Discussion Papers Department of Economics University of Copenhagen

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1 Dscusson Papers Department of Economcs Unversty of Copenhagen No Incomplete Fnancal Markets and Jumps n Asset Prces Hervé Crès, Tobas Markeprand, and Mch Tvede Øster Farmagsgade 5, Buldng 26, DK-1353 Copenhagen K., Denmark Tel.: Fax: ISSN: (onlne)

2 Incomplete fnancal markets and jumps n asset prces Hervé Crès Tobas Markeprand Mch Tvede Abstract A dynamc pure-exchange general equlbrum model wth uncertanty s studed. Fundamentals are supposed to depend contnuously on states of nature. It s shown that: 1. f fnancal markets are complete, then asset prces vary contnuously wth states of nature, and; 2. f fnancal markets are ncomplete, jumps n asset prces may be unavodable. Consequently ncomplete fnancal markets may ncrease volatlty n asset prces sgnfcantly. Keywords: General equlbrum, fnancal markets, jumps n asset prces. JEL-classfcaton: D52, D53, G12. Scences Po, 27 rue Sant-Gullaume, Pars, France; emal: herve.cres@scencespo.fr. Unversty of Copenhagen, Studestraede 6, 1455 Copenhagen K, Denmark; Tel: ; Fax: ; tobas.markeprand@econ.ku.dk. Unversty of Copenhagen, Studestraede 6, 1455 Copenhagen K, Denmark; Tel: ; Fax: ; mch.tvede@econ.ku.dk. 1

3 1 Introducton In the present paper we provde an explanaton of jumps n asset prces based on the nteracton of real markets and fnancal markets. An emprcal characterstc of asset prces s that dstrbutons of prce changes have thck tals,.e., large changes n asset prces are overly represented n observed data. Indeed n Merton (1976), motvated by the observaton that stock prces tend to show far too many outlers, the study of opton prces n case of jumps n the underlyng securty prces was ntated. Thck tals are not consstent wth the standard assumpton of Gaussan processes wdely used n the fnance lterature. Therefore jump processes such as Posson processes seem to be necessary to account for the thck tals (see e.g. Andersen, Benzon & Lund (2002)). In Bansal & Shalastovch (2008) t s mentoned that the frequency of jumps s per year and that around 10 percent of the volatlty n asset prces s explaned by jumps. The consequences of jumps n asset prces are potentally sgnfcant as jumps n asset prces ncrease uncertanty: fundamentals are uncertan and small changes n fundamentals can result n dramatc changes of prces. Several contrbutons am at explanng jumps n asset prces. In Calvet & Fsher (2008) an optmal growth model where endowments and dvdends are uncertan s consdered and t s shown that jumps n the drft and/or volatlty of endowments and dvdends generate jumps n asset prces even though sample paths of endowments and dvdends are contnuous. In Balduzz, Fores & Hat (1997) and Lm, Martn & Teo (1998) partal equlbrum models wth ad hoc behavour of some nvestors are consdered and ths behavour causes supply curves to be non-monotonc leadng to jumps n asset prces. In Bansal & Shalastovch (2008) an optmal growth model wth a representatve consumer, where dvdends are uncertan, nformaton s ncomplete and the consumer can buy a precse sgnal, s consdered and t s shown that from tme to tme the representatve consumer buys the precse sgnal n whch case asset prces jump. Accordng to the market effcency hypothess changes n asset prces must 2

4 be due to changes n dvdends or condtonal expectatons because, as shown n Huang (1985), f both dvdends and condtonal expectatons vary contnuously, then asset prces vary contnuously too. In Calvet & Fsher (2008) and Bansal & Shalastovch (2008) jumps n asset prces are caused by jumps n condtonal expectatons. Some contrbutons am at explorng a possble lnk between ncomplete fnancal markets and volatlty of asset prces. In Geanakoplos (1997) the use of collateral n contracts s shown to nduce an excess volatlty n the prces of the durable goods that are used as collateral, excess volatlty n the sense that the varance s larger wth the use of collateral n contracts than wth complete markets. In Ctanna & Schmedders (2001) fnancal nnovaton s shown to nduce excess volatlty. In Calvet (2001) ncomplete fnancal markets are shown to lead to excess volatlty. The dfference between jumps n asset prces and volatlty of asset prces should be noted. Indeed volatlty of asset prces does not necessarly nvolve jumps, but merely changes of asset prces. In the present paper a dynamc, fnte horzon, pure-exchange general equlbrum model wth uncertanty s studed. Fundamentals are assumed to be contnuous functons of states of nature. We show that: 1. f fnancal markets are complete, then prces (ncludng asset prces), consumpton bundles and portfolos are contnuous functons of the states of nature, and; 2. f fnancal markets are ncomplete, then nether prces, consumpton bundles nor portfolos need to be contnuous functons of states of nature. Therefore ncompleteness of fnancal markets may ncrease volatlty n asset prces sgnfcantly. The paper proceeds as follows: In Secton 2 the set-up, the equlbrum concepts and our mantaned assumptons are ntroduced. In Secton 3, respectve Secton 4, complete fnancal markets, respectve ncomplete fnancal markets, are consdered. Fnally n Secton 5 some fnal remarks are provded. 3

5 2 The model Set-up There s a fnte number T + 1 of dates wth t {0,..., T }. There s uncertanty, the set of states at date t 1 s S = [0, 1] wth s S and π : S T R + s the densty on the set of states S T. There s a fnte number of goods l at every state wth j {1,..., l}. p t : S t R l ++, s a prce system for goods. A collecton of maps p = (p t ), where There s a fnte number m of consumers wth {1,..., m}. Consumers are descrbed by ther dentcal consumpton sets X = (R l ++) T +1, ther endowments ω = (ω t ) t, where endowments at date t s descrbed by a map ω t : S t X, and ther state utlty functons u : X R. A consumpton bundle s a collecton of maps x = (x t ) t, where x t : S t R l ++. An allocaton of goods x = (x ) s a lst of ndvdual consumpton bundles. Walrasan equlbrum Let s t = (s 1,..., s t ) denote the hstory of states up to and ncludng date t, then the problem of consumer s: max u(x 0,..., x T (s T )) π(s T ) ds T x S T s.t. p t (s t ) x t (s t ) ds T S T t Formally ntegrablty assumptons are needed. S T t p t (s t ) ω t (s t ) ds T In a Walrasan equlbrum consumers choose consumpton bundles that solve ther problems and markets clear. Defnton 1 A Walrasan equlbrum s a prce system for goods and an allocaton of goods ( p, x) such that: x s a soluton to the problem of consumer for all, and; markets clear, xt (s t ) = ωt (s t ) for all t and s t. 4

6 Fnancal market equlbrum There s a fnte number n of assets wth k {1,..., n} where the dvdend of asset k at date t s descrbed by a map a t k : St R l. An asset structure s a collecton of assets a = (a k ) k, where a k = (a t k ) t. A prce system for an asset structure s a collecton of maps q = (q t ), where q t : S t R n. A portfolo plan s a collecton of maps z = (z t ) t, where z t : S t R n. An allocaton of assets z = (z ) s a lst of portfolo plans. A prce system (p, q) s a prce system for goods and a prce system for assets. An allocaton (x, z) s an allocaton of goods and an allocaton of assets. Let a t (s t ) be the l n-matrx of dvdends (a 1 t (s t )... a n t (s t )) at date t n state s t, then the problem of consumer s: max u(x 0,..., x T (s T )) π(s T ) ds T (x,z ) S T p 0 x 0 + q 0 z 0 p 0 ω 0 p t (s t ) x t (s t ) + q t (s t ) z t (s t ) s.t. p t (s t ) ω t (s t ) + (q t (s t ) + p t (s t )a t (s t )) z t 1 (s t 1 ) for all t {1,..., T 1} p T (s T ) x T (s T ) p T (s T ) ω T (s T ) + (p T (s T )a T (s T )) z T 1 (s T 1 ) In a fnancal market equlbrum consumers choose consumpton bundles and portfolo plans that solves ther problems and markets clear. Defnton 2 A fnancal market equlbrum s a prce system and an allocaton (( p, q), ( x, z)) such that: ( x, z ) s a soluton to the problem of consumer for all, and; markets clear, xt (s t ) = ωt (s t ) and zt (s t ) = 0 for all t and s t. 5

7 Assumptons The consumers are supposed to satsfy the followng assumptons: (A.1) ω t C 1 (S t, X). (A.2) u C 2 (X, R) wth Du (x ) R lt ++ for all x and v D 2 u (x )v < 0 for all x and v 0. The economy s supposed to satsfy the followng assumptons: (A.3) π C 1 (S T, R ++ ). (A.4) a t k C1 (S t, R l ) for all k and t. Exstence of equlbrum The focus of the present paper s on propertes of equlbra rather than exstence. However a short dscusson of exstence of Walrasan equlbrum and of fnancal market equlbrum for economes wth nfnte dmensonal commodty spaces s provded below. In Bewley (1972) the exstence of a Walrasan equlbrum s shown for consumpton bundles n L and prce systems n L 1. In the proof t s crucal that consumpton sets have non-empty nteror. In Mas-Colell (1986, 1991) exstence of Walrasan equlbrum n more general vector lattces, where the consumpton set does not necessarly have a non-empty nteror, s consdered. The assumpton of unform properness of preferences, whch mples the exstence of supportng prces, replaces the assumpton that consumpton sets have non-empty nteror. The problem wth changes n the dmenson of set of ncome transfers spanned by assets carres over from economes wth fntely many states. Moreover as shown n Mas-Colell & Montero (1996) and Mas-Colell & Zame (1996) there s a problem wth feasblty of consumpton bundles. The assumpton that every feasble portfolo results n a feasble consumpton bundle appears to be needed to ensure exstence of equlbrum. However the 6

8 assumpton s very strong, especally for economes wth at least two goods per state. 3 Complete fnancal markets In the present paper functons that are dentcal except for a set of measure zero are consdered to be dentcal. Defnton 3 A measurable functon f : S T R s contnuous at ŝ T f and only f there exst a neghborhood A of ŝ T and a functon g : S T R, where g 1 (B) s open for B open, such that 1 {s T f(s T ) g(s T )} π(s T ) ds T = 0. A A functon s contnuous f and only f t s contnuous at all ponts. Walrasan equlbrum At Walrasan equlbra, prces and consumpton bundles are dfferentable functons of states of nature. The proof conssts of two steps: n Lemma 1 t s shown that prces and consumpton bundles are contnuous functons, and; n Theorem 1 t s shown that f they are contnuous functons of states, then they are dfferentable functons. Lemma 1 Suppose that ( p, x) s a Walrasan equlbrum. contnuous n s T. Then ( p, x) s Proof: Suppose that ( p, x) s a Walrasan equlbrum, then there exsts λ 1,..., λ m > 0 such that x s the soluton to the followng problem max λ u (x 0,..., x T (s T )) π(s T ) ds T x s.t. x t (s t ) = (1) ω(s t t ) for all t and s t 7

9 The proof that x s contnuous n s T s by backward nducton on t. t = T Suppose that ĉ T 1 = (ˆx 0,..., ˆx T 1 ) and ŝ T are fxed and consder the followng maxmzaton problem max λ u (ĉ T 1 x T, x T ) s.t. x T = ω T (ŝ T ). Then for every ĉ T 1 and ŝ T there exsts a unque contnuous soluton to the maxmzaton problem accordng to assumptons (A.2) and (A.3). Let f T : (X m ) T S T X m be the soluton, then t s contnuous accordng to Berge s maxmum theorem and f c T 1 = ( x 0 (s 0 ),..., x T 1 (s T 1 )), then f T (c T 1, s T ) = x T (s T ). Moreover the functon v T : (X m ) T S T 1 R defned by v T (c T 1, s T 1 ) = s strctly concave n x 0,..., x T 1. u (c T 1, f T (c T 1, s T )) π(s T s T 1 ) ds T t = T 1 Suppose that ĉ T 2 = (ˆx 0,..., ˆx T 2 ) and ŝ T 1 are fxed and consder the followng maxmzaton problem max λ v (ĉ T 2, x T 1, ŝ T 1 ) x T 1 s.t. x T 1 = ω T 1 (ŝ T 1 ). Then for every ĉ T 2 and ŝ T 1 there exsts a unque contnuous soluton to the maxmzaton problem accordng to assumptons (A.2) and (A.3). Let f T 1 : (X m ) T 1 S T 1 X m be the soluton, then t s contnuous accordng to Berge s maxmum theorem and f c T 2 = ( x 0 (s 0 ),..., x T 2 (s T 2 )), then f T 1 (c T 2, s T 1 ) = x T 1 (s T 1 ). Moreover the functon v T 1 : (X m ) T 1 S T 2 R defned by v T 1 (c T 2, s T 2 ) = v T (c T 2, f T 1 (c T 2, s T 1 ), s T 1 ) π(s T 1 s T 2 ) ds T 1 8

10 s strctly concave n c T 2. The steps for t = T 2,..., 0 are smlar to the step for t = T 1. The soluton ( x t ) t, where x t : S t X m, to problem (1) s defned as follows x 0 = f 0 x 1 (s 1 ) = f 1 ( x 0, s 1 ). x T 1 (s T 1 ) = f T 1 ( x 0, x 1 (s 1 ),..., x T 2 (s T 2 ), s T 1 ) x T (s T ) = f T ( x 0, x 1 (s 1 ),..., x T 1 (s T 1 ), s T ). The prce system p s collnear wth the gradents of the consumers, so the prce system s contnuous n s T too. Indeed there exsts τ > 0 such that p t (s t ) = τλ D x tu ( x (s T )) π(s t+1,..., s T s t ) d(s t+1,..., s T ). for all, t and s t. Remark: In the proof of Lemma 1 t s only used that utlty functons are once dfferentable and strctly concave, but t s not used that utlty functons are twce dfferentable wth negatve defnte Hessan matrces. End of remark Theorem 1 Suppose that ( p, x) s a Walrasan equlbrum. Then ( p, x) s dfferentable n s T. Proof: Suppose that ( p, x) s a Walrasan equlbrum, then accordng to Lemma 1 t s contnuous n s T and there exsts λ 1,..., λ m > 0 such that x s the soluton to the followng problem max λ u (x 0,..., x T (s T )) π(s T ) ds T x s.t. x t (s t ) = ω(s t t ) for all t and s t. 9

11 The proof that x s dfferentable n s T s by nducton on t. At step t t s assumed that x 0 s dfferentable n s 0,..., x t 1 s dfferentable n s t 1. t = 0 The frst-order condtons wth respect to x 0 at s 0 are λ D x 0u (x 0,..., x T (s T )) π(s T ) ds T α 0 = 0 for all x 0 ω 0 = 0 The l(m + 1) l(m + 1)-matrx H 0 of dervatves wth respect to x 0 and α 0 of the frst-order condtons s where D 0 D 0 1 I.... D 0 m I I I s a l l-matrx defned by = λ D 2 x 0 x 0u (x 0,..., x T (s T )) π(s T ) ds T D 0 and I s a the l l-dentty matrx. The matrx H 0 has full rank. Therefore accordng to the Implct Functon Theorem x 0 s a dfferentable functon of s 0, because x 1 s a contnuous functon of s 1,..., x T s a contnuous functon of s T. t = T The frst-order condtons wth respect to x T at s T are λ D x T u (x 0,..., x T (s T )) α T = 0 for all x T (s T ) ω T (s T ) = 0 The l(m + 1) l(m + 1)-matrx H T of dervatves wth respect to x T and α T of the frst-order condtons s D T 1 I.... D T m I I 10 I

12 where D T s a l l-matrx defned by D T = λ D 2 x T x u T (x 0,..., x T (s T )). The matrx H T has full rank. Therefore accordng to the Implct Functon Theorem x T s a dfferentable functon of s T, because x 0 s a dfferentable functon of s 0,..., x T 1 s a dfferentable functon of s T 1. The fact that p s dfferentable n s T follows from the proof that p s contnuous n s T n the proof of Lemma 1 and that x s dfferentable n s T. Fnancal market equlbrum: complete markets At fnancal market equlbra, where the allocaton s Pareto optmal, prces of goods and assets, consumpton bundles and portfolos are dfferentable functons of states. Corollary 1 Suppose that ( p, x) s a Walrasan equlbrum and that a = (a k ) k s an asset structure such that (( p, q), ( x, z)) s a fnancal market equlbrum. Then q s dfferentable n s T. Proof: The proof that q s dfferentable n s T s by backward nducton on t. t = T 1 The prce of asset k at date T 1 n state s T 1 s q T k 1(s T 1 ) = p T (s T 1, s T ) a T k (s T 1, s T ) ds T where p T s contnuous n s T accordng to Lemma 1 and a T k s contnuous n s T accordng to assumpton (A.5). Therefore q T k 1 s contnuous n st 1. t = 0 Trval because q k 0 s a number rather than a functon. However the asset prce of asset k at date 0 s q 0 k = ( p 1 (s 1 ) a 1 k(s 1 ) + q1(s k 1 )) ds 1 where p 1 s contnuous n s 1 accordng to Lemma 1 and a 1 k s contnuous n s 1 accordng to assumpton (A.5). Therefore q 0 k s contnuous. 11

13 Remark: In the proof of Corollary 1 t s only used that ( p, x) s contnuous and that a s contnuous, but t s not used that a s dfferentable. End of remark 4 Incomplete fnancal markets Fnancal market equlbrum: ncomplete markets At fnancal market equlbra, where fnancal markets are ncomplete, there may be jumps n prces ncludng asset prces, consumpton bundles and portfolos. The proof s based on an example. Theorem 2 There exsts an economy such that f (( p, q), ( x, z)) s a fnancal market equlbrum, then q s dscontnuous n s T. Proof: Consder an economy wth three dates T = 2, one good per state l = 1, two consumers m = 2 and one asset n = 1. The dvdend of the asset s supposed to be one unt of the good at the last date. Endowments and asset dvdends are supposed to ndependent of the state at the last date. For the densty π : S R ++ suppose that π(s) = 1 for all s S. Endowments at the frst date are supposed to be dentcal ω2 0 = ω1 0 and endowments at the last two dates are supposed to be reverse n the sense that ω2(s) 1 = ω1(1 2 s) and ω2(s) 2 = ω1(1 1 s). Smlarly utlty functons are supposed to be dentcal for the frst date and reverse for the last two dates n the sense that u 2 (x 0, x 1, x 2 ) = u 1 (x 0, x 2, x 1 ). For c 0 let f ( ; c 0 ) : R 2 ++ R ++ R 2 ++ denote the demand functon for the consumer wth endowments e (s) = (ω 1 (s), ω 2 (s)) and utlty functon v ( ; c 0 ) : R 2 ++ R defned by v (x 1, x 2 ; c 0 ) = u (c 0, x 1, x 2 ). Then (p, s) R 2 ++ S s an equlbrum for the Edgeworth box economy E(s; (c 0 ) ) = (e (s), v (, ; c 0 )) f and only f f 1 (p, p e 1 (s); c 0 1) + f 2 (p, p e 2 (s); c 0 2) = e 1 (s) + e 2 (s). 12

14 Clearly (p 1, p 2, s) s an equlbrum for E(s; (c 0 ) ) f and only f (p 2, p 1, 1 s) s an equlbrum for E(1 s; (d 0 ) ), where d 0 1 = c 0 2 and d 0 2 = c 0 1. Suppose that equlbrum prces are normalzed such that the sum of the prces s equal to one and let E R 2 ++ S be the equlbrum set for the collecton of Edgeworth economes (E(s; (c 0 ) ) s, where c 0 = ω 0, so E = { (p, s) (p, s; (ω 0 ) ) s an equlbrum for E(s; (ω 0 ))}. Suppose that E s S-shaped as shown n Fgure 1 and let r : S R 2 ++ be a selecton from E such that r 1 (s) s the lowest equlbrum prce for s < 1/2, r 1 (s) = (1/2, 1/2) for s = 1/2 and r 1 (s) s the hghest equlbrum prce for s > 1/2. In order to construct a fnancal market equlbrum: let the p 1 (s) 1 1 s Fgure 1: The equlbrum set E and the selecton r. allocaton x be defned by x 0 = ω 0, x j (s) = f j (r(s), e (s); ω 0 ) for j {1, 2}; let the portfolo plan z be defned by z 0 = 0 and z 1 (s) = (r 1 (s)/r 2 (s))(ω 1 (s) f 1 (r(s), e (s); ω 0 )) = f 2 (r(s), e (s); ω 0 ) ω 2 (s); let the prce system p be defned by p 2 (s) = p 1 (s) = p 0 = 1, and; let the prce system for assets q be defned by q 1 (s 1 ) = r 2 (s 1 )/r 1 (s 1 ) and q 0 > 0 such that ( u (x (s)) q 0 + q x 0 1 (s) u ) (x (s)) ds = 0. x 1 13

15 Then ((p, q), (x, z)) s a fnancal market equlbrum and the asset prce at date 1 s dscontnuous at s = 1/2. Fnally the portfolo z1 0 of consumer 1 at date 0 s bounded from below by mn s (ω1(s) 1 + q 1 (s)ω1(s))/q 2 1 (s) and from above by mn s (ω2(s) 1 + q 1 (s)ω2(s))/q 2 1 (s). Therefore suppose that (ω1(s), 1 ω1(s)) 2 s bounded from above by ε > 0 for s {0, 1} and that the margnal rates of substtuton at the Pareto optmal allocatons n the Edgeworth box economes for s {0, 1} are bounded away from zero and nfnty. Then for ε suffcently small the set of equlbra for the collecton of Edgeworth box economes s S-shaped for all feasble portfolos so there s a dscontnuty n prces. Remark: The proof of Theorem 2 reveals that any measurable selecton r : S R 2 ++ such that r 1 (s) = 1 r 1 (1 s) and r 2 (s) = 1 r 2 (1 s) s part of a fnancal market equlbrum. Therefore as shown n Mas-Colell (1991) there s a contnuum of fnancal market equlbra. End of remark On the example n the proof of Theorem 2 Let us try, nformally, to argue that the example n the proof of Theorem 2 s robust. In order to consder pertubatons of fundamentals suppose that the set of fundamentals s endowed wth the Whtney topology, endowments and dvdends wth the C 1 -topology and utlty functons wth the C 2 -topology. The S-shape of the equlbrum set E s robust to perturbatons n fundamentals and small changes n portfolos. Therefore every selecton from the equlbrum set s dscontnuous. Hence assets prces are dscontnuous. The robustness of the example n the proof of Theorem 2 shows that the symmetry n the example s not essental, but merely convenent. 14

16 5 Fnal remarks In the present paper we have shown that jumps n asset prces may be unavodable n case of ncomplete fnancal markets. Moreover we have shown that jumps are mpossble n case of complete fnancal markets. Therefore our results mples that ncompleteness of fnancal markets s a possble explanaton of jumps n asset prces. Hence ncompleteness of fnancal markets may ncrease uncertanty sgnfcantly compared to complete fnancal markets. In the example, where asset prces jump, endowments vary contnuously wth states of nature, whle dvdends are constant across states of nature. Thus t should be ponted out that jumps n asset prces have to be seen as the outcome of the nteracton of real markets and fnancal markets. From a fnance perspectve t would be nterestng to calbrate a parametrc model such as an optmal growth model or an overlappng generatons model n order to study whether jumps n asset prces are compatble wth data. From a general equlbrum perspectve a partal answer to the queston of the approprate commodty space for economes wth nfnte dmensonal commodty spaces has been provded. Indeed we have shown that for Walrasan equlbra restrctng attenton to contnuous maps on the underlyng state space as n Chchlnsky & Zhou (1998) s no real restrcton. References Andersen, T., L. Benzon & J. Lund (2002), An emprcal nvestgaton of contnuous-tme equty return models, Journal of Fnance 57, Balduzz, P., S. Fores & D. Hat (1997), Prce Barrers and the dynamcs of asset prces n equlbrum, Journal of Fnancal and Quanttatve Analyss 32,

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