxmze ther expected utltes subject to ther budget constrants and markets for goods as well as assets clear. Defnton A fnancal market equlbrum s a prce system and a lst of ndvdual consumpton plans (( p, q),( x, θ)) such that For all, ( x, θ ) s a soluton to the problem of consumer gven ( p, q). For all t and almost all s t, x t (st ) = ω (s t ) and θ t(st ) = 0. 6

8 Assume portfolo returns are covered by ntal endowments for every consumer, all states and all possble portfolos, then there s an equlbrum. The assumpton that all possble portfolo returns are covered by ntal endowments s very strong. However an example n Mas-Colell & Zame (996) shows that the assumpton s necessary for exstence of fnancal market equlbrum for economes wth non-atomc state spaces. Theorem For an economy E suppose that for all and almost all s, ω (s) j a j (s)δ j X. Then there s a fnancal market equlbrum (( p, q),( x, θ)). Proof: Frst t has to be shown there exsts a compact set C X such that f (( p, q),( x, θ)) s a fnancal market equlbrum, then x t (st ) C for all and t and almost all s t. Second t s shown how uncertanty and fundamentals n the present paper can be transformed nto uncertanty and fundamentals n AMP. Thrd the proof of Theorem 4. n AMP s appled and t s shown how that an a strong form of dscountng used n AMP s not needed because equlbrum consumpton s n a compact set C. Frst: Let c U X be defned by cu h = max s S ω h (s) for all h. Then xth (s t ) cu h for all, t and h and almost all s t. Let α = mn s u (ω (s) h a j (s)δ j ) and β = mn s u (ω (s)). Then for all t and almost all s t the expected utlty from date t and forward s less than or equal to u ( x t (st )) + ρu (c U ) + ρ 2 u (c U ) +... and larger than or equal to α + ρβ + ρ 2 β +... Therefore there exsts c L X such that x th (s t ) c h L for all, t and h and almost all st because lmsup xk x u (x k ) = where x k X and x X accordng to (A.3). Let C X defned by C = then x t (st ) C for all and t and almost all s t. { } c X h : c h [c h L,cU] h Second: The economy s transformed nto an artfcal economy by changng the probablty measure on the set of shocks as well as the endowments and the dvdends at every date. Suppose the probablty measure on the set of shocks at every date s the Lebesgue measure ndependently of the hstory of shocks up to that date. Let φ : S S S be a map from shocks at date t and t to artfcal shocks at date t defned by φ(s t,s t ) = Prob(s s t s t ) so φ(s t,s t ) = π(s,s t )ds Then φ(,s t ) : S S s contnuous, strongly monotone and bjectve. Let ψ : S 2 S be defned by ψ(σ t,s t ) = s t f and only f σ t = φ(s t,s t ), so ψ(,s t ) s the nverse of φ(,s t ). Therefore ψ(,s t ) s contnuous, strongly monotone and bjectve. s s t A hstory of shocks n the orgnal economy s t s transformed nto a hstory of shocks n the artfcal economy σ t (s t ) as follows: σ = φ(s,s 0 ), σ 2 = φ(s 2,s ) and so on. Smlarly 7

9 a hstory of shocks of the artfcal economy σ t s transformed nto a hstory of shocks n the orgnal economy s t (σ t ) as follows: s = ψ(σ,s 0 ), s 2 = ψ(σ 2,s ) = ψ(σ 2,ψ(σ,s 0 )) and so on. Let the endowments n the artfcal economy e t : St R l + at date t be defned by e 0 = ω (σ 0 ) and e t (σt ) = ω (s t (σ t )) for all and t. Let the dvdends n the artfcal economy d t j : St R l + at date t be defned by d t j (σt ) = a j (s t (σ t )) for all j and t. Thrd: The proof of Theorem 4. n AMP can be appled to the artfcal economy wth the Lebesgue measure on the set of shocks at every date t, e = (e t ) as endowments for consumer and d j = (d t j ) as dvdends for asset j, E A = ((e,u,δ ),(d j )). In the frst part of the proof n AMP economes are approxmated by economes wth fntely many dates and fntely many possble shocks per date. For every N N a truncated artfcal economy EN A wth N + dates and N possble shocks per date s defned. For SN n = [(n )/N,n/N[ for n {,...,N } and SN n = [(n )/N,n/N] for n = N the set of possble shocks at every date s {,...,N} wth ν Nt {,...,N} and νn t {,...,N}t. For S N (νn t ) = Sn N... Sn t N let endowments e N = (e t N ) and dvdends d jn = (d t jn ) be defned by e t N (νt N ) = Nt S N (ν t N ) e t (σ t )dσ t d t N (νt N ) = Nt S N (ν t N ) d t j(σ t )dσ t. For every N the artfcal economy wth N+ dates and N possble shocks per date has an equlbrum (( p N, q N ),( x N, θ N )) accordng to Radner (972). Accordng to Lemma 5. n AMP for every and N there exst Lagrange multplers (Λ t N ) such that (u (xn 0 ) u ( x N 0 ρn ))+...+ N N ν N N (u (x N N(ν N N )) u ( x N N(ν N N ))) Λ 0 N p N0 (xn 0 x0 N )+...+ Λ N N(νN N ) p NN (νn N ) (xn(ν N N N ) x N(ν N N N )) νn N +Λ 0 N j + ν N N ν N ν N N q j N0 (θ 0 j N θ 0 j N )+... Λ N N (νn N ) j Λ N(ν N) j Λ N N(ν N N ) j q j NN (νn N )(θn N (νn N ) θ N p N d N j(ν N)(θ N(ν N) θ N(ν N))... N (ν N p NN (νn N ) d N jn(νn N )(θ N j N (νn N ) θ N j N (νn N )). N )) 8

10 Accordng to Lemma 5.2 n AMP the Lagrange multplers satsfy that for every, N and t ρ t N t (u (x t N(ν t N)) u ( x t N(ν t N))) Λ t N(ν t N) p Nt (ν t N) (x t N(ν t N) x t N(ν t N)). Therefore the Lagrange multplers are defned by ρ t N t Du ( x t N(ν t N)) = Λ t N(ν t N) p Nt (ν t N). In Lemma 5.3 n AMP a strong form of dscountng s used to show that for the map λ t N : S t R + defned by λ t N (st ) = N t Λ t N (νt N ) for st S t N (νt N ) the sequence of maps (λ t N ) s unformly bounded n N for all. Below another proof s provded. In the proof t s used that x t N C for every, N and t, but the strong form of dscountng used n AMP s not used. For every t N consder the followng problem for consumer max (z,ψ j ) s.t. ρt N t u ( x t N (νt N )+z ) + ρt+ N t+ u ( x t+ ν Nt+ { pnt (νn t ) z + q j Nt (νt N )ψ j = 0 N (νn t+ )+dt+ N j ψ j ) ψ j δ j θ t j N (νt N ). The soluton s z = 0 and ψ j = 0 for all because (( p N, q N ),( x N, θ N )) s an equlbrum. For every j there s an such that θ j N (νt N ) > δ j because θ j N (νt N ) = 0 and δ j > 0. For every j let ( j) denote an wth θ j N (νt N ) > δ j. The frst-order condtons for the problem for consumer ( j) at z ( j) = 0 and ψ j ( j) = 0 are ρ t N t Du ( j)( x t N( j) (νt N )) µ ( j) p Nt (ν t N ) = 0 µ ( j) q j Nt (νt N ) + ρt+ N t+ Du ( j) ( x t+ N( j) (νt+ N )) d N j(νn) t = 0. ν Nt+ Let l be the vector n R l wth every coordnate equal to one. Then q j Nt (νt N) = ρ N νnt+ Du ( j) ( x t+ N( j) (νt+ N )) d N j(νn t ) Du ( j) ( x t N( j) (νt N )) p Nt (νn) t l l Snce p Nt (ν t N ) l + q Nt (ν t N ) n =, p Nt (ν t N) l = + j ρ νnt+ Du ( j) ( x t+ N( j) (νt+ N Du ( j) ( x t N( j) (νt N )) l N )) d N j(νn t ). 9

11 so Snce and +ρ max,c C, j,s Du (c) a j (s) mn,c C Du (c) l p t N l N t Λ N (νn) t = ρt Du ( x t N (νt N )) l p t N (νt N ), l +ρ mn,c C, j,s Du (c) a j (s) max,c C Du (c) l mn Du (c) l Du ( x t N(νN)) t l max Du (c) l,c C,c C the sequence (λn t ) s unformly bounded n N for every. The rest of the proof follows the last part of the proof of Theorem 4. n AMP. An equlbrum for the artfcal economy s obtaned as a lmt of the sequence of equlbra for the sequence of truncated artfcal economes. Unform boundedness of the sequences (λ t N ) s used to show that the lmt of the sequences of solutons to the problems of the consumers n the truncated artfcal economes are solutons to the problems of the consumers n the artfcal economy. Q.E.D. Remark: Theorem 4. n AMP on exstence of equlbrum rests on the assumpton that there s ε > 0 such that ( ) mn ω h s (s) j a h j(s)δ j > ε for all and h and ( ) ρ ε+ δ j maxa h s j(s) < ε for all j and h. The frst part s equvalent to the assumpton n Theorem whle the second part puts an upper bound on the dscount factor. In the present paper the second part of the assumpton s not used because of (A.2) and (A.3). Contnuous fnancal market equlbra An allocaton s Pareto optmal provded there s no other allocaton for whch no consumer s worse off and at least one consumer s better off. Defnton 2 A Pareto optmal allocaton s an allocaton x for whch there s no other allocaton x wth x t (st ) ω (s t ) for all t and almost all s t. U (x ) U ( x ) for all wth > for at least one. In order for jumps to matter for consumers, prces and consumpton plans have to be measurable dscontnuous. 0

12 Defnton 3 A bounded and measurable map f : S R s (measurable) contnuous provded there s a contnuous map g C(S,R) such that f (s) g(s) ds = 0. Otherwse t s (measurable) dscontnuous. Remark: A map s measurable contnuous provded t can be made contnuous by changng t on a set of measure zero. Indeed the map f : S [0,] defned by f (s) = { 0 for s ratonal otherwse s measurable contnuous, because the set of ratonals has measure zero, so f (s) ds = 0. Therefore random samples from contnuous and measurable contnuous maps (s a, f (s a )) and (s a,g(s a )) cannot be dstngushed. Remark: A map s measurable dscontnuous provded t has a jump. Indeed the map f : S [0,] defned by h(s) = 0 for s [0, 2 ] for s ] 2,] s measurable dscontnuous. For the sequence of maps (g n ) wth g n : S [0,] defned by 0 for s [0, 2 n+ [ n+ g n (s) = 2 s n for s [ 4 2 n+, 2 + n+ ] for s ] 2 + n+,] lm n h(s) gn (s) = 0 because h(s) g n (s) ds = /(2(n+)). Therefore random samples from measurable dscontnuous maps (s a,h(s a )) and contnuous maps (s a,g n (s a )) can be very hard to dstngush. However from a mathematcal pont of vew t s easer to study measurable dscontnuous maps than contnuous maps wth very steep slopes. For ncomplete fnancal markets there s no reason to expect equlbrum prces and consumpton plans to depend contnuously on shocks. However equlbrum prces and consumpton bundles are tme ndependent and depend contnuously on shocks provded the equlbrum allocaton s Pareto optmal. Equlbrum portfolos need not depend contnuously on shocks n case dvdends of assets are collnear.

13 Theorem 2 For an economy E let (( p, q),( x, θ)) be a fnancal market equlbrum. Suppose x s Pareto optmal. Then ( p t, q t ) : S t and x t : S t X m are contnuous for all t. Indeed there are contnuous functons (p,q) : S and x : S X m such that for all t and almost all s t. ( p t (s t ), q(s t )) = (p(s t ),q(s t )) x t (s t ) = x(s t ) Proof: For every r R l ++, c X m wth c r and λ R m ++ the set { } c X m c r and λ u (c ) λ u ( c ) s compact because for all, u s contnuous accordng to (A.2) and lm xk x u (x k ) = accordng to (A.3). Therefore for every s S and λ R m ++ there exsts a unque soluton to the followng problem max c s.t. λ u (c ) c ω (s) because u s contnuous and strctly concave accordng to (A.2). Let c : S R m ++ X m be the soluton to the problem. Then c : S R m ++ X m s contnuous accordng to Berge s maxmum theorem. For every λ R m ++ consder the planner problem max x t s.t. λ U (x t (s t )) x t (s t ) ω (s t ) for all t and almost s t Then the allocaton c(λ) s a soluton to the planner problem f and only f c t (s t,λ) = c(s t,λ) for all t and almost all s t. Next t s shown that x wth U (x ) = for some s nether a soluton to the planner problem nor Pareto optmal. Let ω L X be defned by ω h L = (/m)mn s S ω h (s). For x f U (x) = for some, then x s not a soluton to the planner problem for any λ R m ++ because u (ω L )+ρu (ω L )+ ρ 2 u (ω L ) +... > for all. For x f U (x ) = for all, then x s not Pareto optmal because u (ω L ) + ρu (ω L ) + ρ 2 u (ω L ) +... > for all. For x f U (x ) = for some and U (x ) > for some, then x s not Pareto optmal because x defned by x = (/m)x and x = x +(/m)x for all makes all consumers wth U (x ) > better off and no 2

14 consumer worse off. Next t s shown that x s Pareto optmal f and only f x t (st ) = c (s t, λ) for all t and and almost all s t. Suppose x s not Pareto optmal. Then there exsts another allocaton x such that U (x ) U (x ) for all wth at least one strct nequalty. Therefore x s not a soluton the planner problem for any λ R m ++. Hence f x s a soluton to the planner problem, then x s Pareto optmal. Suppose x s not a soluton to the planner problem for any λ R m ++. Then λ U (c (λ)) > λ U (x ) for all λ R m ++. Let µ : R m ++ R ++ be defned by µ(λ) = m λ (U (c (λ)) U (x )). Then µ : R m ++ R ++ s contnuous. For S m ++ = {λ Rm ++ λ = } let Γ : S m ++ Rm be defned by Γ (λ) = U (c (λ)) U (x ) µ(λ), then: Γ s contnuous; there exsts α R such that Γ (λ) α for all and λ; λ Γ(λ) = 0 for all λ; and, lm λk λ Γ(λ k ) = for all λ S++ m. Therefore there exsts λ S++ m such that Γ( λ) = 0 so U (c ( λ)) > U (x ) for all so x s not Pareto optmal. Hence f x s Pareto optmal, then x s a soluton to the planner problem for some λ S m ++. All n all, x s Pareto optmal f and only f x t (st ) = c (s t, λ) for all t and and almost all s t so there exsts λ S m ++ such that xt (s t ) = c(s t,λ) for all t and almost all s t. Let f : S R l ++ and g : S R n ++ be defned by f (s) = Du (c (s,λ)) g j (s) = ρ Du (c (s,λ)) a j (s )π(s,s)ds. Then ( p, q) satsfes p t (s t ) = q t (s t ) = h f h (s t )+ j g j (s t ) f (s t) h f h (s t )+ j g j (s t ) g(s t) for all t and almost all s t. Therefore p and q are contnuous. Q.E.D. Remark: In Elul (999) fnancal markets are denoted effectvely complete provded equlbrum allocatons are Pareto optmal. Accordng to Theorem 2 f fnancal markets are effectvely complete, then consumers are able to smooth consumpton across shocks such that jumps are elmnated. Recurrent dscontnuous fnancal market equlbra For an equlbrum, there are recurrent jumps provded that for all dates and hstores of shocks s t, prces and consumpton bundles at date t depend dscontnuously on the shock 3

15 s t. Defnton 4 A recurrent dscontnuous fnancal market equlbrum s a fnancal market equlbrum (( p, q),( x, θ)) such that for all t and almost all s t, ( p t, q t, x t )(,s t ) : S X m s dscontnuous. Suppose a fnancal market equlbrum s recurrent dscontnuous. Then for a fxed hstory of shocks s t the relaton between the shock s t at date t and prces and consumpton bundles at date t can be as llustrated below n Fgure. (p t,q t,x t )(s t,s t ) Fgure : jumps n prces and consumpton bundles There s an open set of economes such that for all equlbra of all economes n that set, all endogenous varables are recurrent dscontnuous even though fundamentals depend contnuously on shocks. Theorem 3 There s an open set of economes U, such that every fnancal market equlbrum (( p, q),( x, θ)) for every economy E n U s recurrently dscontnuous. Proof: Suppose (e,u ), where e X for all, s a pure-exchange economy wth l goods and m consumers and at least two regular Walrasan equlbra. See Ghglno & Tvede (997) for detals on how to construct such an economy. Let c and c be two dfferent equlbrum allocatons assocated wth two dfferent regular equlbra of (e,u ). For all let ω : S X be defned by ( 2s)c + (2s)e for s [0, ω (s) = 2 ] (2 2s)e + (2s )c for s ] 2,]. s t 4

16 Then the pure-exchange economy (ω (s),u ) has a unque regular Walrasan equlbrum for s {0, }, because endowments are Pareto optmal for s {0, }, and at least two regular Walrasan equlbra for s = /2. Let W : [0, ] be the Walras correspondence. Then there s no contnuous selecton from the Walras correspondence. Moreover there exsts a neghborhood N of (ω,u ) such that f (ω,u ) N, then there s no contnuous selecton from the Walras correspondence for (ω,u ). Suppose the dscount factor ρ, the number of assets n, the dvdends of the assets (a j ) and the lower bounds on short sales (δ ) are arbtrary and fxed such that mn s ω h(s) j a h j (s)δ j > 0 for all and h. Consder a famly of economes E τ = (X,ρ,(ω,u ),(a τ j )) parametrzed by τ [0,] and defned by a τ j = τa j for all τ [0,]. Accordng to Theorem the economy E τ has a fnancal market equlbrum for every τ. For every par of consumers and, date t, asset j and τ and almost all hstores of shocks s t the vectors Du ( x τt (s t )) and Du ( x τt (s t )) are collnear. For an arbtrary, all t and τ and almost all s t let ft τ : S t R l ++ be defned by f τ t (s t ) = Du ( x τt (s t )). For all t, j and τ and almost all s t, there exsts an such that θ τ j (s t ) > δ j because δ j > 0 for all and θ τ j (s t ) = 0 for all t and almost all s t. For all j, t and τ, any wth θ τt j (s t ) > δ j and almost all s t let gt τ j : S t R n ++ be defned by gt τ j (s t ) = ρ Du ( x τt+ (s t+ )) a τ j(s t+ )π(s t+,s t )ds t+. Then for all t and almost all s t p τ t (s t ) = q τ j t (s t ) = h f τ j t h f τ j t (s t )+ j gt τk (s t ) f τ t (s t ) (s t )+ j gt τ j (s t ) gτ j t (s t ) for all j. Accordng to the proof of Theorem there exsts c L,c U X such that f (( p τ, q τ ),( x τ, θ τ )) s a fnancal market equlbrum for E τ, then x τt (s t ) C for all, t and τ [0, τ] and almost all s t. Therefore asset prces converge unformly to zero as τ converges to zero lm ess sup s τ 0 t sup q t τ j (s t ) = 0. t, j Moreover θ τt j (s t ) [ δ j, + δ j ] for all, t and j and almost all s t because θ τt j (s t ) δ j for all, t and j and almost all s t and θ τt j (s t ) = 0 for all t and j and almost all s t. 5

17 Hence the real dvdends and prces of portfolos converge unformly to zero as τ converges to zero lm τ 0 ess sup s t sup,t j a τ j (s t) θ τt j (s t ) = 0 lm τ 0 ess sup s t sup,t j q t τ j (s t ) θ τt j (s t ) = 0. For all t and almost all s t consder a famly of pure-exchange economes wth ncome transfers parametrzed by s t ( E τ ts t (s t ) = ω (s t )+ j a τ j(s t ) θ τt j ) (s t ),u, j q t τ j (s t ) θ τt j (s t ), where ω (s t )+ j a τ j (s t) θ τt j (s t ) s the endowment of consumer and j q t τ j (s t ) θ τt j (s t ) s the ncome transfer receved by consumer. The equlbrum correspondence W τ : S ts t R l + s defned by v W τ (s ts t t ) f and only f h v h = j q t τ j (s t ) and there exsts (c ) such that c s a soluton to max c u (c) s.t. v c v ( ) ω (s t )+ j a τ j (s t) θ τt j (s t ) j q t τ j (s t ) θ τt j (s t ) for all and c = ω (s t ). Therefore p t τ (s t ) W τ (s ts t t ) for all t and τ and almost all s t. There exsts τ ]0,] such that for all τ ]0, τ[ there s no contnuous selecton from E τ ts t for all t and almost all s t because there s no contnuous selecton from W. Q.E.D. Remark: In the proof of Theorem 3: frst pure-exchange economes wthout fnancal markets are constructed such that all endogenous varables are recurrent dscontnuous; and, second fnancal markets, where assets have small dvdends, are added so the lower bounds on short sales ensures that all endogenous varables are recurrent dscontnuous. In the appendx we present an example of a pure-exchange economy wth fnancal markets and fnte tme horzon where all endogenous varables are dscontnuous and lower bounds on short sales are not bndng. Theorem 3 mples that f fnancal markets are ncomplete, then jumps n both prces and consumpton bundles can be unavodable. In the proof of Theorem 3, t s mportant how endowments depend on shocks, but not how dvdends depend on shocks. Therefore we argue that jumps are generated by the nteracton of real and fnancal markets. 4 Fnal remarks In the present paper we have provded a possble explanaton of jumps n asset prces. Whether there are jumps or not depends on whether markets are ncomplete or not. Indeed, 6

18 for effectvely complete fnancal markets there are no jumps and for ncomplete fnancal markets jumps can be unavodable. The proofs of Theorems -3 reveal that t s straghtforward to extend the analyss to nclude long-lved assets and shock dependent state utlty functons. It would be nce to extend the analyss to economes wth contnuous tme. However, such an extenson s not straghtforward because even exstence of equlbrum for economes wth ncomplete fnancal markets and contnuous tme s a terrtory about whch lttle s known. Appendx: economes wth fnte tme horzon Set-up Consder a pure exchange economy wth fnte horzon and uncertanty. There s a fnte number of dates t {0,...,T }. The set of possble shocks at every date s S = [0,] wth s t S and the shock at the frst date s s 0 S. The evoluton of shocks as tme passes s descrbed by a bounded and measurable tme ndependent transton densty π : S S R ++. There are fnte numbers of goods l wth h {,...,l}, consumers m wth {,...,m} and real assets n wth j {,...,n}. Consumers have dentcal consumpton sets X = (R l ++) T +. Consumers are descrbed by ther endowments ω = (ω t) wth ωt : St R l ++, state utlty functons u : X R and lower bounds on short sales of assets δ R n ++. Assets are descrbed by ther dvdends a j = (a t j ) wth at j : St R l + for all t. An economy s a lst of consumers and a lst of assets E Fnte = ((ω,u,δ ),(a j )). Dscontnuous fnancal market equlbra Theorem 4 There exsts an economy such that f (( p, q),( x, θ)) s a fnancal market equlbrum, then q s dscontnuous n s T. Proof: Consder an economy wth three dates T = 2, one good n each date l =, two consumers m = 2 and one asset n =. The dvdend of the asset s supposed to be one unt of the good at the last date and otherwse before the last date. Endowments and asset dvdends are assumed to depend on the shock s at the second date t = and to be ndependent of the shock at the thrd date t = 2. The probablty dstrbuton on the set of shocks at date t = s the Lebesgue measure π(s) = for all s. Endowments at date t = 0 are supposed to be dentcal ω2 0 = ω0 and endowments at the 7

19 last two dates are supposed to be reverse n the sense that ω 2 (s) = ω2 ( s) ω 2 2 (s) = ω ( s). Smlarly, utlty functons are supposed to be dentcal for the frst date and reverse for the last two dates such that u 2 (x 0,α,β) = u (x 0,β,α) for all postve real numbers x 0,α,β > 0. Denote by E (s;(c 0 )) = (e (s),v ( ;c 0 )) the subeconomy wth consumers {,2}, utlty functons v ( ;c 0 ) : R2 ++ R defned by v (x,x2 ;c0 ) u (c 0,x,x2 ) and endowments e (s) R 2. For c 0 let f ( ;c 0 ) = ( f ( ;c0 ), f 2( ;c0 )) : R2 ++ R ++ R 2 ++ denote the demand functon for the consumer of the sub-economy E (s;(c 0 )), so f (p, p e (s);c 0 ) solves the problem max v (x,x 2 ;c ) (x,x 2 ) s.t. p (x e (s)) 0. Then (s, p) S R 2 ++ s an equlbrum for the sub-economy E (s;(c 0 ) ) f and only f f (p, p e (s);c 0 ) + f 2(p, p e 2 (s);c 0 2 ) = e (s) + e 2 (s). Clearly (s, p, p 2 ) s an equlbrum for E (s;(c 0 )) f and only f ( s, p 2, p ) s an equlbrum for E ( s;(d 0 )), where d0 = c0 2 and d0 2 = c0. For equlbrum prces normalzed such that the sum equals one let W : S S R 2 ++ be the Walras correspondence for the economes (E (s;(c 0 ))) wth c0 = ω 0 for both, so W(s) = { (s, p, p 2 ) (s, p, p 2 ) s an equlbrum for E (s;(ω 0 ))}. Suppose that the graph of W s S-shaped as shown n Fgure 2 and let r : S R 2 ++ be a selecton from W such that r (s) s the lowest equlbrum prce for s < /2, r (s) = (/2,/2) for s = /2 and r (s) s the hghest equlbrum prce for s > /2. In order to construct a fnancal market equlbrum: let the allocaton x be defned by x 0 = ω 0, and x t (s) = f t (r(s),r(s) e (s);ω 0 ) for both and t {,2}; let the portfolo θ be defned by θ 0 = 0 and θ (s) = r (s) ( ω r 2 (s) (s) f (r(s),r(s) e (s);ω 0 ) ) = f 2 (r(s),r(s) e (s);ω 0 ) ω 2 (s); let the prce system ( p, q) be defned by p 0 = p (s) = p 2 (s) = for all s, q (s) = r 2 (s)/r (s) for all s and q 0 > 0 such that ( q 0 u ( x (s)) x 0 + q (s) u ) ( x (s)) x ds = 0. 8

20 p (s) s Fgure 2: The Walras correspondence W and the selecton r. Then (( p, q),( x, θ)) s a fnancal market equlbrum and the asset prce at date s dscontnuous at s = /2. Snce q (s)θ 0 = (x (s) ω (s)) + q (s)(x 2(s) ω2 (s)) for both and almost all s, the portfolo θ 0 of consumer at date 0 satsfes θ 0 [δ L,δ U ] for δ L = nf s (ω (s)/q (s)+ω 2(s)) and δ U = nf s (ω (s)/q (s)+ω 2(s)). Assume ω (s),ω2 (s) < ε for s {0,} and the margnal rates of substtuton at the Pareto optmal allocatons n the sub-economes for s {0,} are bounded away from zero and nfnty. Then lm ε 0 δ L = lm ε 0 δ U = 0. Moreover there s ε > 0 such that f ε ε, then the set of equlbra for the collecton of sub-economes s S-shaped for all feasble date 0 portfolos. Therefore q s dscontnuous n s. Q.E.D. Remark: The proof of Theorem 4 reveals that any measurable selecton r : S R 2 ++ such that r (s) = r ( s) and r 2 (s) = r 2 ( s) s part of a fnancal market equlbrum. Therefore as shown n Mas-Colell (99) there s a contnuum of fnancal market equlbra. Remark: An crucal dfference between the examples n Mas-Colell (99) and n the proof of Theorem 4 s the number of dates. In Mas-Colell (99) there are two dates and there s a sngle asset traded at the frst date. Hence the asset prce at the frst date s unaffected by the shock at the second date so jumps n asset prces are mpossble. In the proof of Theorem 4 there are three dates and a sngle asset traded at the two frst dates. Thus the asset prce on the second date s affected by the shock at that date makng jumps n asset prces possble. References Andersen, T., L. Benzon & J. Lund (2002), An emprcal nvestgaton of contnuous-tme equty return models, Journal of Fnance 57,

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