BENCHMARKED RISK MINIMIZATION

Transcription

1 Mahemaical Finance, Vol. 26, No. 3 July 216), BENCHMARKED RISK MINIMIZAION KE DU Souhwesern Universiy of Finance and Economics, Insiue of Financial Sudies, Chengdu, China ECKHARD PLAEN Universiy of echnology, Sydney, Finance Discipline Group and School of Mahemaical Sciences, Sydney, Ausralia his paper discusses he problem of hedging no perfecly replicable coningen claims using he numéraire porfolio. he proposed concep of benchmarked risk minimizaion leads beyond he classical no-arbirage paradigm. I provides in incomplee markes a generalizaion of he pricing under classical risk minimizaion, pioneered by Föllmer, Sondermann, and Schweizer. he laer relies on a quadraic crierion, requess square inegrabiliy of claims and gains processes, and relies on he exisence of an equivalen risk-neural probabiliy measure. Benchmarked risk minimizaion avoids hese resricive assumpions and provides symmery wih respec o all primary securiies. I employs he real-world probabiliy measure and he numéraire porfolio o idenify he minimal possible price for a coningen claim. Furhermore, he resuling benchmarked i.e., numéraireporfoliodenominaed) profiand lossisonly driven by uncerainy ha is orhogonal o benchmarked-raded uncerainy, and forms a local maringale ha sars a zero. Consequenly, sufficienly differen benchmarked profis and losses, when pooled, become asympoically negligible hrough diversificaion. his propery makes benchmarked risk minimizaion he leas expensive mehod for pricing and hedging diversified pools of no fully replicable benchmarked coningen claims. In addiion, when hedging i incorporaes evolving informaion abou nonhedgeable uncerainy, which is ignored under classical risk minimizaion. KEY WORDS: risk minimizaion, incomplee marke, pricing, hedging, numéraire porfolio, benchmark approach. 1. INRODUCION here has been a growing lieraure ha pays aenion o models ha exhibi anomalies ha canno be accommodaed by classical no-arbirage heory. For insance, an equivalen risk-neural probabiliy measure may no exis in some of hese models, see, e.g., Delbaen and Schachermayer 1995), Loewensein and Willard 2), Fernholz 22), Plaen 22, 26), Fernholz e al. 25), Plaen and Heah 26), Jarrow, Proer, and Shimbo 21), Karazas and Kardaras 27), Heson, Loewensein, and he auhors would like o hank he Referees and he Associae Edior for crucial suggesions. Furhermore, hey express heir hanks o Marin Schweizer and Hardy Hulley for valuable discussions on he opic of he paper. Manuscrip received Augus 21; final revision received December 213. Address correspondence o Eckhard Plaen, Universiy of echnology, Sydney, Finance Discipline Group and School of Mahemaical Sciences, P.O. Box 123, Broadway, NSW 27, Ausralia; eckhard.plaen@us.edu.au. DOI: /mafi.1265 C 214 Wiley Periodicals, Inc. 617

2 618 K. DU AND E. PLAEN Willard 27), Chrisensen and Larsen 27), Fernholz and Karazas 29), Galesso and Runggaldier 21), Fernholz and Karazas 21), Biagini 211), and Davis and Lleo 211). Heah and Plaen 22) and Fernholz e al. 25) demonsraed ha pricing and hedging is sill possible ouside he classical no-arbirage framework. he resuls in Fernholz 22), Plaen 22), Fernholz e al. 25), and Plaen and Heah 26) indicae ha for realisic long-erm modeling one has, mos likely, o abandon he classical no-arbirage paradigm. A general framework for pricing and hedging in incomplee markes, which can handle also models ouside he classical no-arbirage framework, is provided by he benchmark approach, described in Plaen 22, 26) and Plaen and Heah 26). A similar framework for pricing and hedging in complee markes beyond he classical heory has been suggesed in Fernholz e al. 25). Under he benchmark approach, asse prices are modeled under he real-world probabiliy measure and he corresponding numéraire is he numéraire porfolio NP). his porfolio, which was originally sudied by Kelly 1956), maximizes expeced log uiliy. When he NP is aken as numéraire, pricing can be convenienly performed under he real-world probabiliy measure, see Long 199) and Plaen 22). Under he benchmark approach he exisence of an equivalen risk-neural probabiliy measure is no required. he benchmark approach generalizes he classical risk-neural approach. he normalized benchmarked savings accoun, wih he NP as benchmark, is hen he Radon Nikodym derivaive of he puaive risk-neural measure. he pricing and hedging of no fully replicable coningen claims has been a challenging ask. Sraegies ha aim o replicae such a coningen claim generae usually a flucuaing profi and loss process. he risk minimizaion approach of Föllmer and Sondermann 1986), furher developed in Föllmer and Schweizer 1989) and Schweizer 1991, 2), minimizes flucuaions of discouned profi and loss processes by using a quadraic crierion under an assumed risk-neural probabiliy measure. In principle, i inroduces an accoun, which moniors in unis of he savings accoun he adaped inflow and ouflow of capial o and from he hedge porfolio. he resuling discouned profi and loss process forms a local maringale under he assumed equivalen risk-neural probabiliy measure, referred o as he minimal equivalen maringale measure, and is under his measure orhogonal o discouned raded wealh. his provides an inuiively appealing mehodology for pricing and hedging coningen claims ha canno be perfecly replicaed. Hereafer, we refer o he above approach as he Föllmer Sondermann Schweizer approach or he classical risk minimizaion approach. Despie he appealing properies of classical risk minimizaion, his approach creaes some asymmeries among primary securiies by using he domesic savings accounas numéraire, and i makes he resricive assumpion on he exisence of he minimal equivalen maringale measure. he exisence of such measure in he case wih jumps is no easily esablished, see Schweizer 2). Moreover, cerain second momens have o exis. We will see ha diversificaion of hedge errors in a large rading book occurs according o he law of large numbers under he real-world probabiliy measure. In Schweizer 2), see also Biagini e al. 211), he concep of local risk minimizaion, inroduced in Schweizer 1991), was generalized wih a view oward he real-world probabiliy measure. However, asymmeries wih respec o primary securiy accouns and second-momen condiions sill remain presen in his generalizaion. his paper proposes he concep of benchmarked risk minimizaion for pricing and hedging coningen claims which canno be perfecly replicaed in an incomplee semimaringale marke. I does no assume he exisence of an equivalen risk-neural

3 BENCHMARKED RISK MINIMIZAION 619 probabiliy measure or minimal equivalen maringale measure. I generalizes he pricing under classical risk minimizaion, and allows one o price and hedge in models beyond he risk-neural paradigm. Symmery wih respec o all primary securiy accouns will be secured, and second-momen assumpions will be avoided. he resuling pricing rule is ha of real-world pricing, wih he NP as numéraire and he real-world probabiliy measure as pricing measure. Under benchmarked risk minimizaion he minimal possible price for a coningen claim is obained. When a minimal equivalen maringale measure exiss, hen real-world pricing coincides wih he pricing under classical risk minimizaion. he remaining benchmarked NP denominaed) profi and loss process forms a local maringale. I sars a zero and is orhogonal o he benchmarked primary securiy accouns, in he sense ha he producs wih hese are local maringales. A benchmarked risk minimizing BRM) hedging sraegy minimizes he flucuaions of he benchmarked nonhedgeable par of a benchmarked coningen claim. Moreover, during hedging i akes evolving informaion abou he nonhedgeable par of he claim ino accoun, whereas classical risk minimizaion ignores such informaion. his is an imporan propery of BRM sraegies. he oal benchmarked profi and loss of a rading book becomes asympoically negligible when holding an increasing number of sufficienly differen benchmarked profi and loss processes. he paper is organized as follows: Secion 2 presens a general semimaringale marke. In Secion 3, he real-world pricing formula is derived. Secion 4 considers benchmarked profi and loss processes. he concep of benchmarked risk minimizaion is inroduced in Secion 5. Secion 6 links maringale represenaions and benchmarked risk minimizaion. Secion 7 derives he respecive hedging sraegy. In Secion 8 a quadraic crierion is illusraed wih is link o benchmarked risk minimizaion. Secion 9 discusses real-world and risk-neural pricing. Finally, Secion 1 emphasizes differences in hedging beween benchmarked risk minimizaion and classical risk minimizaion. 2. FINANCIAL MARKE In his paper, we consider a semimaringale financial marke in coninuous ime. Assume a filered probabiliy space,f, F, P) ha saisfies he usual condiions, as described in Proer 25). Here, he sigma field F models he informaion available a ime [, ). he filraion F = F ) [, ) describes he evoluion of marke informaion over ime. P denoes he real-world probabiliy measure. In his marke we consider d {1, 2,...} adaped, nonnegaive asses, which we call primary securiy accouns, where all ineress and dividends are reinvesed. We assume ha a NP exiss such ha every nonnegaive primary securiy accoun process Ŝ j ={Ŝ j, [, )}, j {1,...,d}, when expressed in unis of he NP, forms a righ-coninuous, inegrable F, P)-local maringale and, hus, an F, P)-supermaringale, see, e.g., Plaen 22) and Plaen and Heah 26). Karazas and Kardaras 27) provide general condiions for he exisence of a NP. Hereafer, we refer o prices, when denominaed in unis of he NP, as benchmarked prices. Denoe by [Ŝ] ={[Ŝ] = [Ŝ i, Ŝ j ] ) i, d j=1, [, )} hemarixvalued opional covariaion process of he vecor process of benchmarked primary securiy accouns Ŝ ={Ŝ = Ŝ 1,...,Ŝd ), [, )}. We denoe by S i,i he ih primary securiy accoun value a ime [, ), when denominaed in unis of he ih securiy iself, i {1,...,d}. In he case of he ih currency denominaion, S i,i denoes he savings accoun in unis of his currency. In he

4 62 K. DU AND E. PLAEN case when he ih shares are used for denominaion, S i,i denoes he respecive share savings accoun in unis of hose shares. hen he NP value S i,δ, when denominaed a ime in unis of he ih securiy, can be expressed by he raio 2.1) S i,δ = Si,i Ŝ i, for [, ), i {1,...,d}. Consequenly, he jh primary securiy accoun S i, j,when denominaed a ime in unis of he ih securiy, can be obained as he produc 2.2) S i, j = Ŝ j S i,δ, for i, j {1,...,d}, [, ). he marke paricipans can combine primary securiy accouns o form porfolios. Denoe by δ ={δ = δ 1,...,δd ), [, )} he sraegy, where δ j, j {1,...,d},represens he number of unis of he jh primary securiy accoun ha are held a ime in a corresponding porfolio. When denominaed in unis of he NP, his porfolio is denoed by he benchmarked porfolio process Ŝ δ ={Ŝ δ, [, )}, where 2.3) Ŝ δ = δ Ŝ, for [, ). If changes in he value of a porfolio are only due o changes in he values of he primary securiy accouns, hen no exra funds flow in or ou of he porfolio, and he corresponding porfolio and sraegy are called self-financing. his propery can be expressed by he equaion 2.4) Ŝ δ = Ŝ δ + δ s d Ŝ s, for all [, ), where he sochasic inegral in 2.4) is a vecor Iô inegral. Since each benchmarked primary securiy accoun process Ŝ j, j {1,...,d}, is a local maringale, he benchmarked self-financing porfolio Ŝ δ is also a local maringale. A benchmarked nonnegaive, self-financing porfolio is, herefore, a supermaringale by Faou s lemma. his confirms he defining propery of he NP of being he sricly posiive porfolio ha when used as benchmark makes all benchmarked nonnegaive porfolios supermaringales, see Long 199), Becherer 21), and Karazas and Kardaras 27). Dynamic rading sraegies ha may no be self-financing are crucial for risk managemen. Obviously, no all sraegies can be allowed. I is sensible o focus in he following on sraegies ha are consisen wih he fac ha he NP is he bes performing porfolio in he sense ha hey yield benchmarked nonnegaive price processes ha are supermaringales. Now, le us inroduce a class of admissible sraegies ha can form nonself-financing porfolios. DEFINIION 2.1. A dynamic rading sraegy v, iniiaed a ime =, is an R d+1 -valued sochasic process v ={v = η,ϑ 1,...,ϑd ), [, )}, where ϑ ={ϑ = ϑ 1,..., ϑ d), [, )} describes he number of unis invesed in he benchmarked primary securiy accouns o form a ime he self-financing par ϑ Ŝ of he associaed porfolio. he righ coninuous benchmarked price process ˆV v ={ˆV v, [, )} of he associaed porfolio is a supermaringale and given by he sum 2.5) ˆV v = ϑ Ŝ + η

5 BENCHMARKED RISK MINIMIZAION 621 a ime [, ). Here, ϑ isassumedobeanr d -valued, predicable process saisfying 2.6) ϑ u d[ŝ] u ϑ u < for all [, ). he adaped, scalar process η ={η, [, )}, saring wih iniial value η =, moniors he benchmarked nonself-financing par of he benchmarked price process ˆV v,soha 2.7) ˆV v = ˆV v + ϑ s dŝ s + η, for [, ), where he sochasic inegral in 2.7) is a vecor Iô inegral. Wih he above noion of a dynamic rading sraegy one can model a wide range of benchmarked price processes. Laer, we will resric he above class of admissible dynamic rading sraegies when inroducing he concep of benchmarked risk minimizaion. We emphasize ha a dynamic rading sraegy generaes via is self-financing par ϑ Ŝ he benchmarked gains from rade 2.8) ϑ s d Ŝ s = ϑ Ŝ ˆV v. I does his in a manner ha does no require ouside funds and also does no generae exra funds. In general, capial has o be added or removed from a porfolio so ha is benchmarked value maches he evoluion of a given benchmarked price process ˆV v.we will see ha for risk managemen purposes i is enough o monior, here in unis of he NP, he cumulaive amoun η, which has o be added or removed from he porfolio o machadesiredprice ˆV v a ime [, ). he predicabiliy of he inegrand in he benchmarked gains from rade 2.8) expresses he real informaional consrain ha he allocaion expressed in ϑ is no allowed o anicipae he movemens of Ŝ. his predicabiliy is also heoreically needed for he inegrand in 2.8) o yield a proper vecor Iô inegral wih respec o he vecor of benchmarked primary securiy accoun processes. he monioring process η in 2.7) needs only o be adaped, which is less resricive han he predicabiliy required for he componens of he process ϑ. Via he process η he invesor moniors in unis of he NP he cumulaive virual capial inflow and ouflow from he porfolio. In previous work by Föllmer and Sondermann 1986) and Schweizer 2), a similar adaped process was employed for describing he holdings in heir numéraire, he domesic savings accoun. his choice of numéraire creaes some asymmery in he requesed measurabiliy properies among all primary securiy accouns. he dynamic rading sraegy, inroduced in Definiion 2.1, employs he NP as numéraire and moniors he inflow and ouflow of exra capial in unis of he NP. his choice of numéraire brings all primary securiy accouns ino comparable posiions, including he domesic savings accoun. Noe, if here is no inflow or ouflow of capial in a dynamic rading sraegy, hen one deals wih a self-financing porfolio, as described in 2.4). More generally, when allowing exra capial inflows and ouflows, one obains direcly from Definiion 2.1 he following resul:

6 622 K. DU AND E. PLAEN COROLLARY 2.2. For a dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )}, as inroduced in Definiion 2.1, he benchmarked porfolio is given by 2.9) wih ˆV v = Ŝ δ = δ Ŝ, 2.1) δ = ϑ + η δ ). Here δ ) = δ 1 )...δd )), denoes he vecor of numbers of unis of he respecive primary securiy accouns held in he NP a ime. We have for he benchmarked NP he rivial equaliy 2.11) Ŝ δ = δ )Ŝ = 1, for [, ). We remark ha δ in 2.1) is, in general, no predicable since η needs only o be adaped. Furhermore, we noe ha a dynamic rading sraegy has sill some ambiguiy in wha consiues for a given price process is self-financing par and wha is monioring par. his ambiguiy will be removed in Secion 5 when inroducing he concep of benchmarked risk minimizaion. 3. REAL-WORLD PRICING he main aim of hedging is risk minimizaion for he delivery of a argeed payoff via some dynamic rading sraegy. Fix a bounded sopping ime >, and le L 1 F ) denoe he se of inegrable F -measurable random variables. DEFINIION 3.1. For a bounded sopping ime, ) a nonnegaive payoff Ĥ L 1 F ), denominaed in unis of he NP, is called a benchmarked coningen claim. Since one can decompose a general payoff ino is nonnegaive and negaive par, here is no real resricion imposed when considering in Definiion 3.1 nonnegaive payoffs. DEFINIION 3.2. We say, a dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )} delivers he benchmarked coningen claim Ĥ if 3.1) ˆV v = Ĥ P-a.s. A benchmarked coningen claim is called replicable if here exiss a self-financing dynamic rading sraegy v wih η = P-a.s for all [, ], which delivers he claim. here may exis several self-financing sraegies ha deliver a given benchmarked coningen claim. Examples can be found in Plaen 22), Fernholz e al. 25), and Plaen and Heah 26). he defining propery of he NP ensures ha all nonnegaive, selffinancing porfolios, when benchmarked, are supermaringales. We show in Appendix A ha in a se of nonnegaive supermaringales, which replicae a given benchmarked coningen claim, he minimal nonnegaive supermaringale is he maringale. his crucial fac yields he following resul:

7 BENCHMARKED RISK MINIMIZAION 623 PROPOSIION 3.3. If for a given benchmarked coningen claim Ĥ a self-financing benchmarked porfolio Ŝ δĥ exiss, saisfying he real-world pricing formula 3.2) Ŝ δĥ = EĤ F ), for all [, ] P-a.s., hen his porfolio provides he leas expensive hedge for Ĥ. Proof. his resul follows direcly from he applicaion of Lemma A.1 in Appendix A. Noe ha equaion 3.2) provides he minimal possible price for a fully replicable claim. In general, coningen claims may be no fully replicable. We will show in Secion 5 ha he above real-world pricing formula 3.2) also makes perfec sense for nonreplicable claims. 4. BENCHMARKED PROFI AND LOSS Risk can be reduced by hedging and diversificaion. Hedging a nonreplicable coningen claim usually resuls in a hedge error. his paper aims o idenify he leas expensive way of delivering coningen claims hrough hedging, while minimizing he flucuaions of he benchmarked hedge error. he following noion will allow us o keep rack of benchmarked hedge errors. DEFINIION 4.1. For a dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )}, wih benchmarked price ˆV v a ime [, ), he benchmarked profi and loss P&L) process Ĉ v ={Ĉ v, [, )} is defined as 4.1) Ĉ v = ˆV v ϑ u d Ŝ u ˆV v, for [, ). One obains direcly from Definiion 4.1 wih Definiion 2.1 he following saemen: COROLLARY 4.2. For a dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )} he corresponding benchmarked P&L process Ĉ v ={Ĉ v, [, )} coincides wih he adaped process η ={η, [, )} ha moniors he cumulaive inflow and ouflow of exra capial. Inuiively, he adaped process η can be inerpreed as benchmarked hedge error. For convenience in his paper, for a given dynamic rading sraegy v he hedging and, hus, he benchmarked P&L process Ĉ v are assumed o sar a he iniial ime =. herefore, he benchmarked P&L has iniial value Ĉ v = η = and moniors a ime wih Ĉ v = η he adaped accumulaed benchmarked capial ha flew in or ou of he respecive porfolio ha maches he benchmarked price process ˆV v unil his ime. In oher words, Ĉ v represens he benchmarked exernal coss incurred by he dynamic rading sraegy v over he ime period [, ] afer he hedge was se up a he iniial ime zero. If one has o deliver a general claim, one faces a flucuaing benchmarked P&L process and, hus, an inrinsic risk ha needs o be conrolled. For implemening sysemaically such a conrol one can inroduce a crierion o obain a desirable behavior of he benchmarked P&L process. he quesion is, wha crierion would be mos appropriae from a risk managemen poin of view? o ge an idea abou wha crierion o choose, we look a he broader picure and prove he following moivaing resul:

8 624 K. DU AND E. PLAEN PROPOSIION 4.3. Consider benchmarked coningen claims Ĥ,l,l {1, 2,...}, wih respecive price processes ˆV v l, l {1, 2,...}, and benchmarked P&L processes Ĉ v l, l {1, 2,...}, where he laer form independen square inegrable maringales wih E Ĉv l ˆV v l ) 2 ) K < for l {1, 2,...} and [, ], [, ). Assume, for simpliciy, ha a he iniial ime he considered well-diversified rading book of a financial insiuion holds equal fracions of he managed iniial benchmarked wealh Û in he firs m of he coningen claims, such ha is oal benchmarked wealh a ime [, ] accouns o Û = Û m ˆV v l m l=1 ˆV v l. he oal benchmarked P&L ˆR m ) of he rading book has hen a ime [, ] he value ˆR m ) = Û m m l=1 Ĉ v l ˆV v, l and i follows for increasing number m of claims in he rading book ha he oal benchmarked P&L vanishes almos surely, ha is, P-a.s. lim ˆR m ) = m he proof of his raher illuminaing fac is given in Appendix B. I shows ha he benchmarked P&L of a rading book wih increasing number of claims can be asympoically removed, which can be inerpreed as he process of diversificaion. he insigh ha such removal is, in principle, possible is crucial. We emphasize, for he above resul o hold, i is imporan ha he benchmarked P&Ls are locally in ime close o maringales. We reflec his by requesing below ha benchmarked P&Ls should be local maringales under he dynamic rading sraegies ha we will admi. his means a benchmarked P&L is locally in he mean self-financing. Mean self-financing urns ou o be an exremely useful noion, which was inroduced in Schweizer 1991) when using he savings accoun as numéraire and employing an assumed risk-neural probabiliy measure as pricing measure. Under he benchmark approach we use he NP as numéraire and he real-world probabiliy measure for aking expecaions. Hence, he following noion will be employed when inroducing in he nex secion he concep of benchmarked risk minimizaion: DEFINIION 4.4. A dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )} is called locally real-world mean self-financing if is monioring process η is a local maringale. his noion mainains symmery wih respec o all primary securiy accouns, including he domesic savings accoun. I uses he real-world probabiliy measure P and avoids he resricive assumpion on he exisence of an equivalen risk-neural probabiliy measure. 5. BENCHMARKED RISK MINIMIZAION I is no immediaely obvious how o price and hedge a general coningen claim in an incomplee marke, even when aking ino accoun he observaions made above. For complee markes he pricing and hedging of coningen claims can be performed in a sraighforward manner also for models where no equivalen risk-neural probabiliy

9 BENCHMARKED RISK MINIMIZAION 625 measure exiss. his was observed in Fernholz e al. 25), and under he benchmark approach demonsraed in Heah and Plaen 22), Plaen 22), and Plaen and Heah 26). Concepually, here exis many ways o hedge a nonreplicable claim, and a wide range of lieraure has emerged. he pricing in incomplee markes and some pricing of nonreplicable coningen claims have been discussed, for insance, in secions 11.4 and 11.5 in Plaen and Heah 26). Inuiively appealing and pracically useful is he already menioned concep of classical risk minimizaion for which an excellen survey is given in Schweizer 2). Under classical risk minimizaion along he lines of Föllmer Sondermann Schweizer, he hedging is implemened via a savings accoun discouned porfolio under an assumed equivalen risk-neural probabiliy measure. he flucuaions of he discouned P&L processes are measured and minimized via a quadraic crierion, where a good sraegy urns ou o be mean self-financing under he assumed risk-neural probabiliy measure, see Schweizer 2). Mos imporanly, he Föllmer Sondermann Schweizer approach links he opimizaion problem of risk minimizaion o he well-known Kunia Waanabe decomposiion, see Schweizer 2). his crucial decomposiion became known as Föllmer Schweizer decomposiion in he conex of pricing and hedging in incomplee markes. he Föllmer Schweizer decomposiion has been exensively sudied by several auhors, where we refer o Schweizer 2) for a lis of references. his paper is of concepual naure, and proposes a pricing and hedging approach for nonreplicable claims in incomplee markes in he spiri of classical risk minimizaion, bu under he real-world probabiliy measure wih he NP as numéraire. I generalizes he pricing suggesed by classical risk minimizaion. However, he hedging will be, in general, differen. Recall from Definiion 2.1 ha dynamic rading sraegies form benchmarked nonnegaive price processes ha are consisen wih he fac ha he NP is he bes performing porfolio, in he sense ha benchmarked price processes form supermaringales. Noe also ha, a his sage, for a given benchmarked price process a corresponding locally real-world mean self-financing dynamic rading sraegy remains poenially exposed o some ambiguiy concerning wha forms is self-financing par and wha consiues is monioring par, see equaion 2.1). his ambiguiy will be removed by focusing below on benchmarked P&Ls wih flucuaions ha are orhogonal o hose of he benchmarked primary securiy accouns under he real-world probabiliy measure. his means, inuiively, hese flucuaions, when denominaed in unis of he benchmark, have no chance o be removed via hedging. o formalize his idea we inroduce he following noion: DEFINIION 5.1. A dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, )} has an orhogonal benchmarked P&L η ={η, [, )} if η is orhogonal o he benchmarked primary securiies in he sense ha η Ŝ forms a vecor local maringale. In some sense, all hedgeable benchmarked uncerainy is removed from an orhogonal benchmarked P&L. o summarize he so far idenified desirable properies of dynamic rading sraegies, le us define he following se: DEFINIION 5.2. For a benchmarked coningen claim Ĥ,leVĤ denoe he se of locally real-world mean self-financing dynamic rading sraegies, which deliver Ĥ wih orhogonal benchmarked P&Ls.

10 626 K. DU AND E. PLAEN here may exis several dynamic rading sraegies in VĤ ha could deliver he benchmarked coningen claim Ĥ. o finalize our search for a suiable crierion, we assume ha a marke paricipanalwaysprefers more for less. he following definiion selecs hen he mos economical price process, which is he leas expensive possible price process. DEFINIION 5.3. A dynamic rading sraegy ṽ ={ṽ = η 1, ϑ 1,..., ϑ d ), [, ]} VĤ, wih corresponding benchmarked price process ˆVṽ, is called BRM if for all dynamic rading sraegies v VĤ, wih corresponding price process ˆV v, he price process ˆVṽ is minimal in he sense ha 5.1) ˆVṽ ˆV v P-a.s. for all [, ]. As required by he inequaliy 5.1), and similarly as in Secion 3 we can exploi he fac ha he maringale among he nonnegaive supermaringales conained in VĤ yields he minimal possible benchmarked price process; see Lemma 11.1 in Appendix A. herefore, we obain direcly he following resul: COROLLARY 5.4. For given Ĥ L 1 F ) a BRM dynamic rading sraegy v ={v = η,ϑ 1,...,ϑd ), [, ]} forms wih he corresponding benchmarked price process ˆV v a maringale, ha is, i saisfies he real-world pricing formula 5.2) ˆV v = EĤ F ) P-a.s. for [, ]. his is an inuiively appealing and pracically useful conclusion. Obviously, formula 5.2) exends he real-world pricing formula 3.2) o he case of no fully replicable benchmarked coningen claims. Noe ha according o he real-world pricing formula 5.2), he benchmarked price process is unique and does no depend on he ime when he hedge is iniiaed. However, he benchmarked P&L process depends on he iniiaion ime of he hedge, see 4.1). Is specificaion follows from he reques ha i should be a local maringale ha is orhogonal o all benchmarked primary securiy accouns, see Definiions 5.1 and 5.2. Benchmarked risk minimizaion does no require he exisence of an equivalen riskneural probabiliy measure. I aims for he minimal possible price process. Furhermore, in a rading book wih an increasing number of sufficienly differen coningen claims i can poenially remove nonhedgeable risk via diversificaion, as indicaed in Proposiion 4.3. Moreover, i provides symmery wih respec o all primary securiy accouns. Finally, resricive square inegrabiliy assumpions are avoided. Since he proposed concep of benchmarked risk minimizaion requires only very weak assumpions, i permis he handling of more general financial marke models and more general coningen claims han covered under classical risk minimizaion. Is main requiremen is he exisence of he NP, which is a very weak assumpion, as shown in Karazas and Kardaras 27). 6. REGULAR BENCHMARKED CONINGEN CLAIMS o uilize efficienly he above inroduced concep of BRM sraegies, i will be exremely useful o have access o corresponding maringale represenaions for benchmarked

11 BENCHMARKED RISK MINIMIZAION 627 coningen claims, similar o he Föllmer Schweizer decomposiion in classical risk minimizaion, see Schweizer 2). We emphasize, in his paper we will use maringale represenaions for benchmarked coninen claims under he real-world probabiliy measure. Unforunaely, maringale represenaions canno be easily mahemaically guaraneed for general semimaringale markes. Sysemaic resuls in his direcion can be found, for insance, in Karazas and Shreve 1991) and Jacod, Meleard, and Proer 2). Forunaely, maringale represenaions exis for mos inegrable benchmarked coningen claims in Markovian marke models and for mos coninuous marke models, as will be demonsraed in he nex secion. A represenaion of a benchmarked coningen claim, which separaes is self-financing hedgeable par from is orhogonal monioring par, is crucial for hedging. We inroduce he following noion: DEFINIION 6.1. We call a benchmarked coningen claim Ĥ L 1 F )regularifi has for all [, ] a represenaion of he following form: 6.1) Ĥ = EĤ F ) + ϑ Ĥ s)d Ŝ s + ηĥ ) ηĥ ) P-a.s, involving some predicable vecor process ϑ Ĥ ={ϑ Ĥ ) = ϑ 1 ),...,ϑ ḓ )), Ĥ H [, ]} saisfying 2.6), and some local maringale ηĥ ={ηĥ ), [, ]} wih ηĥ ) =. Furhermore, he produc process ZĤ ={ZĤ ) = ηĥ )Ŝ, [, ]} forms a vecor local maringale. By combining Definiion 5.3, Corollary 5.4, and Definiion 6.1, benchmarked risk minimizaion allows us o obain in a sraighforward manner he following resul: COROLLARY 6.2. For a regular benchmarked coningen claim Ĥ L 1 F ) wih represenaion 6.1) here exiss a BRM sraegy v ={v = ηĥ ),ϑ 1 ),...,ϑ ḓ )), Ĥ H [, ]} VĤ wih corresponding benchmarked price process ˆV v Ĥ, saisfying 2.9), which delivers he benchmarked coningen claim, ha is, ˆV v Ĥ = Ĥ P-a.s. he benchmarked price a ime [, ] is deermined by he real-world pricing formula 6.2) ˆV v Ĥ = EĤ F ), yielding wihin he se VĤ of admissible sraegies he minimal possible price process. he resuling benchmarked P&L a ime [, ] is given by 6.3) Ĉ v Ĥ = ηĥ ). Wha remains is o idenify for a given marke model and given regular benchmarked coningen claim he respecive represenaion of he form 6.1). o esablish such represenaion, as a firs sep one can calculae he condiional expecaion 6.2), eiher by explici calculaions or via some numerical mehods. In a second sep, one can idenify he holdings ϑ Ĥ in he self-financing par of he calculaed benchmarked price process ˆV v Ĥ. he vecor ϑ Ĥ ), characerizing he unis o be held in he primary securiy accouns, follows by making he local maringale ηĥ ) = ˆV v Ĥ ϑ )Ŝ Ĥ orhogonal o he benchmarked primary securiy accouns. his means, he produc ηĥ )Ŝ needs o form a drifless vecor process. Noe ha due o he possible presence of redundan primary securiy accouns ϑ Ĥ may no be unique. he final hird sep calculaes hen he unis of he benchmark o be accumulaed in he benchmarked P&L.

12 628 K. DU AND E. PLAEN 7. HEDGING REGULAR CLAIMS We emphasize BRM sraegies do no reques square inegrabiliy of benchmarked quaniies. he benchmarked self-financing par and also he benchmarked P&L do only need o form local maringales. herefore, due o he avoidance of he reques on he exisence of an equivalen risk-neural probabiliy measure, he proposed concep has wide applicabiliy. As we will show in Secion 9, i generalizes imporan pricing rules and allows us o go far beyond he classical no-arbirage modeling world. In paricular, marke models wih jumps can be covered ha may have infinie jump aciviy and random jump sizes. Ineresing properies of BRM sraegies emerge when sudying paricular ypes of models. I is impossible o presen and discuss in his paper ineresing resuls ha emerge for models wih jumps. A forhcoming paper will focus on such resuls and also on models where an equivalen risk neural probabiliy measure does no exis. In he remainder of his paper we focus on BRM sraegies for coninuous models. Wihou loss of generaliy, consider a benchmarked coningen claim Ĥ wih fixed mauriy, where all is uncerainy is modeled by he coninuous local maringales W 1, W 2,...,W d. hese local maringales are assumed o be orhogonal o each oher in he sense ha heir pairwise producs form local maringales. Furhermore, each benchmarked primary securiy accoun value Ŝ j, j {1,...,d}, saisfies a sochasic differenial equaion of he form 7.1) d Ŝ j = Ŝ j d 1 k=1 θ j,k dw k, for wih Ŝ j >. Here, θ j,k ={θ j,k, [, ]} forms for each j {1,...,d} and k {1,...,d 1} a predicable process such ha he Iô inegrals corresponding o 7.1) exis. Noe ha he local maringale W d does no appear as uncerainy of he benchmarked primary securiy accouns. However, we allow i o model uncerainy of he benchmarked coningen claim Ĥ. his means ha he claim Ĥ will no be fully hedgeable. o idenify below he corresponding BRM sraegy, denoe by = [ i,k ] i,k=1 d he d d marix wih elemens { i,k θ i,k for k {1,...,d 1} 7.2) =, 1 for k=d for i {1,...,d} and [, ]. PROPOSIION 7.1. In he seing of his secion assume o be inverible for Lebesguealmos every [, ]. Furhermore, a ime [, ] he condiional expecaion ˆV of he benchmarked coningen claim is assumed o have a represenaion of he form 7.3) d 1 ˆV = ˆV + xs k dwk s + xs d dwd s, k=1 where x 1,...,x d are predicable processes. hen Ĥ is a regular benchmarked coningen claim wih EĤ F ) = ˆV for all [, ], and he corresponding BRM sraegy is given by 7.4) ϑ Ĥ ) = diagŝ ) 1 ) 1 ξ,

13 BENCHMARKED RISK MINIMIZAION 629 wih 7.5) ξ = x 1,..., xd 1, ˆV ηĥ )) and 7.6) ηĥ ) = x d s dwd s. he proof of his resul is given in Appendix C. For insance, in a mulifacor Markovian diffusion model, which models he benchmarked coningen claim Ĥ and he benchmarked primary securiy accouns, one obains in a sraighforward manner a represenaion of he form 7.3) via he Feynman Kac formula and by using he Kolmogorov backward equaion for ˆV as a funcion of he Markovian sae variables. he applicaion of he Iô formula o he pricing funcion provides direcly he represenaion 7.3). he quesion arises, how does he above pricing and hedging relae o he well-known hedging under he risk-neural approach? Noe ha when W 1...,W d are independen sandard Brownian moions, hen θ i,k becomes he marke price of risk a ime wih respeco he kh Brownian moion for he denominaion of he securiies in unis of he ih primary securiy accoun. By he Iô formula i follows from 7.1) ha he jh primary securiy accoun, when denominaed in unis of he ih primary securiy accoun, denoed by S i, j he sochasic differenial equaion d S i, j = S i, j d 1 k=1 θ i,k ) θ j,k θ i,k ) d + dw k, = Ŝj Ŝ i, saisfies i, j,k for [, ]andi, j {1,...,d}. his shows ha he volailiy b wih respec o he kh Brownian moion for he j h primary securiy accoun, when denominaed in unis of he ih primary securiy accoun, has he form 7.7) i, j,k b = θ i,k ) θ j,k. his also means, when we selec, wihou loss of generaliy, he dh primary securiy accoun as domesic savings accoun, hen he volailiy marix b d for he dh securiy denominaion has he form 7.8) b d = [ b d,j,k ] d 1,d 1, j,k=1 for [, ]. I is well known how one can hedge claims in he dh securiy denominaion. In his seing, he key assumpion is ha he volailiy marix b d is an inverible marix, see Karazas and Shreve 1998). he following resul shows ha he marix b d is indeed inverible under our assumpions. PROPOSIION 7.2. For [, ) he marix is inverible if and only if b d is inverible. We provide he proof for his resul in Appendix D.

14 63 K. DU AND E. PLAEN 8. A QUADRAIC CRIERION he concep of benchmarked risk minimizaion avoids resricive assumpions, which makes i widely applicable. However, is assumpions may appear raher absrac o some readers. herefore, we show now ha i can be inerpreed, under appropriae assumpions, as he minimizaion of a disance, which is he expeced square of he benchmarked P&L. he orhogonaliy of he benchmarked P&L corresponds hen o he minimizaion of is disance o benchmarked raded wealh. o illusrae he link of BRM sraegies o he indicaed quadraic crierion, le us consider a regular benchmarked coningen claim Ĥ, wih, ) fixed, and represenaion 6.1). We assume in 6.1) ha he erms ϑ s)d Ŝ Ĥ s and ηĥ ) form independen, square inegrable maringales. In addiion, assume ha also Ŝ 1,...,Ŝd and ηĥ are muually independen, square inegrable maringales. he laer propery guaranees ha ηĥ is orhogonal o benchmarked-raded primary securiy accouns, in he sense of Definiion 5.1. Assume now ha Ĥ is square inegrable so ha a square inegrable maringale is formed by he condiional expecaion EĤ F ). he second momen of he benchmarked P&L represens he above-menioned disance. Obviously, i can be inerpreed as a measure for he risk of he hedge. his disance would be zero if he claim could be perfecly replicaed. Now, le us minimize he above-menioned disance, ha is, we minimize E Ĉ δ )2) = E Ĥ δ s d Ŝ s Ŝ δ )2 ), by employing self-financing sraegies δ ={δ = δ 1,...,δd ), [, ]}, where δ s d Ŝ s is a square inegrable maringale, independen of ηĥ. By exploiing he maringale represenaion 6.1), he orhogonaliy of ηĥ o benchmarked-raded wealh and he assumed independence and square inegrabiliy properies, i follows ha E Ĉ δ )2) = E EĤ F ) Ŝ δ + = E EĤ F ) Ŝ δ ) 2 + E ) ) 2 ) ϑ s) δ Ĥ s d Ŝ s + ηĥ ) ) ) 2 ϑ s) δ Ĥ s d[ŝ]s + EηĤ )) 2 ). When minimizing he righ-hand side of he above equaion i becomes obvious ha he minimum can only be obained when seing he benchmarked iniial price o Ŝ δ = EĤ F ), which represens he price ˆV v Ĥ obained by he real-world pricing formula 6.2). Furhermore, aking he minimum requires choosing he second summand such ha δ = ϑ Ĥ ) forall [, ]. We canno reduce he hird summand in he above equaion. herefore, he minimal disance equals he minimal second momen for he benchmarked P&L, which becomes EηĤ )) 2 ). o exend his discussion, one could pool an increasing number of independen benchmarked P&Ls of he above ype in a rading book. his seup would saisfy he assumpions of Proposiion 4.3, and he resuling oal benchmarked P&L would vanish almos surely. In his manner, a well-diversified insiuion can, in principle, remove asympoically he nonhedgeable uncerainy from is rading book. he concep of benchmarked risk minimizaion idenifies he hedging sraegies yielding minimal flucuaions

15 BENCHMARKED RISK MINIMIZAION 631 of benchmarked P&Ls and, hus, allows one o perform sysemaically diversificaion in an opimal manner. 9. REAL-WORLD AND RISK-NEURAL PRICING Le us inerpre Ŝ 1 as he benchmarked savings accoun process of he domesic currency. Obviously, i is a local maringale bu may no be a rue maringale. We can sae he following resul: PROPOSIION 9.1. For a benchmarked coningen claim Ĥ = H S 1,δ,wihS 1,δ denoing he value of he NP denominaed in domesic currency a mauriy, he real-world price coincides wih he risk neural price if he benchmarked savings accoun Ŝ 1 is a rue maringale, and he Radon Nikodym derivaive = dq dp F for he puaive risk-neural measure Q equals he normalized benchmarked savings accoun = Ŝ1 Ŝ 1,for [, ]. Proof. he real-world pricing formula 6.2) can be rewrien for he discouned price process S δĥ S δĥ = ŜδĤ Ŝ1 = S1,δ S 1,1 E Here, S 1,1 H S 1,δ by using Bayes s rule in he form F ) = E Ŝ1 H Ŝ 1 S 1,1 F ) = E H S 1,1 ) F = E Q H S 1,1 ) F. = Ŝ 1S1,δ, [, ] denoes he savings accoun denominaed in unis of he domesic currency. he las equaliy on he righ-hand side of he above equaion follows by he Bayes rule and provides he well-known risk-neural pricing formula, where E Q denoes expecaion under Q. he Radon Nikodym derivaive for he minimal equivalen maringale measure Q, as defined in Schweizer 1995), is characerized by he normalized benchmarked savings accoun = dq dp F = Ŝ1. herefore, under he exisence of he minimal equivalen Ŝ 1 maringale measure, classical risk minimizaion in he sense of Föllmer Sondermann Schweizer yields he same price as benchmarked risk minimizaion, which is a saisfying resul. his does no mean ha one obains also he same hedging sraegy, as we will see in he nex secion. We remark ha, i has been shown in Plaen and Heah 26, secion 9.2), ha for H independen of S 1,δ, he real-world pricing formula yields he acuarial pricing formula, which has been widely used by acuaries wihou formal proof. his paper provides a foundaion for acuarial pricing via benchmarked risk minimizaion in incomplee markes and for nonhedgeable claims in a wide range of marke models. Finally, we remark ha i has been shown in secion 11.4 of Plaen and Heah 26) ha also some form of uiliy indifference pricing is equivalen o real-world pricing. 1. DIFFERENCES BEWEEN CLASSICAL AND BRM HEDGING We have seen in he previous secion ha when a minimal equivalen maringale measure exiss, hen real-world pricing yields he same prices as classical risk minimizaion. However, his does no mean ha afer a hedge has been iniiaed using ha price ha boh approaches yield he same hedging sraegy. We demonsrae below

16 632 K. DU AND E. PLAEN ha he hedging sraegy for no fully hedgeable claims is differen under he wo approaches. he reason for his difference is he fac ha he BRM sraegy generaes he benchmarked P&L in such a way ha i becomes orhogonal o he benchmarked primary securiy accouns under he real-world probabiliy measure. his is, in general, differen o requesing ha he discouned profi and loss is orhogonal o he discouned primary securiy accouns under he minimal equivalen maringale measure. PROPOSIION 1.1. BRM hedging of no fully replicable coningen claims is, in general, differen o hedging under classical risk minimizaion using he minimal equivalen maringale measure. Proof. As proof for he above saemen we provide an illusraive example. Consider in he seing of Secion 7 wih d = 2 a random payou H, denominaed in unis of he domesic savings accoun S 1,1, such ha Ĥ = H S 1,1 = H S 1,δ Ŝ 1 is a regular benchmarked coningen claim. More precisely, we assume ha his discouned payou H has he represenaion { H = exp } 2 + W2, where W 2 denoes he nonhedgeable Brownian moion. On he oher hand, we have in our example as primary securiy accouns he discouned savings accoun S 1,1 = 1 and he risky securiy S 1,2. he laer is in our example also he discouned NP, where we assume { } S 1,2 = exp 2 W1, for [, ], where W 1 denoes he hedgeable Brownian moion. he benchmarked savings accoun equals hen Ŝ 1 = 1, and he benchmarked NP is rivially Ŝ 2 S 1,2 = 1for [, ]. Obviously, from he perspecive of classical risk minimizaion, he claim H is no hedgeable in his marke. According o Schweizer 2), he minimal equivalen maringale measure Q has he Radon Nikodym derivaive = Ŝ 1. herefore, he discouned iniial price for he claim amouns o V = E Q H ) = E H ) = E )EH ) = EH ) = 1. he hedging sraegy under classical risk minimizaion would purchase a he iniial ime one uni of he savings accoun and would keep i unil mauriy. he BRM sraegy would, by Proposiion 7.1, obain he same iniial price V = ˆV Ĥ S 1,δ = S 1,2 ˆV Ĥ = S 1,2 E H S 1,δ ) = S 1,2 EH )EŜ 1 ) = EH ) = 1. Noe ha he condiional expecaion of he benchmarked coningen claim equals wih exponenial maringales 1.1) ˆV = Ĥ = EĤ F ) = EH F )EŜ 1 F ) = H Ŝ 1, H = EH F ) = exp { } 2 + W2

17 BENCHMARKED RISK MINIMIZAION 633 and 1.2) { Ŝ 1 = exp } 2 + W1, for [, ]. By he Iô formula, we obain wih 1.1) and 1.2) he maringale represenaion 1.3) ˆV = H Ŝ 1 + H s d Ŝs 1 + Ŝs 1 dh s = H Ŝ 1 + H s Ŝs 1 dw1 s + Ŝs 1 H sdws 2, for [, ]. Consequenly, we have in Proposiion 7.1 x 1 = H Ŝ 1 2 2marix has by 7.2) and 1.2) he form ) ) =, 1 for [, ]. he which equals is inverse 1 = for [, ]. Furhermore, by 7.6), 1.3), and 7.5) we have ηĥ ) = Ŝ1 s H sdws 2 and ξ ) = H Ŝ 1, H Ŝ1 η Ĥ )). By 7.4), 1.4), and Ŝ 2 = 1 we ge 1.5) ϑ 1 Ĥ ) = H and ϑ 2 Ĥ ) = ηĥ ). We observe ha his hedging sraegy is differen o he classical risk minimizing one. According o 1.5) he number of unis held in he savings accoun equals H = EH F ), which is he bes forecas for he payoff H. Under classical risk minimizaion one holds always H unis in he savings accoun and nohing in any oher securiy. Under he minimal equivalen maringale measure we sill have a maringale for ηĥ )Ŝ 1) 1, which is he discouned P&L of he BRM sraegy. However, when muliplied wih he discouned NP S 1,δ = Ŝ 1) 1, which is a raded securiy, he produc does no form a local maringale under he minimal equivalen maringale measure. his means, we do no have he kind of orhogonaliy ha classical risk minimizaion requess. Alernaively, one can say, he discouned profi and loss H H a ime [, ] ofheföllmer Schweizer decomposiion, when muliplied by he benchmarked savings accoun Ŝ 1, does no yield he benchmarked P&L of he represenaion 6.1) of he regular benchmarked coningen claim. he claim we considered has a represenaion of a regular benchmarked coningen claim, as shown in 1.3). his proves Proposiion 1.1. his example demonsraes ha BRM sraegies ake evolving informaion abou he nonhedgeable uncerainy ino accoun by using is bes forecas, whereas classical risk minimizaion ignores such informaion. his is a key feaure of he proposed concep of benchmarked risk minimizaion. o saisfy invesors, who would like o minimize he second momen of heir discouned P&L under he minimal equivalen maringale measure, one can direcly generalize

18 634 K. DU AND E. PLAEN classical risk minimizaion under he benchmark approach. Since his generalizaion is beyond he scope of his paper i will be described in forhcoming work. 11. CONCLUSION his paper proposes he concep of benchmarked risk minimizaion for pricing and hedging of no fully replicable coningen claims in incomplee markes. Benchmarked risk minimizaion goes beyond classical risk minimizaion, originally developed by Föllmer, Sondermann, and Schweizer. Under he proposed concep a wider range of coningen claims can be priced and hedged in a richer modeling world. I does no require an equivalen risk-neural probabiliy measure or square inegrabiliy properies. he main assumpion is exremely weak. I only requires ha he NP exiss. he NP is employed as numéraire and benchmark. he resuling price represens he minimal possible price. he benchmarked profi and loss is a local maringale and orhogonal o benchmarkedraded wealh in he sense ha he produc of benchmarked profi and loss wih each benchmarked primary securiy accoun forms a local maringale. When using benchmarked risk minimizaion, he oal benchmarked profi and loss of a large rading book wih increasing number of sufficienly differen coningen claims can, in principle, be removed asympoically. In his sense, benchmarked risk minimizaion yields he minimal possible price and allows one o remove he nonhedgeable risk via diversificaion. In he case when classical risk minimizaion can be applied, benchmarked risk minimizaion yields he same price process, however, i employs a hedging sraegy which akes evolving informaion abou he nonhedgeable uncerainy of he claim ino accoun, whereas classical risk minimizaion ignores such informaion. APPENDIX A LEMMA A.1. Consider for a bounded sopping ime [, ) a benchmarked coningen claim Ĥ L 1 F ) and a supermaringale Y ={Y, [, ]} wih Y = Ĥ P-a.s., as well as a maringale X ={X, [, ]} wih X = Y = Ĥ P-a.s. hen i follows for all [, ] he inequaliy P-a.s. X Y Proof. Following secion 3 in Chaper II of Revuz and Yor 1999), we can prove he above resul as follows: By he supermaringale propery of Y we have A.1) Y EĤ F ) P-a.s. for all [, ]. On he oher hand, by he maringale propery of X i follows A.2) X = EĤ F ), for all [, ]. Consequenly, one has by A.1) and A.2) for all [, ] he inequaliy A.3) Y X

19 BENCHMARKED RISK MINIMIZAION 635 P-a.s., which proves he saemen of he above lemma. APPENDIX B Proof of Proposiion 4.3. We apply Kolmogorov s srong law of large numbers, see Chaper IV, secion 3 in Shiryaey 1984). By he maringale propery of Ĉ v l,wehave ) = foralll {1, 2,...} and [, ]. E Ĉv l ˆV v l By he square inegrabiliy of Ĉv l ˆV v l 1 l E 2 l=1 i follows Ĉv l ˆV v l ) 2 K l= 1 l 2 <, [, ]. hus, we obain for [, ] by he srong law of large numbers ha ˆR m ) = 1 m Û m l=1 Ĉ v l ˆV v l converges P-almos surely o zero, which proves Proposiion 4.3. APPENDIX C Proof of Proposiion 7.1. Denoe by ϑ he self-financing par of a BRM sraegy v ˆVĤ. Accordingly, he self-financing par of he regular benchmarked claim Ĥ can be wrien as C.1) ϑ Ŝ = ϑ diagŝ )1 = ˆV x d s dwd s. Ŝ forms he self-financing par, one has by 7.1) he sochas- Furhermore, because ϑ ic differenial ϑ d Ŝ = ϑ diagŝ )θ dw ) = ϑ diagŝ )θ dw, where θ = [θ i,k ] d,d 1 1,1 is a d d 1marix,W = W 1,...,Wd 1 ).Bymachinghe self-financing par wih he maringale represenaion 7.3), one has C.2) ϑ diagŝ )θ = x, where x = x 1,...,xd 1 ). By C.1) and C.2), one has wih ξ = x, ˆV xd s dwd s ) ha ϑ diagŝ ) = ξ. Since is inverible, one obains he relaionship 7.4), which proves Proposiion 7.1

20 636 K. DU AND E. PLAEN APPENDIX D Proof of Proposiion 7.2. Observe ha he marix has he form θ 1,1... θ 1,d 1 1 = θ d,1... θ d,d 1 1 We perform in he following operaions ha leave an inverible marix inverible: We are using he firs unil he d 1)h row from, subrac from each he dh row, and hen ake he negaive elemens in he resuling firs unil he d 1)h row. One obains afer hese operaions by 7.7) and 7.8) he marix = θ 1,1... θ d,d 1 θ 1,d θ d 1,1... θ d,d 1 θ d 1,d 1 = θ d,1... θ d,d 1 1 θ d,1 θ d,1 θ d,1 b d.... θ d,d 1 1 he marix on he righ-hand side of he equaion has in is upper lef par he volailiy marix for he dh denominaion of he securiies. Since we have in he marix a he righhand side of he equaion a 1 in he lower righ-hand corner, i is clear ha he marix b d has full rank if and only if he marix has full rank. hus, i follows ha has full rank if and only if b d has full rank. Accordingly, he inveribiliy of is given if and only if b d is inverible. REFERENCES BECHERER, D. 21): he Numéraire Porfolio for Unbounded Semimaringales, Finance. Soch. 5, BIAGINI, F. 211): Evaluaing Hybrid Producs: he Inerplay beween Financial and Insurance Markes. Preprin, LMU Munich. BIAGINI, F., A. CREAROLA, ande.plaen 211): Local Risk-Minimizaion under he Benchmark Approach. Preprin, LMU Munich. CHRISENSEN,M.M.,andK.LARSEN 27): NoArbirageandheGrowhOpimalPorfolio, Soch. Anal. Appl. 251), DAVIS, M. H. A., ands. LLEO 211): Fracional Kelly Sraegies for Benchmarked Asse Managemen, in he Kelly Capial Growh Invesmen Crierion,L.C.MacLean,E.O.horp, and W.. Ziemba, eds., Singapore: World Scienific, pp DELBAEN, F.,andW.SCHACHERMAYER 1995): Arbirage Possibiliies in Bessel Processes and heir Relaions o Local Maringales, Probab. heory Relaed Fields 12, FERNHOLZ, D.,andI.KARAZAS 21): Probabilisic Aspecs of Arbirage, in Conemporary Quaniaive Finance, C. Chiarella and A. Novikov, eds., Heidelberg: Springer Verlag, pp FERNHOLZ, R. 22): Sochasic Porfolio heory, New York: Springer-Verlag. FERNHOLZ, R.,andI.KARAZAS 29): Sochasic Porfolio heory: An Overview, in Mahemaical Modeling and Numerical Mehods in Finance, Handbook of Numerical Analysis, Vol. XV, A. Bensoussan and Q. Zhang, eds., Oxford: Norh-Holland, pp