Completeness of a General Semimartingale Market under Constrained Trading

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1 Compleeness of a General Semimaringale Marke under Consrained Trading Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 666, USA Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 925 Évry Cedex, France Marek Rukowski School of Mahemaics Universiy of New Souh Wales Sydney, NSW 252, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology -66 Warszawa, Poland December 5, 24 The research of he firs auhor was suppored in par by NSF Gran 2285 and by Moody s Corporaion gran The research of he second auhor was suppored in par by Moody s Corporaion gran

2 2 Marke Compleeness under Consrained Trading Inroducion In his noe, we provide a raher deailed and comprehensive sudy of he basic properies of selffinancing rading sraegies in a general securiy marke model driven by disconinuous semimaringales. Our main goal is o analyze he issue of replicaion of a generic coningen claim using a self-financing rading sraegy ha is addiionally subjec o an algebraic consrain, referred o as he balance condiion. Alhough such porfolios may seem o be arificial a he firs glance, hey appear in a naural way in he analysis of hedging sraegies wihin he reduced-form approach o credi risk. Le us menion in his regard ha in a companion paper by Bielecki e al. 24b we also include specific defaulable asses in our porfolio, and we show how o use consrained porfolios o derive replicaing sraegies for defaulable coningen claims e.g., credi derivaives. The reader is also referred o Bielecki e al. 24a, where he case of coninuous semimaringale markes was sudied, for some background informaion regarding he probabilisic and financial se-up, as well as he erminology used in his noe. The main emphasis is pu here on he relaionship beween compleeness of a securiy marke model wih unconsrained rading and compleeness of an associaed model in which only rading sraegies saisfying he balance condiion are allowed. 2 Trading in Primary Asses Le Y, Y 2,..., Y k represen cash values a ime of k primary asses. We posulae ha he prices Y, Y 2,..., Y k follow possibly disconinuous semimaringales on some probabiliy space Ω, F, P, endowed wih a filraion F saisfying he usual condiions. Thus, for example, general Lévy processes, as well as jump-diffusions are covered by our analysis. Noe ha obviously F Y F, where F Y is he filraion generaed by he prices Y, Y 2,..., Y k of primary asses. As i is usually done, we se X = X for any sochasic process X, and we only consider semimaringales wih càdlàg sample pahs. We assume, in addiion ha a leas one of he processes Y, Y 2,..., Y k, say Y, is sricly posiive, so ha i can be chosen as a numeraire asse. We consider rading wihin he ime inerval [, T ] for some finie horizon dae T >. We emphasize ha we do no assume he exisence of a risk-free asse a savings accoun. 2. Unconsrained Trading Sraegies Le φ = φ, φ 2,..., φ k be a rading sraegy; in paricular, each process φ i is predicable wih respec o he reference filraion F. The componen φ i represens he number of unis of he ih asse held in he porfolio a ime. Then he wealh V φ a ime of he rading sraegy φ = φ, φ 2,..., φ k equals V φ = φ i Y i, [, T ], i= and φ is said o be a self-financing sraegy if V φ = V φ + i= φ i u dy i u, [, T ]. 2 Le Φ be he class of all self-financing rading sraegies. By combining he las wo formulae, we obain he following expression for he dynamics of he wealh process of a sraegy φ Φ dv φ = V φ φ i Y i Y dy + φ i dy i.

3 T.R. Bielecki, M. Jeanblanc and M. Rukowski 3 The represenaion above shows ha he wealh process V φ depends only on k componens of φ. Noe also ha, in our seing, he process V φ k φi Y i Y is predicable. Remarks. Le us noe ha Proer 2 assumes ha he componen of a sraegy φ ha corresponds o he savings accoun which is a coninuous process is merely opional. The ineresed reader is referred o Proer 2 for a horough discussion of oher issues relaed o he regulariy of sample pahs of processes φ, φ 2,..., φ k and V φ. Choosing Y as a numeraire asse, and denoing V φ = V φy, Y i, = Y i Y, we ge he following well-known resul showing ha he self-financing feaure of a rading sraegy is invarian wih respec o he choice of a numeraire asse. Lemma 2. i For any φ Φ, we have V φ = V φ + φ i u dy i, u, [, T ]. 3 ii Conversely, le X be a F T -measurable random variable, and le us assume ha here exiss x R and F-predicable processes φ i, i = 2, 3,..., k such ha T X = YT x + φ i u dyu i,. Then here exiss a F-predicable process φ such ha he sraegy φ = φ, φ 2,..., φ k is selffinancing and replicaes X. Moreover, he wealh process of φ saisfies V φ = V Y, where he process V is given by formula 4 below. Proof. The proof of par i is given, for insance, in Proer 2. We shall hus only prove par ii. Le us se V = x + φ i u dyu i,, [, T ], 4 and le us define he process φ as φ = V where V = V Y. From 4, we have dv = k From he equaliy i follows ha φ i Y i, = Y V φi dy i, dv = dv Y = V dy + Y dv = V dy +, and hus φ i Y i, + d[y, V ] φ i Y dy i, + d[y, Y i, ]. dy i = dy i, Y = Y i, dy + Y dy i, + d[y, Y i, ], dv = V dy + = V φ i dy i Y i, dy φ i Y i, dy + φ i dy i,

4 4 Marke Compleeness under Consrained Trading and our aim is o prove ha The las equaliy holds if dv = i= φ i dy i. φ = V φ i Y i, = V φ i Y i,, 5 i.e., if V = k φi Y i,, which is he case from he definiion 4 of V. Noe also ha from he second equaliy in 5 i follows ha he process φ is indeed F-predicable. Finally, he wealh process of φ saisfies V φ = V Y for every [, T ], and hus V T φ = X. 2.2 Consrained Trading Sraegies In his secion, we make an addiional assumpion ha he price process Y k is sricly posiive. Le φ = φ, φ 2,..., φ k be a self-financing rading sraegy saisfying he following consrain: φ i Y i = Z, [, T ], 6 for some l k and a predeermined, F-predicable process Z. In he financial inerpreaion, equaliy 6 means ha he porfolio φ should be rebalanced in such a way ha he oal wealh invesed in securiies Y l+, Y l+2,..., Y k should mach a predeermined sochasic process for insance, we may assume ha i is consan over ime or follows a deerminisic funcion of ime. For his reason, he consrain 6 will be referred o as he balance condiion. Our firs goal is o exend par i in Lemma 2. o he case of consrained sraegies. Le Φ l Z sand for he class of all self-financing rading sraegies saisfying he balance condiion 6. They will be someimes referred o as consrained sraegies. Since any sraegy φ Φ l Z is self-financing, we have V φ = and hus we deduce from 6 ha Le us wrie Y i, V φ = = Y i Y, Y i,k i= φ i Y i i= φ i Y i = = V φ l i= i= φ i Y i, φ i Y i + Z. = Y i Y k, Z = Z Y. The following resul exends Lemma.7 in Bielecki e al. 24a from he case of coninuous semimaringales o he general case. I is apparen from Proposiion 2. ha he wealh process V φ of a sraegy φ Φ l Z depends only on k 2 componens of φ. Proposiion 2. The relaive wealh V φ = V φy of a sraegy φ Φ l Z saisfies V φ = V φ + l φ i u dy i, u + φ i u u Y i, dy i, Y k, dy k, u + Zu Y k, dyu k,. 7 Proof. Le us consider discouned values of price processes Y, Y 2,..., Y k, wih Y aken as a numeraire asse. By virue of par i in Lemma 2., we hus have V φ = V φ + φ i u dy i, u. 8

5 T.R. Bielecki, M. Jeanblanc and M. Rukowski 5 The balance condiion 6 implies ha and hus φ i Y i, = Z, φ k = Y k, Z By insering 9 ino 8, we arrive a he desired formula 7. φ i Y i,. 9 Le us ake Z =, so ha φ Φ l. Then he balance condiion becomes k and 7 reduces o dv φ = l φ i dy i, Case of Coninuous Semimaringales φ i Y i, dy i, Y k, dy k, φi Y i =,. For he sake of noaional simpliciy, we denoe by Y i,k, he process given by he formula Y i,k, = dyu i, Y i, dyu k, Y k, so ha 7 becomes V φ = V φ + l φ i u dy i, u + φ i u dyu i,k, Zu + Y k, dyu k,. 2 In Bielecki e al. 24a, we posulaed ha he primary asses Y, Y 2,..., Y k follow sricly posiive coninuous semimaringales, and we inroduced he auxiliary processes Ŷ i,k, = Y i,k e αi,k,, where α i,k, = ln Y i,k, ln Y,k = Y i,k u Y,k u d Y i,k, Y,k u. In Lemma.7 in Bielecki e al. 24a see also Vaillan 2, we have shown ha, under coninuiy of Y, Y 2,..., Y k, he discouned wealh of a self-financing rading sraegy φ ha saisfies he consrain k φi Y i = Z can be represened as follows: V φ = V φ + l φ i u dy i, u + φ i,k, u dŷ u i,k, Zu + Yu k, dyu k,, 3 φ i,k, where we wrie = φ i Y,k e αi,k,. The following simple resul reconciles expression 2 esablished in Proposiion 2. wih represenaion 3 derived in Bielecki e al. 24a. Lemma 2.2 Assume ha he prices Y, Y i and Y k follow sricly posiive coninuous semimaringales. Then we have and dy i,k, Y i,k, = Y,k u e αi,k, u = Y,k dy i,k Y i,k dŷ i,k, u dα i,k,.

6 6 Marke Compleeness under Consrained Trading Proof. In he case of coninuous semimaringales, formula becomes Y i,k, = dyu i, Y u i, Yu k, dyu k, = On he oher hand, an applicaion of Iô s formula yields dŷ i,k, dy i, u Y i,k u dy,k u. = e αi,k, dy i,k Y,k d Y i,k, Y,k and hus Y,k e αi,k, dŷ i,k, u = Y,k dy i,k Y,k d Y i,k, Y,k. One checks easily ha for any wo coninuous semimaringales, say X and Y, we have Y dx Y d X, Y = dx Y X dy, provided ha Y is sricly posiive. To conclude he derivaion of he firs formula, i suffices o apply he las ideniy o processes X = Y i,k and Y = Y,k. For he second formula, noe ha dy i,k, = Y,k e αi,k, dŷ i,k, = Y,k e αi,k, dy i,k e αi,k, = Y,k dy i,k Y i,k dα i,k,, as required. I is obvious ha he processes Y i,k, and Ŷ i,k, are uniquely specified by he join dynamics of Y, Y i and Y k. The following resul shows ha he converse is also rue. Corollary 2. The price Y i i he join dynamics of processes Y, Y k and Ŷ i,k,, or ii he join dynamics of processes Y, Y k and Y i,k,. a ime is uniquely specified by he iniial value Y i and eiher Proof. Since Ŷ i,k, = Y i,k e αi,k,, we have α i,k, = ln Y i,k, ln Y,k = ln Ŷ i,k,, ln Y,k, and hus Y i = Y k Ŷ i,k, e αi,k, = Y k Ŷ i,k, e ln Ŷ i,k,,ln Y,k. This complees he proof of par i. For he second par, noe ha he process Y i, saisfies Y i, = Y i, + Y i,k, Yu i, + Yu k, dyu k,. 4 I is well known ha he SDE X = X + H + X u dy u, where H and Y are coninuous semimaringales wih H = has he unique, srong soluion given by he formula X = E Y X + Eu Y dh u Eu Y d Y, H u. Upon subsiuion, his proves ii.

7 T.R. Bielecki, M. Jeanblanc and M. Rukowski 7 3 Replicaion wih Consrained Sraegies The nex resul is essenially a converse o Proposiion 2.. Also, i exends par ii of Lemma 2. o he case of consrained rading sraegies. As in Secion 2.2, we assume ha l k, and Z is a predeermined, F-predicable process. Proposiion 3. Le a F T -measurable random variable X represen a coningen claim ha seles a ime T. Assume ha here exis F-predicable processes φ i, i = 2, 3,..., k such ha l T X = YT x + φ i dy i, + φ i dy i,k, Z + Y k, dy k,. 5 Then here exis he F-predicable processes φ and φ k such ha he sraegy φ = φ, φ 2,..., φ k belongs o Φ l Z and replicaes X. The wealh process of φ equals, for every [, T ], l V φ = Y x + φ i u dyu i, + φ i u dyu i,k, Zu + Y k, dyu k,. 6 Proof. As expeced, we firs se noe ha φ k is F-predicable and V = x + l φ k = Y k Z φ i u dy i, u + φ i Y i φ i u dyu i,k, + Arguing along he same lines as in he proof of Proposiion 2., we obain Zu Y k, dyu k,. 7 Now, we define φ = V V = V + φ i u dy i, u. φ i Y i, = Y V where V = V Y. As in he proof of Lemma 2., we check ha φ = V φ i Y i,, φ i Y i, and hus he process φ is F-predicable. I is clear ha he sraegy φ = φ, φ 2,..., φ k is selffinancing and is wealh process saisfies V φ = V for every [, T ]. In paricular, V T φ = X, so ha φ replicaes X. Finally, equaliy 7 implies 6, and hus φ Φ l Z. Noe ha equaliy 5 is a necessary by Proposiion 2. and sufficien by Proposiion 3. condiion for he exisence of a consrained sraegy replicaing a given coningen claim X. 3. Modified Balance Condiion I is emping o replace he consrain 6 by a more convenien condiion: φ i Y i = Z, [, T ], 8

8 8 Marke Compleeness under Consrained Trading where Z is a predeermined, F-predicable process. If a self-financing rading sraegy φ saisfies he modified balance condiion 8 hen for he relaive wealh process we obain cf. 7 V φ = V φ + l φ i u dy i, u + φ i u u Y u i, dy i, Y k, dy k, u Zu + Y k, dyu k,. 9 Noe ha in many cases he inegrals above are meaningful, so ha a counerpar of Proposiion 2. wih he modified balance condiion can be formulaed. To ge a counerpar of Proposiion 3., we need o replace 5 by he equaliy X = Y T x + l φ i dy i, + φ i Y i, dy i, where φ 3, φ 4,..., φ k are F-predicable processes. We define and we se V = x + l φ i u dy i, u + φ k = Y k Z φ i u dy i, Y k, u Y u i, Y k, φ i Y i, φ = V dy k, dy k, u + + φ i Y i,. Z Y k, dy k, Zu Y k, dyu k,,, 2 Suppose, for he sake of argumen, ha he processes φ and φ k defined above are F-predicable. Then he rading sraegy φ = φ, φ 2,..., φ k is self-financing on [, T ], replicaes X, and saisfies he consrain 8. Noe, however, ha he predicabiliy of φ and φ k is far from being obvious, and i is raher difficul o provide non-rivial and pracically appealing sufficien condiions for his propery. 3.2 Synheic Asses Le us fix i, and le us analyze he auxiliary process Y i,k, given by formula. We claim ha his process can be inerpreed as he relaive wealh of a specific self-financing rading sraegy associaed wih Y, Y 2,..., Y k. Specifically, we will show ha for any i = 2, 3,..., k he process Ȳ i,k,, given by he formula Ȳ i,k, = Y Y i,k, = Y u Y i, dy i, Y k, dy k, u represens he price of a synheic asse. For breviy, we shall frequenly wrie Ȳ i insead of Ȳ i,k,. Noe ha he process Ȳ i is no sricly posiive in fac, Ȳ i =., 3.2. Equivalence of Primary and Synheic Asses Our goal is o show ha rading in primary asses is formally equivalen o rading in synheic asses. The firs resul shows ha he process Ȳ i can be obained from primary asses Y, Y i and Y k hrough a simple self-financing sraegy. This jusifies he name synheic asse given o Ȳ i. Lemma 3. For any fixed i = 2, 3,..., k, le a F T -measurable random variable Ȳ T i be given as T ȲT i = YT Y i,k, T = YT dy i, Y i, dy k,. 2 Y k,

9 T.R. Bielecki, M. Jeanblanc and M. Rukowski 9 Then here exiss a sraegy φ Φ ha replicaes he claim Ȳ T i. Moreover, we have, for every [, T ], V φ = Y Y i,k, = Y u Y i, dy i, Y k, dy k, u = Ȳ i. 22 Proof. To esablish he exisence of a sraegy φ wih he desired properies, i suffices o apply Proposiion 3.. We fix i and we sar by posulaing ha φ i = and φ j = for any 2 j k, j i. Then equaliy 2 yields 5 wih X = Ȳ T i, x =, l = and Z =. Noe ha he balance condiion becomes φ j Y j = Y i + φ k Y k =. Le us define φ and φ k by seing Noe ha we also have j=2 Y k φ k = Y i, φ = V Y i, φ k Y k,. φ = V Y i, φ k Y k, = V. Hence, φ and φ k are F-predicable processes, he sraegy φ = φ, φ 2,..., φ k is self-financing, and i saisfies 6 wih l = and Z =, so ha φ Φ. Finally, equaliy 22 holds, and hus V T φ = Ȳ T i. Noe ha o replicae he claim Ȳ T i = Ȳ i,k, T, i suffices o inves in primary asses Y, Y i and Y k. Essenially, we sar wih zero iniial endowmen, we keep a any ime one uni of he ih asse, we rebalance he porfolio in such a way ha he oal wealh invesed in he ih and kh asses is always zero, and we pu he residual wealh in he firs asse. Hence, we deal here wih a specific sraegy such ha he risk of he ih asse is perfecly offse by rebalancing he invesmen in he kh asse, and our rades are financed by aking posiions in he firs asse. Noe ha he process Y i, saisfies he following SDE cf. 4 Y i, = Y i, + Ȳ i, Y i, + Y k, dyu k,, 23 which is known o possess a unique srong soluion. Hence, he relaive price Y i, a ime is uniquely deermined by he iniial value Y i, and processes Ȳ i, and Y k,. Consequenly, he price Y i a ime of he ih primary asse is uniquely deermined by he iniial value Y i, he prices Y, Y k of primary asses, and he price Ȳ i of he ih synheic asse. We hus obain he following resul. Lemma 3.2 Filraions generaed by he primary asses Y, Y 2,..., Y k and by he price processes Y, Y 2,..., Y l, Ȳ l+,..., Ȳ, Y k coincide. Lemma 3.2 suggess ha for any choice of he underlying filraion F such ha F Y F, rading in asses Y, Y 2,..., Y k is essenially equivalen o rading in Y, Y 2,..., Y l, Ȳ l+,..., Ȳ, Y k. Le us firs formally define he equivalence of marke models. Definiion 3. We say ha he wo unconsrained models, M and M say, are equivalen wih respec o a filraion F if boh models are defined on a common probabiliy space and every primary asse in M can be obained by rading in primary asses in M and vice versa, under he assumpion ha rading sraegies are F-predicable. Noe ha we do no assume ha models M and M have he same number of primary asses. The nex resul jusifies our claim of equivalence of primary and synheic asses.

10 Marke Compleeness under Consrained Trading Corollary 3. Models M = Y, Y 2,..., Y k ; Φ and M = Y, Y 2,..., Y l, Ȳ l+,..., Ȳ, Y k ; Φ are equivalen wih respec o any filraion F such ha F Y F. Proof. In view of Lemma 3., i suffices o show ha he price process of each primary asse Y i for i = l, l +,..., k can be mimicked by rading in Y, Ȳ i and Y k. To see his, noe ha for any fixed i = l, l +,..., k, we have see he proof of Lemma 3. Ȳ i = V φ = φ Y + Y i + φ k Y k wih Consequenly, and dȳ i = dv φ = φ dy + dy i + φ k dy k. Y i = φ Y + Ȳ i φ k Y k dy i = φ dy + dȳ i φ k dy k. This shows ha he sraegy φ,, φ k in Y, Ȳ i and Y k is self-financing and is wealh equals Y i Replicaing Sraegies wih Synheic Asses In view of Lemma 3., he replicaing rading sraegy for a coningen claim X, for which 5 holds, can be convenienly expressed in erms of primary securiies Y, Y 2,..., Y l and Y k, and synheic asses Ȳ l+, Ȳ l+2,..., Ȳ. To his end, we represen 5-6 in he following way: where Ȳ i, X = Y T = Ȳ i /Y V φ = Y x + = Y i,k, x + l, and l φ i dy i, + φ i u dy i, u + φ i dȳ i, + φ i u dȳ u i, + Z Y k, dy k, Zu Y k, dyu k, Corollary 3.2 Le X be a F T -measurable random variable such ha 24 holds for some F-predicable process Z and some F-predicable processes φ 2, φ 3,..., φ. Le ψ i = φ i for i = 2, 3,..., k, and ψ = V l ψ i Y i, ψ k = Z Y k, = Z Y k, ψ i Ȳ i, ψ k Y k, = V l ψ i Y i, ψ i Ȳ i, ψ k Y k,. Then ψ = ψ, ψ 2,..., ψ k is a self-financing rading sraegy in asses Y,..., Y l, Ȳ l+,..., Ȳ, Y k. Moreover, ψ saisfies ψ k Y k = Z, [, T ], and i replicaes X. Proof. In view of 24, i suffices o apply Proposiion 3. wih l = k. 4 Model Compleeness We shall now examine he relaionship beween he arbirage-free propery and compleeness of a marke model in which rading is resriced a priori o self-financing sraegies saisfying he balance condiion.

11 T.R. Bielecki, M. Jeanblanc and M. Rukowski 4. Minimal Compleeness of an Unconsrained Model Le M = Y, Y 2,..., Y k ; Φ be an arbirage-free marke model. Unless explicily saed oherwise, Φ sands for he class of all F-predicable, self-financing sraegies. Noe, however, ha he number of raded asses and heir selecion may be differen for each paricular model. Consequenly, he dimension of a sraegy φ Φ will depend on he number of raded asses in a given model. For he sake of breviy, his feaure is no refleced in our noaion. Definiion 4. We say ha a model M is complee wih respec o F if any bounded F T -measurable coningen claim X is aainable in M. Oherwise, a model M is said o be incomplee wih respec o F. Definiion 4.2 A model M = Y, Y 2,..., Y k ; Φ is minimally complee wih respec o F if M is complee, and for any i =, 2,..., k he reduced model M i = Y, Y 2,..., Y i, Y i+,..., Y k ; Φ is incomplee wih respec o F, so ha for each i here exiss a bounded, F T -measurable coningen claim, which is no aainable in he model M i. In his case, we say ha he degree of compleeness of M equals k. Le us sress ha rading sraegies in he reduced model M i are predicable wih respec o F, raher han wih respec o he filraion generaed by price processes Y, Y 2,..., Y i, Y i+,..., Y k. Hence, when we move from M o M i, we reduce he number of raded asse, bu we preserve he original informaion srucure F. Minimal compleeness of a model M means ha all primary asses Y, Y 2,..., Y k are needed if we wish o generae he class of all bounded F T -measurable claims hrough F-predicable rading sraegies. The following lemma is hus an immediae consequence of Definiion 4.2. Lemma 4. Assume ha a model M is complee, bu no minimally complee, wih respec o F. Then here exiss a leas one primary asse Y i, which is redundan in M, in he sense ha i corresponds o he wealh process of some rading sraegy in he reduced model M i. Complee models ha are no minimally complee do no seem o describe adequaely he reallife feaures of financial markes in fac, i is frequenly argued ha he real-life markes are no even complee. Also, from he heoreical perspecive, here is no advanage in keeping a redundan asse among primary securiies. For his reasons, in wha follows, we shall resric our aenion o marke models M ha are eiher incomplee or minimally complee. Lemma 4.2 shows ha he degree of compleeness is a well-defined noion, in he sense ha i does no depend on he choice of raded asses, provided ha he model compleeness is preserved. Lemma 4.2 Le a model M = Y, Y 2,..., Y k ; Φ be minimally complee wih respec o F. Le M = Ỹ, Ỹ 2,..., Ỹ k ; Φ, where he processes Ỹ i = V φ i, i =, 2,..., k represen he wealh processes of some rading sraegies φ, φ 2,..., φ k Φ. If a model M is complee wih respec o F hen i is also minimally complee wih respec o F, and hus is degree of compleeness equals k. Proof. The proof relies on simple algebraic consideraions. By assumpion, for every i =, 2,..., k, we have dỹ i = φ ij dy j, j= for some family φ ij, i, j =, 2,..., k of F-predicable sochasic processes. By assumpion, he marke M is complee. To check ha i is minimally complee, i suffices o show ha he marke M = Ỹ 2,..., Ỹ k ; Φ is incomplee he same proof will work for any reduced model M i. Suppose, on he conrary, ha M is complee wih respec o F. In paricular, he price of each primary asse Y l, l =, 2,..., k can be replicaed in M by means of some rading sraegy ψ l = ψ l2,..., ψ lk.

12 2 Marke Compleeness under Consrained Trading In oher words, here exiss a family ψ li, l =, 2,..., k, i = 2, 3,..., k of F-predicable sochasic processes such ha dy l = ψ li dỹ i. 26 Since ψ, ψ 2,..., ψ k are F-predicable processes wih values in R, i is raher clear ha here exiss a family α, α 2,..., α of F-predicable processes such ha we have, for every [, T ], ψ = ψ 2, ψ 3,..., ψ k = Consequenly, using 26, we obain dy = ψ i dỹ i = j=2 j=2 α j ψ j2 αψ j ji dỹ i =, ψ j3,..., ψ jk = α j j=2 αψ j j. j=2 ψ ji dỹ i = j=2 α j dy j. We conclude ha Y is redundan in M, and hus he reduced model M is complee. This conradics he assumpion ha M is minimally complee. By combining Lemma 4.2 wih Corollary 3., we obain he following resul. Corollary 4. A model M = Y, Y 2,..., Y k ; Φ is minimally complee if and only if a model M = Y, Y 2,..., Y l, Ȳ l+,..., Ȳ, Y k ; Φ has his propery. As one migh easily guess, he degree of a model compleeness depends on he relaionship beween he number of primary asses and he number of independen sources of randomness. In he wo models examined in Secions 5. and 5.2 below, we shall deal wih k = 4 primary asses, bu he number of independen sources of randomness will equal wo and hree for he firs and he second model, respecively. 4.2 Compleeness of a Consrained Model Le M = Y, Y 2,..., Y k ; Φ be an arbirage-free marke model, and le us denoe by M l Z = Y, Y 2,..., Y k ; Φ l Z he associaed model in which he class Φ is replaced by he class Φ l Z of consrained sraegies. We claim ha if M is arbirage-free and minimally complee wih respec o he filraion F = F Y, where Y = Y, Y 2,..., Y k, hen he consrained model M l Z is arbiragefree, bu i is incomplee wih respec o F. Conversely, if he model M l Z is arbirage-free and complee wih respec o F, hen he original model M is no minimally complee. To prove hese claims, we need some preliminary resuls. The following definiion exends he noion of equivalence of securiy marke models o he case of consrained rading. Definiion 4.3 We say ha he wo consrained models are equivalen wih respec o a filraion F if hey are defined on a common probabiliy space and he class of all wealh processes of F-predicable consrained rading sraegies is he same in boh models. Corollary 4.2 The consrained model is equivalen o he consrained model M l Z = Y, Y 2,..., Y k ; Φ l Z M Z = Y, Y 2,..., Y l, Ȳ l+,..., Ȳ, Y k ; Φ Z.

13 T.R. Bielecki, M. Jeanblanc and M. Rukowski 3 Proof. I suffices o make use of Corollaries 3. and 3.2. Noe ha he model M Z is easier o handle han M l Z. For his reason, we shall sae he nex resul for he model M l Z which is of our main ineres, bu we shall focus on he equivalen model M Z in he proof. Proposiion 4. i Assume ha he model M is arbirage-free and minimally complee. Then for any F-predicable process Z and any l =, 2,..., k he consrained model M l Z is arbirage-free and incomplee. ii Assume ha he consrained model M l Z associaed wih M is arbirage-free and complee. Then M is eiher no arbirage-free or no minimally complee. Proof. The arbirage-free propery of M l Z is an immediae consequence of Corollary 4.2 and he fac ha Φ Z Φ. In view of Corollary 4., i suffices o check ha he minimal compleeness of M implies ha M Z is incomplee. By assumpion, here exiss a bounded, F T -measurable claim X ha canno be replicaed in M k = Y, Y 2,..., Y l, Ȳ l+,..., Ȳ ; Φ i.e., when rading in Y k is no allowed. Le us consider he following random variable We claim ha Y canno be replicaed in we have l V T φ = V φ + i= Y = X + Z Y k dy k. M Z. Indeed, for any rading sraegy φ Φ Z, φ i dy i + φ i dȳ i + Z Y k dy k, and hus he exisence of a replicaing sraegy for Y in M Z will imply he exisence of a replicaing sraegy for X in M k, which conradics our assumpion. Par ii is a sraighforward consequence of par i. I is worh noing ha he arbirage-free propery of M l Z does no imply he same propery for M. As a rivial example, we may ake l = k and Z =, so ha rading in he asse Y k is in fac excluded in M l Z, bu i is allowed in he larger model M. 5 Jump-Diffusion Case In order o make he resuls of Secions 2-4 more angible, we shall now analyze he case of jumpdiffusion processes. For he sake of concreeness and simpliciy, we shall ake k = 4. Needless o say ha his assumpion is no essenial, and he similar consideraions can be done for any sufficienly large number of primary asses. We consider a model M = Y, Y 2,..., Y 4 ; Φ wih disconinuous asse prices governed by he SDE dy i = Y i µi d + σ i dw + κ i dm 27 for i =,..., 4, where W = W, W 2,..., W d, [, T ], is a d-dimensional sandard Brownian moion and M = N λ, [, T ], is a compensaed Poisson process under he acual probabiliy P. Le us sress ha W and N are a Brownian moion and a Poisson process wih respec o F, respecively. This means, in paricular, ha hey are independen processes. We shall assume ha F = F W,N is he filraion generaed by W and N. The coefficiens µ i, σ i = σi, σ2 i,..., σd i and κ i in 27 can be consan, deerminisic or even sochasic predicable wih respec o he filraion F. For simpliciy, in wha follows we shall assume ha hey are consan. In addiion, we posulae ha κ >, so ha Y > for every

14 4 Marke Compleeness under Consrained Trading [, T ], provided ha Y >. Finally, le Z be a predeermined F-predicable process. Recall ha Φ Z is he class of all self-financing sraegies ha saisfy he balance condiion 4 φ i Y i = Z, [, T ]. 28 Our goal is o presen examples illusraing Proposiion 4. and, more imporanly, o show how o proceed if we wish o replicae a coningen claim using a rading sraegy saisfying he balance condiion. I should be acknowledged ha in he previous secions we have no deal a all wih he issue of admissibiliy of rading sraegies, and hus some relevan echnical assumpions were no menioned. Also, an imporan ool of an equivalen maringale measure was no ye employed. 5. Complee Consrained Model In his subsecion, i i assumed hro ha d =, so ha we have wo independen sources of randomness, a one-dimensional Brownian moion W and a Poisson process N. We shall verify direcly ha, under naural addiional condiions, he model M Z is arbirage-free and complee wih respec o F, bu he original model M is no minimally complee, so ha a redundan primary asse exiss in M. Lemma 5. Assume ha δ := de A, where [ σ2 σ A = 4 κ 2 κ 4 σ 3 σ 4 κ 3 κ 4 ]. Then here exiss a unique probabiliy measure P, equivalen o P on Ω, F T, and such ha he relaive prices Ȳ 2, = Ȳ 2 /Y and Ȳ 3, = Ȳ 3 /Y of synheic asses Ȳ 2 and Ȳ 3 are P-maringales. Proof. Le us wrie Ŵ = W σ and M = M λκ. By sraighforward calculaions, he relaive value of he synheic asse Ȳ i saisfies, for i = 2, 3, dȳ i, = dy i,4, = Y i, µ i µ 4 d + σ i σ 4 dw σ d + κ i κ 4 dm λκ d, 29 + κ or equivalenly, dȳ i, = Y i, µ i µ 4 d + σ i σ 4 dŵ + κ i κ 4 d M. + κ By virue of Girsanov s heorem, here exiss a unique probabiliy measure P, equivalen o P on Ω, F T, and such ha he processes Ŵ and M follow F-maringales under P. Under our assumpion δ := de A, he equaions µ 4 µ i = σ 4 σ i θ + κ 4 κ i + κ νλ, i = 2, 3, 3 uniquely specify θ and ν. Using once again Girsanov s heorem, we show ha here exiss a unique probabiliy measure P, equivalen o P on Ω, F T, and such ha he processes W = Ŵ θ = W σ + θ and M = M λν = N λ + κ + ν are F-maringales under P. We hen have, for i = 2, 3 and every [, T ], dȳ i, = Y i, σ i σ 4 d W + κ i κ 4 d M. + κ Noe ha N follows under P a Poisson process wih he consan inensiy λ + κ + ν, and hus M is he compensaed Poisson process under P. Moreover, under he presen assumpions, he processes W and M are independen under P.

15 T.R. Bielecki, M. Jeanblanc and M. Rukowski 5 From now on, we posulae ha δ = de A and κ i > for every i =, 2,..., 4. Under his assumpion, he filraion F coincides wih he filraion F Y generaed by primary asses. In he nex resul, we provide sufficien condiions for he exisence of a replicaing sraegy saisfying he balance condiion 28. Essenially, Proposiion 5. shows ha he model M Z = Y, Ȳ 2, Ȳ 3, Y 4 ; Φ Z is complee wih respec o F. Proposiion 5. Le X be a F T -measurable coningen claim ha seles a ime T. Assume ha he random variable X, given by he formula X = X Y T Z Y 4 dy 4,, 3 is square-inegrable under P, where P is he unique probabiliy measure equivalen o P on Ω, F T such ha he relaive prices Ȳ 2, and Ȳ 3, are P-maringales. Then X can be replicaed in he model M Z. Proof. To prove he exisence of a replicaing sraegy for X in he class Φ Z, we may use eiher Proposiion 3. if we wish o work wih raded asses Y, Y 2, Y 3, Y 4 or Corollary 3.2 and Lemma 5. if we prefer o work wih Y, Ȳ 2, Ȳ 3, Y 4. The second choice seems o be more convenien, and hus we shall focus on he exisence a rading sraegy ψ = ψ, ψ 2,..., ψ 4 wih he properies described in Corollary 3.2. In view of 24 and Corollary 3.2, i suffices o check ha here exis a consan x, and F-predicable processes φ 2 and φ 3 such ha X = x + 3 φ i dȳ i,. 32 To show ha such processes exis, we shall use Lemma 5.. I is crucial o observe ha he pair W, M, which was obained in he proof of Lemma 5.2 from he original pair W, M by means of Girsanov s ransformaion, enjoys he predicable represenaion propery see, for example, Jacod and Shiryaev 22, Secions III.4 and III.5. Since X is square-inegrable under P, here exiss a consan x and F-predicable processes ξ and ς such ha Observe ha and d W = δ d M = + κ δ Hence, upon seing κ 3 κ 4 X = x + ξ d W + ς d M. dȳ 2, Y 2, σ 2 σ 4 κ 2 κ 4 dȳ 3, Y 3, σ 3 σ 4 φ 2 = ξ Θ 2 + ς Ψ 2, φ 3 = ξ Θ 3 + ς Ψ 3, 3, dȳ Y 3, =: Θ 2 dȳ 2, + Θ 3 dȳ 3, 2, dȳ Y 2, =: Ψ 2 dȳ 2, + Ψ 3 dȳ 3,. we obain he desired represenaion 32 for X. To complee he proof of he proposiion, i suffices o make use of Corollary 3.2. Remarks. If we ake he class Φ 2 Z of consrained sraegies, insead of he class Φ Z, hen we need o show he exisence of F-predicable processes φ 2 and φ 3 such ha X = x + φ 2 dy 2, + φ 3 dȳ 3,. 33

16 6 Marke Compleeness under Consrained Trading To his end, i suffices o focus on an equivalen probabiliy measure under which he relaive prices Y 2, and Ȳ 3, are F-maringales, and o follow he same seps as in he proof of Proposiion 5.. In view of Lemma 5., he reduced model M4 = Y, Ȳ 2, Ȳ 3 ; Φ admis a maringale measure P corresponding o he choice of Y as a numeraire asse, and hus i is arbirage-free, under he usual choice of admissible rading sraegies e.g., he so-called ame sraegies. By virue of formula 7 in Proposiion 2., for he arbirage-free propery of he model M Z o hold, i suffices, in addiion, ha he process follows a maringale under P. Z u Y 4 dy 4, u, [, T ], Noe, however, ha he above-menioned propery does no imply, in general, ha he probabiliy measure P is a maringale measure for he relaive price Y 4,. Since dy 4, = Y 4, µ 4 µ d + σ 4 σ dw σ d + κ 4 κ dm λκ d, 34 + κ a maringale measure for he relaive prices Ȳ 2,, Ȳ 3, and Y 4, exiss if and only if for he pair θ, ν ha solves 3, we also have ha µ µ 4 = σ 4 σ θ + κ 4 κ + κ νλ. This holds if and only if de  =, where  is he following marix  = µ µ 4 σ σ 4 κ κ 4 µ 2 µ 4 σ 2 σ 4 κ 2 κ 4. µ 3 µ 4 σ 3 σ 4 κ 3 κ 4 Hence, he model M or, equivalenly, he model M is no arbirage-free, in general. In fac, M is arbirage-free if and only if he primary asse Y 4 is redundan in M. The following resul summarizes our findings. Proposiion 5.2 Le M be he model given by 27. Assume ha κ i > for every i =, 2,..., 4 and δ = de A. Moreover, le he process Z u Y 4 dy 4, u follow a maringale under P. Then he following saemens hold. i The model M Z is arbirage-free and complee, in he sense of Proposiion 5.. ii If he model M is arbirage-free hen i is complee, in he sense ha any F T -measurable random variable X such ha XY T is square-inegrable under P is aainable in his model, bu M is no minimally complee. Example 5. Consider, for insance, a call opion wrien on he asse Y 4, so ha X = YT 4 K+, and le us assume ha Z = Y. 4 Under assumpions of Proposiion 5.2, models M and M Z are arbirage-free and he asse Y 4 is redundan. I is hus raher clear ha he opion can be hedged by dynamic rading in primary asses Y, Y 2, Y 3 and by keeping a any ime one uni of Y 4. Of course, he same conclusion applies o any European claim wih Y 4 as he underlying asse. 5.2 Incomplee Consrained Model We now assume ha d = 2, so ha he number of independen sources of randomness is increased o hree. In view of 27, we have, for i =,..., 4, dy i = Y i µi d + σ i dw + σ 2 i dw 2 + κ i dm.

17 T.R. Bielecki, M. Jeanblanc and M. Rukowski 7 We are going o check ha under he se of assumpions making he unconsrained model M arbirage-free and minimally complee, he consrained model M l Z is also arbirage-free, bu i is incomplee. To his end, we firs examine he exisence and uniqueness of a maringale measure associaed wih he numeraire Y. Lemma 5.2 Assume ha de Ã, where he marix à is given as σ σ4 σ 2 σ4 2 κ κ 4 à = σ2 σ4 σ2 2 σ4 2 κ 2 κ 4. σ3 σ4 σ3 2 σ4 2 κ 3 κ 4 Then here exiss a unique probabiliy measure P, equivalen o P on Ω, F T, and such ha he relaive prices Ȳ 2, = Ȳ 2 /Y, Ȳ 3, = Ȳ 3 /Y of synheic asses Ȳ 2, Ȳ 3, and he relaive price Y 4, of he primary asse Y 4 follow maringales under P. Proof. Le us wrie Ŵ = W σ = W, W 2 σ, σ 2 and M = M λκ. By sraighforward calculaions, he relaive values Ȳ i,, i = 2, 3 and Y 4, saisfy dȳ i, = Y i, µ i µ 4 d + σ i σ 4 dŵ + κ i κ 4 d M + κ and dy 4, = Y 4, µ 4 µ d + σ 4 σ dw σ d + κ 4 κ dm λκ d. + κ By virue of Girsanov s heorem, here exiss a unique probabiliy measure P, equivalen o P on Ω, F T, and such ha he processes Ŵ and M follow F-maringales under P. Now, le θ = θ, θ 2 and ν be uniquely specified by he condiions µ 4 µ i = σ i σ 4 θ + κ i κ 4 + κ νλ, i = 2, 3, 4. Anoher applicaion of Girsanov s heorem yields he exisence of a unique probabiliy measure P, equivalen o P on Ω, F T, such ha he processes W = Ŵ θ = W σ + θ and M = M λν = N λ + κ + ν are F-maringales under P. We hen have, for i = 2, 3 and every [, T ], while dȳ i, = Y i, σ i σ 4 d W + κ i κ 4 d M + κ 35 dy 4, = Y 4, σ 4 σ d W + κ 4 κ d M κ Noe ha N follows under P a Poisson process wih he consan inensiy λ + κ + ν, and hus M is he compensaed Poisson process under P. Moreover, under he presen assumpions, he processes W and M are independen under P. I is clear ha he inequaliy de à is a necessary and sufficien condiion for he arbiragefree propery of he model M. Under his assumpion, we also have F = F Y and, as can be checked easily, he model M is minimally complee. In he nex resul, we provide sufficien condiions for he exisence of a replicaing sraegy saisfying he balance condiion 28 wih some predeermined process Z. In paricular, i is possible o deduce from Proposiion 5.3 ha he model M Z is incomplee wih respec o F.

18 8 Marke Compleeness under Consrained Trading Proposiion 5.3 Assume ha de Ã. Le X be a F T -measurable coningen claim ha seles a ime T. Assume ha he random variable X, given by he formula X := X Y T Z Y 4 dy 4,, 37 is square-inegrable under P, where P is he unique probabiliy measure, equivalen o P on Ω, F T, such ha he relaive prices Ȳ 2,, Ȳ 3, and Y 4, follow maringales under P. Then X can be replicaed in M Z if and only if he process φ 4 given by formula 4 below vanishes idenically. Proof. We shall use similar argumens as in he proof of Proposiion 5.. In view of Corollary 3.2, we need o check ha here exis a consan x, and F-predicable processes φ 2 and φ 3 such ha X = x + 3 φ i dȳ i,. 38 Noe ha he pair W, M inroduced in he proof of Lemma 5.2 has he predicable represenaion propery. Since X is square-inegrable under P, here exiss a consan x and F-predicable processes ξ and ς such ha X = x + ξ d W + ς d M. 39 In view of 35-36, we have d W d W 2 d M = Ã Y 2, dȳ 2, Y 3, dȳ 3, Y 4, dy 4, so ha here exis F-predicable processes Ψ i, Λ i, Θ i, i = 2, 3, 4 such ha Le us se, for i = 2, 3, 4,, d W = Θ 2 dȳ 2, + Θ 3 dȳ 3, + Θ 4 dy 4,, d W 2 = Λ 2 dȳ 2, + Λ 3 dȳ 3, + Λ 4 dy 4,, 4 d M = Ψ 2 dȳ 2, + Ψ 3 dȳ 3, + Ψ 4 dy 4,. φ i = ξ Θ i + ξ 2 Λ i + ς Ψ i, [, T ]. 4 Suppose firs ha φ 4 = for every [, T ]. Then, by combining 39, 4 and 4, we end up wih he desired represenaion 38 for X. To show he exisence of a replicaing sraegy for X in M Z, i suffices o apply Corollary 3.2. If, on he conrary, φ 4 does no vanish idenically, equaliy 38 canno hold for any choice of φ 2 and φ 3. The fac ha φ 4 is non-vanishing for some claims follows from Proposiion 4.. In general, i.e., when he componen φ 4 does no vanish, we ge he following represenaion X Y T = x + 3 φ i dȳ i, + φ 4 dy 4,, 42 where we se φ 4 = φ 4 + Z Y 4. Hence, as expeced any coningen claim saisfying a suiable inegrabiliy condiion is aainable in he unconsrained model M. Example 5.2 To ge a concree example of a non-aainable claim in M Z, le us ake X = YT 4 K+ and Z = Y. 4 Then, for K = Y 4, we obain X = Y 4 YT 4+ YT, and hus we formally deal wih he pu opion wrien on Y 4 wih srike Y 4. We claim ha X does no admi represenaion 38. Indeed, equaliy 38 implies ha he hedge raio of a pu opion wih respec o he underlying asse equals zero. This may happen only if he underlying asse is redundan so ha hedging can be done wih oher primary asses, and his is no he case in our model.

19 T.R. Bielecki, M. Jeanblanc and M. Rukowski 9 References T.R. Bielecki, M. Jeanblanc and M. Rukowski 24a Hedging of defaulable claims. In: Paris- Princeon Lecures on Mahemaical Finance 23, R.A. Carmona, E. Cinlar, I. Ekeland, E. Jouini, J.E. Scheinkman, N. Touzi, eds., Springer-Verlag, Berlin Heidelberg New York, pp T.R. Bielecki, M. Jeanblanc and M. Rukowski 24b Compleeness of a reduced-form credi risk model wih disconinuous asse prices. Working paper. J. Jacod and A.N. Shiryaev 987 Limi Theorems for Sochasic Processes. Springer-Verlag, Berlin Heidelberg New York. P. Proer 2 A parial inroducion o financial asse pricing heory. Sochasic Processes and Their Applicaions 9, N. Vaillan 2 A beginner s guide o credi derivaives. Working paper, Nomura Inernaional.