Arbitrage-Free Pricing Of Derivatives In Nonlinear Market Models

Transcription

1 Arbirage-Free Pricing Of Derivaives In Nonlinear Marke Models Tomasz R. Bielecki a, Igor Cialenco a, and Marek Rukowski b Firs Circulaed: January 28, 217 This Version: April 4, 218 Forhcoming in Probabiliy, Uncerainy and Quaniaive Risk Absrac: The objecive of his paper is o provide a comprehensive sudy no-arbirage pricing of financial derivaives in he presence of funding coss, he counerpary credi risk and marke fricions affecing he rading mechanism, such as collaeralizaion and capial requiremens. To achieve our goals, we exend in several respecs he nonlinear pricing approach developed in El Karoui and Quenez [27] and El Karoui e al. [26], which was subsequenly coninued in Bielecki and Rukowski [8]. Keywords: hedging, fair price, funding cos, margin agreemen, marke fricion, BSDE MSC21: 91G4, 6J28 Conens 1 Inroducion 3 2 Nonlinear Marke Model Conracs wih Trading Adjusmens Self-financing Trading Sraegies Funding Adjusmen Financial Inerpreaion of Trading Adjusmens Wealh Process Trading in Risky Asses Cash Marke Trading Shor Selling of Risky Asses Repo Marke Trading Collaeralizaion Rehypohecaed Collaeral Segregaed Collaeral Iniial and Variaion Margins Counerpary Credi Risk Closeou Payoff Counerpary Credi Risk Decomposiion Local and Global Valuaion Problems a Deparmen of Applied Mahemaics, Illinois Insiue of Technology 1 W 32nd Sr, Building E1, Room 28, Chicago, IL 6616, USA s: bielecki@ii.edu T.R. Bielecki and cialenco@ii.edu I. Cialenco URLs: hp://mah.ii.edu/~bielecki and hp://mah.ii.edu/~igor b School of Mahemaics and Saisics, Universiy of Sydney, Sydney, NSW 26, Ausralia and Faculy of Mahemaics and Informaion Science, Warsaw Universiy of Technology, -661 Warszawa, Poland marek.rukowski@sydney.edu.au, URL: hp://sydney.edu.au/science/people/marek.rukowski.php 1

2 2 T.R. Bielecki, I. Cialenco and M. Rukowski 3 No-Arbirage Properies of Nonlinear Markes No-arbirage Pricing Principles Discouned Wealh and Admissible Sraegies No-arbirage wih Respec o he Null Conrac No-arbirage for he Trading Desk Dynamics of he Discouned Wealh Process Sufficien Condiions for he Trading Desk No-Arbirage Hedger s Fair Pricing and Marke Regulariy Replicaion on [, T ] and he Gained Value Marke Regulariy on [, T ] Replicable Conracs in Regular Markes Nonreplicable Conracs Nonregular Marke Model Replicaion and Marke Regulariy on [, T ] Pricing by Replicaion in Regular Markes Hedger s Ex-dividend Price a Time Exi Price Offseing Price A BSDE Approach o Nonlinear Pricing BSDE for he Gained Value BSDE for he Ex-dividend Price BSDE for he CCR Price Nonlinear Valuaion Versus Marke Pracice 44

3 Derivaives Pricing in Nonlinear Models 3 1 Inroducion The paper conribues o he nonlinear arbirage-free pricing heory, which arises in a naural way due o he salien feaures of real-world rades, such as: rading consrains, differenial funding coss, collaeralizaion, counerpary credi risk, and capial requiremens. Our aim is o exend in several respecs he nonlinear hedging and pricing approach developed in El Karoui and Quenez [27] and El Karoui e al. [26] who used a BSDE approach, by accouning for he complexiy of overhe-couner financial derivaives and specific feaures of he rading environmen afer he global financial crisis. This work builds also upon he earlier paper by Bielecki and Rukowski [8] where, however, he imporan issue of no-arbirage was no sudied in deph. The paper is srucured as follows: In Secion 2, we inroduce self-financing rading sraegies in he presence of differenial funding raes and adjusmen processes. We consider general conracs wih cash flow sreams, raher han simple coningen claims wih a single payoff eiher a he conrac s mauriy or upon early exercise. We also inroduce in Secion 2.9 he conceps of local and global valuaion problems. This disincion is crucial since i demonsraes ha resuls obained in Secions 3 and 4 are capable of covering also financial models and valuaion problems ha canno be addressed hrough classical BSDEs, which are nowadays commonly used o deal wih nonlinear financial markes. Secion 3 is devoed o a comprehensive examinaion of he issue of exisence of arbirage opporuniies for he hedger and for he rading desk in a nonlinear rading framework and wih respec o a predeermined class of conracs. We inroduce he concep of no-arbirage wih respec o he null conrac and a sronger noion of no-arbirage for he rading desk. We hen proceed o he issue of unilaeral fair valuaion of a given conrac by he hedger who is endowed wih an iniial capial. We examine he link beween he concep of no-arbirage for he rading desk and he financial viabiliy of prices compued by he hedger. In Secion 4, we propose and analyze he concep of a regular marke model, which can be seen as an exension of he noion of a nonlinear pricing sysem, which was inroduced by El Karoui and Quenez [27]. The goal is o idenify a class of nonlinear markes, which are arbirage-free for he rading desk and, in addiion, enjoy he desirable propery ha if a given conrac can be replicaed, hen he cos of replicaion is also he fair price for he hedger. Secion 5 focuses on replicaion of a conrac in a regular marke model. We propose four alernaive definiions of no-arbirage prices, namely, he gained value, he ex-dividend price, he exi price, and he offseing price. Generally speaking, i is no expeced ha hese prices will coincide, since hey correspond o differen valuaion problems for he hedger. However, when he rading arrangemens in he underlying model are such ha he valuaion problem is local hen, under some suiable echnical condiions, we show ha he gained value and he ex-dividend price coincide. In Secion 6, we presen a BSDEs approach o he valuaion and hedging and we give examples of BSDEs for he gained value and he ex-dividend price. Finally, we briefly address in Secion 7 he issue of valuaion adjusmens in linear and nonlinear markes and we make some commens on he prevailing marke pracice of performing separae compuaions of he so-called clean price and he oal valuaion adjusmen, and subsequenly adding hem o obain he full price charged o cusomers. Alhough we focus on he issue of fair unilaeral valuaion from he perspecive of he hedger, i is clear ha idenical definiions and valuaion mehods are applicable o his counerpary as

4 4 T.R. Bielecki, I. Cialenco and M. Rukowski well. Hence, in principle, i is possible o use our resuls o examine he inerval of fair bilaeral prices in a regular marke model. Paricular insances of such unilaeral and bilaeral valuaion problems were previously sudied in Nie and Rukowski [4, 41, 43] where i was shown ha a non-empy inerval of eiher fair bilaeral prices or bilaerally profiable prices can be obained in some nonlinear markes for conracs wih eiher an exogenous or an endogenous collaeralizaion. I should be acknowledged ha here exiss a vas body of lieraure devoed o valuaion and hedging of financial derivaives under differenial funding coss, collaeralizaion, he counerpary credi risk and oher rading adjusmens see, for insance, Bichuch e al. [4], Brigo and Pallavicini [11], Brigo e al. [9, 1], Burgard and Kjaer [12, 13], Crépey [17, 18], Mercurio [39], Pallavicini e al. [45, 44], and Pierbarg [47]. In view of limied space, we canno presen here hese works in deail. Le us only menion ha mos of hese papers deal wih linear markes wih credi risk possibly also wih differenial funding raes, whereas he general heory developed in his work aims o address problems where he emphasis is pu on a nonlinear characer of valuaion in markes wih imperfecions. In conras, Albanese e al. [1], Albanese and Crépey [2], and Crépey e al. [2] propose o address he issue of valuaion adjusmens hrough an alernaive approach, which is based on he global valuaion paradigm referencing o he balance shee of he bank, is inernal srucure, and long-erm ineress of bank s shareholders. The issue of nonlineariy of rading does no appear in heir approach, since he classical hedging argumens are no longer employed o deermine he value of a new conrac, which is added o he exising porfolio of bank s asses. For furher commens on some of he above-menioned papers, we refer o Secion 7. 2 Nonlinear Marke Model We sar by re-examining and exending he nonlinear rading seup inroduced in Bielecki and Rukowski [8]. Throughou he paper, we fix a finie rading horizon dae T > for our marke model. Le Ω, G, G, P be a filered probabiliy space saisfying he usual condiions of righconinuiy and compleeness, where he filraion G = G [,T ] models he flow of informaion available o he hedger and his counerpary. For convenience, we assume ha he iniial σ-field G is rivial. All processes inroduced in wha follows are implicily assumed o be G-adaped and, as usual, any semimaringale is assumed o be a càdlàg process. Le us inroduce he noaion for he prices of all raded asses in our model. Risky asses. We denoe by S = S 1,..., S d he collecion of he ex-dividend prices of a family of d risky asses wih he corresponding cumulaive dividend sreams D = D 1,..., D d. The process S i represens he ex-dividend price of any raded securiy, such as, sock, sovereign or corporae bond, sock opion, ineres rae swap, currency opion or swap, credi defaul swap, ec. Funding accouns. We denoe by B i,l resp. B i,b he lending resp. borrowing funding accoun associaed wih he ih risky asse, for i = 1, 2,..., d. The financial inerpreaion of hese accouns varies from case o case. For an overview of rading mechanisms for risky asses, we refer o Secion 2.6. In he special case when B i,l = B i,b, we will use he noaion B i and we call i he funding accoun for he ih risky asse. Cash accouns. The lending cash accoun B,l and he borrowing cash accoun B,b are used for unsecured lending and borrowing of cash, respecively. For breviy, we will someimes wrie B l and B b insead of B,l and B,b. Also, when he borrowing and lending cash raes are equal, he single cash accoun is denoed by B or, simply, B. Noe, however, ha since an unlimied borrowing/deposiing of cash in he bank accoun is no a realisic feaure of a rading model, i is no assumed in wha follows.

5 Derivaives Pricing in Nonlinear Models 5 For breviy, we denoe by B = B i,l, B i,b, i =, 1,..., d he collecion of all cash and funding accouns. 2.1 Conracs wih Trading Adjusmens We will consider financial conracs beween wo paries, called he hedger and he counerpary. In wha follows, all he cash flows will be viewed from he prospecive of he hedger, wih he convenion ha a posiive cash flow means ha he hedger receives he corresponding amoun, and a negaive cash flow meaning ha he hedger makes a paymen. A bilaeral financial conrac or simply a conrac is given as a pair C = A, X where he meaning of each erm is explained below. A sochasic processes A represens he cumulaive cash flows from ime ill he conracs s mauriy dae, which is denoed as T. In he financial inerpreaion, he process A is assumed o model he cumulaive promised cash flows of a given conrac, which are eiher paid ou from he hedger s wealh or added o his wealh via he value process of his porfolio of raded asses including posiive or negaive holdings of cash, ha is, len or borrowed money. Noe ha he price of he conrac C exchanged a is iniiaion ha is, a ime is no included in A. For example, if he conrac sipulaes ha he hedger will receive he possibly random and of arbirary signs cash flows a 1, a 2,..., a m a imes 1, 2,..., m, T ], hen A is given by A = m 1 [l, a l. l=1 Le A, denoe a basic conrac originaed a ime wih X =. Then he only cash flow exchanged beween he counerparies a ime is he price of he conrac and hus he remaining cumulaive cash flows of A, are given as A u := A u A for u [, T ]. In paricular, he equaliy A = is valid for any basic conrac A, and any dae [, T. All fuure cash flows a l for l such ha l > are predeermined, in he sense ha hey are explicily specified by he conrac covenans. As a simple example of cash flows, consider he siuaion where he hedger sells a ime he European call opion on he risky asse S i. Then m = 1, 1 = T, and he erminal payoff from he perspecive of he hedger equals a 1 = ST i K+. More generally, for every [, T, he process A is given by A u = ST i K+ 1 [T, u for every u [, T ]. To accoun for addiional feaures of a paricular conrac a hand, we find i convenien o posulae ha he cash flows A resp. A of a basic conrac are complemened by rading adjusmens, which are represened by he process X resp. X given as X = X 1,..., X n ; α 1,..., α n ; β 1,..., β n. The role of X is o describe addiional clauses of a given conrac, such as rehypohecaed or segregaed collaeral, as well as o accoun for he impac of aypical rading arrangemens on he value process of he hedger s porfolio. For each adjusmen process X k, he process α k X k represens addiional incoming or ougoing cash flows for he hedger, which are eiher sipulaed in he clauses of he conrac or imposed by a hird pary for insance, he regulaor. To each process X k, k = 1, 2,..., n we also specify he remuneraion process β k, which is used o deermine he ne ineres paymens if any associaed wih he process X k. I should be noed ha he processes X 1,..., X n and he associaed remuneraion processes β 1,..., β n do no represen raded asses, alhough hey impac he dynamics of he value of a porfolio see 2.4. I is raher clear ha he processes α and β may depend on he respecive adjusmen process. Therefore, when he adjusmen process is Y, raher han X, one should wrie αy and βy in order o avoid confusion. However, for breviy, we will keep he shorhand noaion α and β when he adjusmen process is denoed as X. For furher commens on rading adjusmen, we refer o Secion 2.3 and

6 6 T.R. Bielecki, I. Cialenco and M. Rukowski 2.4. Las, bu no leas, we will need o define a suiable modificaion of he promised cash flows A resuling from he counerpary credi risk; see Definiion 2.8 where he concep of counerpary risky cumulaive cash flows is inroduced. In essence, he unilaeral valuaion of a given conrac is he process of finding a any dae he range of is fair prices p, as seen from he viewpoin of eiher he hedger or he counerpary. Alhough i will be posulaed ha he wo paries in a conrac adop he same valuaion paradigm, due o he asymmery of cash flows, differenial rading coss, and possibly also differen rading opporuniies, hey will ypically obain differen ranges for he respecive fair unilaeral prices of a bilaeral conrac. The dispariy in unilaeral valuaion execued independenly by he wo paries is a consequence of he nonlineariy of he wealh dynamics in rading sraegies, so ha i will ypically occur even wihin he framework of a complee nonlinear marke, where he perfec replicaion of a conrac can be achieved by he counerparies. An imporan issue of deerminaion of he range of fair bilaeral prices in a general nonlinear framework is lef for a fuure work; for resuls on bilaeral pricing in specific nonlinear marke models, see Nie and Rukowski [4, 41, 43]. 2.2 Self-financing Trading Sraegies The concep of a porfolio refers o he family of primary raded asses, ha is, risky asses, cash accouns, and funding accouns for risky asses. Formally, by a porfolio on he ime inerval [, T ], we mean an arbirary R 3d+2 -valued, G-adaped process ϕ u u [,T ] denoed as ϕ = ξ 1,..., ξ d ; ψ,l, ψ,b, ψ 1,l, ψ 1,b,..., ψ d,l, ψ d,b, 2.1 where he componens represen he posiions in risky asses S i, D i, i = 1, 2,..., d, cash accouns B,l, B,b, and funding accouns B i,l, B i,b, i = 1, 2,..., d for risky asses. I is posulaed hroughou ha ψu j,l, ψu j,b and ψu j,l ψu j,b = for all j =, 1,..., d and u [, T ]. If he borrowing and lending raes are equal, hen we wrie ψ j = ψ j,l + ψ j,b. I is also assumed hroughou ha he processes ξ 1,..., ξ d are G-predicable. We say ha a porfolio ϕ is consrained if a leas one of he componens of he process ϕ is assumed o saisfy some explicily saed consrains, which direcly affec he choice of ϕ. For insance, we will need o impose condiions ensuring ha he funding of each risky asse is done using he corresponding funding accoun. Anoher example of an explici consrain is obained when we se ψu,b = for all u [, T ], meaning ha an ourigh borrowing of cash from he accoun B,b is prohibied. For examples of markes wih various kinds of porfolio consrains, we refer o Carassus e al. [14], Fahim and Huang [28], Karazas and Kou [32, 33], and Pulido [48] and he references herein. The concep of a consrained porfolio should be conrased wih he noion of admissibiliy of a rading sraegy ha may involve some addiional condiions imposed on he wealh process and hus indirecly also on he class of admissible processes ϕ see Definiion 3.2. Noe ha porfolio consrains are no a maer of choice, since hey are due o genuine real-life resricions imposed on raders. This should be conrased wih he idea of admissibiliy of a rading sraegy, which is a mahemaical arifac needed o preclude unrealisic arbirage opporuniies like doubling sraegies, which may be presen wihin a sochasic model when coninuous rading is allowed. Noe in his regard ha here is no need o be concerned wih he admissibiliy under he realisic assumpion ha only a finie number of rading imes is available o raders. We are now in a posiion o sae some sandard echnical assumpions underpinning our furher developmens. Assumpion 2.1. We work hroughou under he following sanding assumpions:

7 Derivaives Pricing in Nonlinear Models 7 i for every i = 1, 2,..., d, he price S i of he ih risky asse is a semimaringale and he cumulaive dividend sream D i is a process of finie variaion wih D i = ; ii he cash and funding accouns B j,l and B j,b are sricly posiive and coninuous processes of finie variaion wih B j,l = Bj,b = 1 for j =, 1,..., d; iii he cumulaive cash flow process A of any conrac is a process of finie variaion; iv he adjusmen processes X k, k = 1, 2,..., n and he auxiliary processes α k, k = 1, 2,..., n are semimaringales; v he remuneraion processes β k, k = 1, 2,..., n are sricly posiive and coninuous processes of finie variaion wih β k = 1 for every k. In he nex definiion, he G -measurable random variable x represens he endowmen of he hedger a ime [, T whereas p, which a his sage is an arbirary G -measurable random variable, sands for he price a ime of C = A, X, as seen by he hedger. Recall ha A denoes he cumulaive cash flows of he conrac A ha occur afer ime, ha is, A u := A u A for all u [, T ]. Hence A can be seen as a conrac wih he same remaining cash flows as he original conrac A, excep ha A sars and is raded a ime. By he same oken, we denoe by X he adjusmen process relaed o he conrac A. Le C be a predeermined class of conracs. As expeced, i is assumed hroughou ha he null conrac N =, is raded in any marke model a any ime, ha is, N C for every [, T see Assumpion 3.1. I should be noed ha he prices p for conracs belonging o he class C are ye unspecified and hus here is a cerain degree of freedom in he foregoing definiions. Noe also ha we use he convenion ha u :=,u] for any u. Definiion 2.2. A quadruple x, p, ϕ, C is a self-financing rading sraegy on [, T ] associaed wih he conrac C = A, X if he porfolio value V p x, p, ϕ, C, which is given by saisfies for all u [, T ] V p u x, p, ϕ, C := ξus i u i + j= ψu j,l Bu j,l + ψu j,b Bu j,b 2.2 V p u x, p, ϕ, C = x + p + G u x, p, ϕ, C, 2.3 where he adjused gains process Gx, p, ϕ, C is given by G u x, p, ϕ, C := + u ξ i v ds i v + dd i v + αux k u k u j= u ψ j,l v db j,l v X k v β k v 1 dβ k v + A u. + ψ j,b v dbv j,b For a given pair x, p, we denoe by Φ,x p, C he se of all self-financing rading sraegies on [, T ] associaed wih he conrac C. When sudying he valuaion of a conrac C for a fixed, we will ypically assume ha he hedger s endowmen x is given and we will search for he range of hedger s fair prices p for C. Therefore, when dealing wih he hedger wih a fixed iniial endowmen x a ime, we will consider he following se of self-financing rading sraegies Φ,x C = C C p G Φ,x p, C. Noe, however, ha he definiion of he marke model does no assume ha he quaniy x is predeermined. 2.4

8 8 T.R. Bielecki, I. Cialenco and M. Rukowski Definiion 2.3. The marke model is he quinuple M = S, D, B, C, ΦC where ΦC sands for he se of all self-financing rading sraegies associaed wih he class C of conracs, ha is, ΦC = [,T x G Φ,x C. In principle, he marke model defined above exhibis nonlinear feaures, in he sense ha eiher he porfolio value process V p x, p, ϕ, C is no linear in x, p, ϕ, C or he class of all self-financing sraegies is no a vecor space or, ypically, boh. Therefore, we refer o his seup as o a generic nonlinear marke model. In conras, by a linear marke model we will undersand in his paper he version of he model defined above in which all rading adjusmens are null i.e., X k = for all k = 1, 2,..., n, here are no differenial funding raes i.e., B j,b = B j,l for all j =, 1,..., d and no porfolio consrains are imposed. In paricular, in he linear marke model he class of all self-financing rading sraegies is a vecor space and he value process V p x, p, ϕ, C is a linear mapping in x, p, ϕ, C. Noe, however, ha he las propery is usually los when an admissibiliy condiion is imposed on he class of rading sraegies since, ypically, a rading sraegy is deemed o be admissible if i is discouned wealh is bounded from below or nonnegaive hence he class of admissible rading sraegies is no longer a vecor space. To alleviae noaion, we will frequenly wrie x, p, ϕ, C insead of x, p, ϕ, C when working on he inerval [, T ]. Noe ha yield he following equaliies for any rading sraegy x, p, ϕ, C Φ,x C x, p, ϕ, C = ξs i i + V p j= ψ j,l Bj,l + ψj,b Bj,b = x + p + αx k k. 2.5 Recall ha in he classical case of a fricionless marke, i is common o assume ha he iniial endowmens of raders are null. Moreover, he price of a derivaive has no impac on he dynamics of he gains process. In conras, when porfolio s value is driven by nonlinear dynamics, he iniial endowmen x a ime, he iniial price p and he adjusmen cash flows of a conrac may all affec he dynamics of he gains process and hus he classical approach is no longer valid. 2.3 Funding Adjusmen The concep of he funding adjusmen refers o he spreads of funding raes wih regard o some benchmark cash rae. In he presen seup, i can be defined relaive o eiher B l or B b. If he lending and borrowing raes are no equal, hen 2.3 can be wrien as follows V p x,p, ϕ, C = x + p j= ψu j,l dbu,l + ψu j,b dbu,b ξ i u ds i u + dd i u + α k X k + A Xu k + Bu,l 1 dbu,l Xu k Bu,b 1 dbu,b ψ i,l u Bi,l u 1 db,l Xu k + β u k,l 1 d β u k,l u + Bu,l d B i,l u X k u β k,b u + ψ i,b u Bi,b u 1 db,b 1 d β k,b u u + Bu,b d i,b B u

9 Derivaives Pricing in Nonlinear Models 9 where B j,l/b := B j,l/b B,l/b 1 and β k,l/b := β k B,l/b 1. The quaniy γ := ψ i,l u Bi,l u 1 db,l Xu k + β u k,l 1 d β u k,l u + Bu,l d B i,l u X k u β k,b u + ψ i,b u Bi,b u 1 db,b 1 d β k,b u u + Bu,b d i,b B u is called he funding adjusmen. If he borrowing and lending raes are equal, hen he expression for he funding adjusmen simplifies o γ = ψu i Bi u 1 dbu + Bu d B u i X k u β k u 1 d β k u. When he cash accoun B is used for funding and remuneraion for adjusmen processes, ha is, when B i = B for i = 1, 2,..., d and β k = B for k = 1, 2,..., n, hen he funding adjusmen vanishes, as was expeced. 2.4 Financial Inerpreaion of Trading Adjusmens In his sudy, we will devoe significan aenion o erms appearing in he dynamics of V p x, ϕ, A, X, which correspond o he rading adjusmen process X. Definiion 2.4. The sochasic process ϖ = n ϖk, where for k = 1, 2,..., n, is called he cash adjusmen. ϖ k := α k X k X k uβ k u 1 dβ k u 2.6 In general, he financial inerpreaion of he cash adjusmen erm ϖ k is as follows: he erm α k X k represens he par of he kh adjusmen ha he hedger can eiher use for his rading purposes when α k X k > or has o pu aside for insance, pledge o his counerpary as a collaeral or hold in a separae accoun as a regulaory capial when α k X k <. Formally, he quaniy X k β k 1 can be seen as he number of shares of he remuneraion process β k ha he hedger should hold a ime in order o cover ineres paymens associaed wih he adjusmen process X k. Hence he inegral Xk uβu k 1 dβu k represens he cumulaive ineres eiher paid or received by he hedger due o he presence of he kh rading adjusmen. Le us illusrae a few alernaive inerpreaions of cash adjusmens given by 2.6. We hereafer wrie X k = β k 1 X k. Le us firs assume ha α k = 1, for all. The erm X k X u k dβu k indicaes ha he cash adjusmen ϖ k is affeced by boh he curren value X k of he adjusmen process and by he cos of funding of his adjusmen given by he inegral X u k dβu. k Such a siuaion occurs, for example, when X k represens he capial charge or he rehypoecaed collaeral. The inegraion by pars formula gives ϖ k = X k X u k dβu k = X k + βu k d X u, k 2.7 where he inegral βk u d X k u has he following financial inerpreaion: Xk u is he number of unis of he funding accoun β k u ha are needed o fund he amoun X k u of he adjusmen

10 1 T.R. Bielecki, I. Cialenco and M. Rukowski process. Hence d X k u is he infiniesimal change of his number and β k u d X k u is he cos of his change, which has o be absorbed by he change in he value of he rading sraegy. Observe ha he erm β k u d X k u may be negaive, meaning ha a cash relieve siuaion is aking place. In he special case when α k = 1 and β k = 1 for all, we obain ϖ k = X k for all. We deal here wih he cash adjusmen X k on which here is no remuneraion since manifesly X k u dβ k u =. This siuaion may arise, for example, if he bank does no use any exernal funding for financing his adjusmen, bu relies on is own cash reserves, which are assumed o be kep idle and neiher yield ineres nor require ineres payous. Le us now assume ha α k = for all. Then he erm ϖ k = X u k dβu k indicaes ha he cash value of he adjusmen X k does no conribue o he porfolio value. Only he remuneraion of he adjusmen process X k, which is given by he inegral X u k dβu, k conribues o he porfolio s value. This happens, for example, when he adjusmen process represens he collaeral posed by he counerpary and kep in he segregaed accoun. The above consideraions lead o he following lemma, which gives a convenien represenaion for he cash adjusmen process when α k is equal o eiher 1 or. In mos pracical siuaions, a general case can also be deal wih using Lemma 2.5 and a suiable redefiniion of adjusmen processes. Lemma 2.5. Le he non-negaive inegers n 1, n 2, n 3 be such ha n 1 + n 2 + n 3 = n. If α k = 1 for k = 1, 2,..., n 1 + n 2, β k = 1 for k = n 1, n 1 + 1,..., n 1 + n 2 and α k = for k = n 1 + n 2, n 1 + n 2 + 1,..., n 1 + n 2 + n 3, hen he cash adjusmen process ϖ saisfies, for all [, T ], n 1 n 1 ϖ = X k + β k u d X k u + 1 +n 2 k=n 1 +1 X k n 1 +n 2 +n 3 k=n 1 +n 2 +1 X k u dβ k u Wealh Process Le x, p, ϕ, C be an arbirary self-financing rading sraegy. Then he following naural quesion arises: wha is he wealh of a hedger a ime, say V x, p, ϕ, C? I is clear ha if he hedger s iniial endowmen equals x, hen his iniial wealh equals x + p when he sells a conrac C a he price p a ime. By conras, he iniial value of he hedger s porfolio, ha is, he oal amoun of cash he invess a ime in his porfolio of raded asses, is given by 2.5 meaning ha he rading adjusmens a ime need o be accouned for in he iniial porfolio s value. However, according o he financial inerpreaion of rading adjusmens, hey have no bearing on he hedger s iniial wealh and hus he relaionship beween he hedger s iniial wealh and he iniial porfolio s value reads V x, p, ϕ, C = V p x, p, ϕ, C αx k k. Analogous argumens can be used a any ime [, T ], since he hedger s wealh a ime should represen he value of his porfolio of raded asses ne of he value of all rading adjusmens see 2.1. Furhermore, one needs o focus on he acual ownership as opposed o he legal ownership of each of he adjusmen processes X 1,..., X n, of course, provided ha hey do no vanish a ime. Alhough his general rule is cumbersome o formalize, i will no presen any difficulies when applied o a paricular conrac a hand. For insance, in he case of he rehypohecaed cash collaeral see Secion 2.7.1, he hedger s wealh a ime should be compued by subracing he collaeral amoun C from he porfolio s

11 Derivaives Pricing in Nonlinear Models 11 value. This is consisen wih he acual ownership of he cash amoun delivered by eiher he hedger or he counerpary a ime. For example, if C + > hen he legal owner of he amoun C + a ime could be eiher he holder or he counerpary depending on he legal covenans of he collaeral agreemen bu he hedger, as a collaeral aker, is allowed o use he collaeral amoun for his rading purposes. If here is no defaul before T, he collaeral aker reurns he collaeral amoun o he collaeral provider. Hence he amoun C + should be accouned for when dealing wih he hedger s porfolio, bu should be excluded from his wealh. In general, we have he following definiion of he wealh process. Definiion 2.6. The wealh process of a self-financing rading sraegy x, p, ϕ, C is defined, for every u [, T ], by V u x, p, ϕ, C := Vu p x, p, ϕ, C αux k u k 2.9 or, more explicily, V u x, p, ϕ, C = ξus i u i + j= ψu j,l Bu j,l + ψu j,b Bu j,b αux k u. k 2.1 Le us observe ha here is a lo of flexibiliy in he choice of he adjusmen processes X k and corresponding processes α k. However, we will always assume ha hese processes are specified such ha he above argumens of inerpreing he acual ownership of he capial and hus also of he wealh process V x, p, ϕ, A, X hold rue. As an immediae consequence of Definiions 2.2 and 2.6, i follows ha he wealh process V of any self-financing rading sraegy x, p, ϕ, C admis he dynamics, for u [, T ], V u x, p, ϕ, C = x + p + u u ξ i v ds i v + D i v + j= X k v β k v 1 dβ k v + A u. u ψ j,l v db j,l v + ψ j,b v dbv j,b 2.11 One could argue ha i would be possible o ake equaions 2.1 and 2.11 as he definiion of a self-financing rading sraegy and subsequenly deduce ha equaliy 2.3 holds for he porfolio s value V p x, p, ϕ, C, which is hen given by 2.9. We conend his alernaive approach would no be opimal, since condiions in Definiion 2.2 are obained hrough a sraighforward analysis of he rading mechanism and physical cash flows, whereas he financial jusificaion of equaions is less appealing. Clearly, he wealh processes of a self-financing rading sraegy is characerized in erms of wo equaions 2.1 and Observe ha, using 2.1, i is possible o eliminae one of he processes ψ j,l or ψ j,b from 2.11 and hus o characerize he wealh process in erms of a single equaion. One obains in ha way a ypically nonlinear BSDE, which can be used o formulae various valuaion problems for a given conrac. 2.6 Trading in Risky Asses Noe ha we do no posulae ha he processes S i, i = 1, 2,..., d are posiive, unless i is explicily saed ha he process S i models he price of a sock. Hence by he long cash posiion resp. shor cash posiion, we mean he siuaion when ξ i S i resp. ξ i S i, where ξ i is he number of hedger s posiions in he risky asse S i a ime.

12 12 T.R. Bielecki, I. Cialenco and M. Rukowski Cash Marke Trading For simpliciy of presenaion, we assume ha S i for all [, T ]. Assume firs ha he purchase of ξ i > shares of he ih risky asse is funded using cash. Then, we se ψ i,b = for all [, T ] and hus he process B i,b becomes irrelevan. Le us now consider he case when ξ i <. If we assume ha he proceeds from shor selling of he risky asse S i can be used by he hedger his is ypically no rue in pracice, we also se ψ i,l = for all [, T ], and hus he process B i,l becomes irrelevan as well. Hence, under hese sylized cash rading convenions, here is no need o inroduce he funding accouns B i,l and B i,b for he ih risky asse. Since dividends D i are passed over o he lender of he asse, hey do no appear in he erm represening he gains/losses from he shor posiion in he risky asse. In he simples case of no marke fricions and rading adjusmens, and wih he single risky asse S 1, under he presen shor selling convenion, 2.3 becomes V p x, p, ϕ, C = x + p + ξu 1 dsu 1 + ddu 1 + ψu,l dbu,l + ψu,b dbu,b + A. More pracical shor selling convenions for risky asses are discussed in he foregoing subsecions Shor Selling of Risky Asses Le us now consider he classical way of shor selling of a risky asse borrowed from a broker. In ha case, he hedger does no receive he proceeds from he sale of he borrowed shares of a risky asse, which are held insead by he broker as he cash collaeral. The hedger may also be requesed o pos addiional cash collaeral o he broker and, in some cases, he may be paid ineres on his margin accoun wih he broker. 1 To represen hese rading arrangemens for he ih risky asse, we se ψ i,l =, α i = α i+d = and X i = 1 + δ i ξ i S i, X i+d = δ i ξ i S i where β i specifies he ineres if any on he hedger s margin accoun wih he broker, δ i represens an addiional cash collaeral, and β i+d specifies he ineres rae paid by he hedger for financing he addiional collaeral. Le us assume, for insance, ha here is only one risky asse, S 1, which is eiher sold shor or purchased using cash as in Secion Then we obain he following expression for he porfolio value whereas equaion 2.3 becomes V p V p x, p, ϕ, C = ξ1 + S 1 + ψ,l B,l + ψ,b B,b 2.12 x, p, ϕ, C = x + p + ξu 1 dsu 1 + ddu β 1 u δ 1 uξ 1 u S 1 u dβ 1 u ψu,l dbu,l + ψu,b dbu,b + A β 2 u 1 δ 1 uξ 1 u S 1 u dβ 2 u In paricular, if here is no specific ineres rae for remuneraion of an addiional collaeral, hen we se X 2 = and hus he las erm in 2.13 should be omied. I is worh noing ha 2.12 can be seen as a special case of he following exended version of 2.2 V p u x, p, ϕ, C := h i ξus i u i + j= ψu j,l Bu j,l + ψu j,b Bu j,b 1 The ineresed reader may consul he web pages hp:// and hps: // for more deails on he mechanics of shor-sales.

13 Derivaives Pricing in Nonlinear Models 13 wih d = 1 and h 1 x = x + for x R Repo Marke Trading Le us firs consider he cash-driven repo ransacion, he siuaion when shares of he ih risky asse owned by he hedger are used as collaeral o raise cash. 2 To represen his ransacion, we se ψ i,b = B i,b 1 1 h i,b ξ i + S, i 2.14 where B i,b specifies he ineres paid o he lender by he hedger who borrows cash and pledges he risky asse S i as collaeral, and he consan h i,b represens he haircu for he ih asse pledged. A synheic shor-selling of he risky asse S i using he repo marke is obained hrough he securiy-driven repo ransacion, ha is, when shares of he risky asse are posed as collaeral by he borrower of cash and hey are immediaely sold by he hedger who lends he cash. Formally, his siuaion corresponds o he equaliy ψ i,l = B i,l 1 1 h i,l ξ i S i, 2.15 where B i,l specifies he ineres amoun paid o he hedger by he borrower of he cash amoun 1 h i,l ξ i S i and h i,l is he corresponding haircu. If only one risky asse is raded and ransacions are exclusively in repo marke, hen we obain V p x, p, ϕ, C = x + p + ξu 1 dsu 1 + ddu ψu,l dbu,l + ψu,b dbu,b Bu 1,l 1 1 h 1,l ξu 1 Su 1 dbu 1,l Bu 1,b 1 1 h 1,b ξu 1 + Su 1 dbu 1,b + A In pracice, i is reasonable o assume ha he long and shor repo raes for a given risky asse are idenical, ha is, B i = B i,l = B i,b. In ha case, we may and do se ψ i = 1 h i B i 1 ξ i S i, so ha equaions 2.14 and 2.15 reduce o jus one equaion 1 h i ξ i S i + ψ i B i = According o his inerpreaion of B i, equaliy 2.17 means ha rading in he ih risky asse is done using he symmeric repo marke and ξ i shares of a risky asse are pledged as collaeral a ime, meaning ha he collaeralizaion rae equals 1. Under 2.17, equaion 2.16 reduces o V p 2.7 Collaeralizaion x, p, ϕ, C = x + p + ξu 1 dsu 1 + ddu 1 + B 1 u 1 1 h 1 ξ 1 us 1 u db 1 u + A. ψu,l dbu,l + ψu,b dbu,b 2.18 We consider he siuaion when he hedger and he counerpary ener a conrac and eiher receive or pledge collaeral wih value denoed by C, which is assumed o be a semimaringale. Generally speaking, he process C represens he value of he margin accoun. We le C = X 1 + X We refer o hps:// for a deailed descripion of mechanics of repo rading.

14 14 T.R. Bielecki, I. Cialenco and M. Rukowski where X 1 := C + = C 1 {C }, and X 2 := C = C 1 {C<}. By convenion, he amoun C + is he cash value of collaeral received a ime by he hedger from he counerpary, whereas C represens he cash value of he collaeral pledged by him and hus ransferred o his counerpary. For simpliciy of presenaion and consisenly wih he prevailing marke pracice, i is posulaed hroughou ha only cash collaeral may be delivered or received for oher collaeral convenions, see, e.g., Bielecki and Rukowski [8]. According o ISDA Margin Survey 214, abou 75% of noncleared OTC collaeral agreemens are seled in cash and abou 15% in governmen securiies. We also make he following naural assumpion regarding he value of he margin accoun a he conrac s mauriy dae. Assumpion 2.7. The G-adaped collaeral amoun process C saisfies C T =. Typically his means ha he collaeral process C will have a jump a ime T from C T o. The posulaed equaliy C T = is simply a convenien way of ensuring ha any collaeral amoun posed is reurned in full o he pledger when he conrac maures, provided ha defaul evens have no occurred prior o or a mauriy dae T. As soon as he defaul evens are also modeled, we will need o specify closeou payoffs see Secion Le us firs make some commens from he hedger s perspecive regarding he crucial feaures of he margin accoun. The financial pracice may require o hold he collaeral amouns in segregaed margin accouns, so ha he hedger, when he is a collaeral aker, canno make use of he collaeral amoun for rading. Anoher collaeral convenion mosly encounered in pracice is rehypohecaion around 9% of cash collaeral of OTC conracs are rehypohecaed, which refers o he siuaion where he hedger may use he collaeral pledged by his counerparies as collaeral for his conracs wih oher counerparies. Obviously, if he hedger is a collaeral provider, hen a paricular convenion regarding segregaion or rehypohecaion is immaerial for he dynamics of he value process of his porfolio. We refer he reader o Bielecki and Rukowski [8] and Crépey e al. [19] for a deailed analysis of various convenions on collaeral agreemens. Here we will examine some basic aspecs of collaeralizaion someimes also called margining in our conex. In general, he cash adjusmens due o collaeralizaion are ϖ C := α 1 C + α 2 C β 1 u 1 C + u dβ 1 u + β 2 u 1 C u dβ 2 u, 2.2 where he remuneraion processes β 1 and β 2 deermine he ineres raes paid or received by he hedger on collaeral amouns C + and C, respecively. The auxiliary processes α 1 and α 2 inroduced in 2.2 are used o cover alernaive convenions regarding rehypohecaion and segregaion of margin accouns. Noe ha we always se α 2 = 1 for all [, T ] when considering he porfolio of he hedger, since a paricular convenion regarding rehypohecaion or segregaion is manifesly irrelevan for he pledger of collaeral Rehypohecaed Collaeral As i is cusomary in he exising lieraure, we assume ha rehypohecaion of cash collaeral means ha i can be used by he hedger for his rading purposes wihou any resricions. To cover his sylized version of a rehypohecaed collaeral for he hedger, i suffices o se α 1 = α 2 = 1 for all [, T ], so ha for he hedger we obain α 1 X 1 + α 2 X 2 = C. Consequenly, he cash adjusmen corresponding o he margin accoun equals ϖ = ϖ 1 + ϖ 2 = 2 X k + β k u d X k u. 2.21

15 Derivaives Pricing in Nonlinear Models Segregaed Collaeral Under segregaion, he collaeral received by he hedger is kep by he hird pary, so ha i canno be used by he hedger for his rading aciviies. In ha case, we se α 1 = and α 2 = 1 for all [, T ] and hus α 1 X 1 + α 2 X 2 = C. Hence he corresponding cash adjusmen erm ϖ equals ϖ = ϖ 1 + ϖ 2 = X Iniial and Variaion Margins X 1 u dβ 1 u + β 2 u d X 2 u In marke pracice, he oal collaeral amoun is usually represened by wo componens, which are ermed he iniial margin also known as he independen amoun and he variaion margin. In he conex of self-financing rading sraegies, his can be easily deal wih by inroducing wo or more collaeral processes for a given conrac A. I is worh menioning ha each of he collaeral processes specified in he clauses of a conrac is usually subjec o a differen convenion regarding segregaion and/or remuneraion. 2.8 Counerpary Credi Risk The counerpary credi risk in a financial conrac arises from he possibiliy ha a leas one of he paries in he conrac may defaul prior o or a he conrac s mauriy, which may resul in failure of his pary o fulfil all heir conracual obligaions leading o financial loss suffered by eiher one of he wo paries in he conrac. We will model defaulabiliy of he wo paries o he conrac in erms of heir defaul imes. We denoe by τ h and τ c he defaul imes of he hedger and his counerpary, respecively. We require ha τ h and τ c are non-negaive random variables defined on Ω, G, G, P. If τ h > T holds a.s. resp. τ c > T, a.s. hen he hedger resp. he counerpary is considered o be defaul-free in regard o he conrac under sudy. Hence he counerpary risk is a relevan aspec for he conrac mauring a T provided ha Pτ T > where τ := τ h τ c is he momen of he firs defaul. From now on, we posulae ha he process A models all promised or nominal cash flows of he conrac, as seen from he perspecive of he rading desk wihou accouning for he possibiliy of defauls of rading paries. In oher words, A represens cash flows ha would be realized in case none of he wo paries has defauled prior o or a he conrac s mauriy. We will someimes refer o A as o he counerpary risk-free cash flows and we will call he conrac wih cash flows A he counerpary risk-free conrac. The key concep in he conex of counerpary risk is he counerpary risky conrac, which will be examined in he foregoing subsecion Closeou Payoff Recall ha τ denoes he momen of he firs defaul. On he even {τ < }, we define he random variable Υ as Υ = Q τ + A τ C τ, 2.23 where Q is he Credi Suppor Annex CSA closeou valuaion process of he conrac A, A τ = A τ A τ is he jump of A a τ corresponding o a possibly null promised bulle dividend a τ, and C τ is he value of he collaeral process C a ime τ. In he financial inerpreaion, Υ + is he amoun he counerpary owes o he hedger a ime τ, whereas Υ is he amoun he hedger owes o he counerpary a ime τ. I accouns for he legal value Q τ of he conrac, plus he

16 16 T.R. Bielecki, I. Cialenco and M. Rukowski bulle dividend A τ o be received/paid a ime τ, less he collaeral amoun C τ since i is already held by eiher he hedger if C τ > or he counerpary if C τ <. We refer he reader o Secion in Crépey e al. [19] for he deailed discussion of he specificaion of Υ. One of he key financial aspecs of he counerpary credi risk is he closeou payoff, which occurs if a leas one of he paries defauls eiher before or a he mauriy of he conrac. I represens he cash flow exchanged beween he wo paries a he firs-pary-defaul ime. The following definiion of he closeou payoff, as usual given from he perspecive of he hedger, is aken from Crépey e al. [19]. The random variables R c and R h aking values in [, 1] represen he recovery raes of he counerpary and he hedger, respecively. Definiion 2.8. The CSA closeou payoff K is defined as K := C τ + 1 {τ c <τ h }R c Υ + Υ + 1 {τ h <τ c }Υ + R h Υ + 1 {τ h =τ c }R c Υ + R h Υ The counerpary risky cumulaive cash flows process A is given by A = 1 {<τ}a + 1 { τ} A τ + K, [, T ] Le us make some commens on he form of he closeou payoff K. Firs, he erm C τ is due o he fac ha he legal ile o he collaeral amoun comes ino force only a he ime of he firs defaul. The hree erms appearing afer C τ in 2.24 correspond o he CSA convenion ha he cash flow a he firs defaul from he perspecive of he hedger should be equal o Q τ + A τ. Le us consider, for insance, he even {τ c < τ h }. If Υ + >, hen we obain K = C τ + R c Q τ + A τ C τ Q τ + A τ, where he equaliy holds whenever R c = 1. If Υ >, hen we ge K = C τ Q τ A τ + C τ = Q τ + A τ. Finally, if Υ =, hen K = C τ = Q τ + A τ. Similar analysis can be done on he remaining wo evens in Remark 2.9. Of course, here is no counerpary credi risk presen under he assumpion ha Pτ > T = 1. Le us consider he case where Pτ > T < 1. We denoe by P e he counerpary risk-free ex-dividend price of he conrac a ime. If we se R c = R h = 1, hen we obain A τ = A τ + Q τ. Hence he counerpary credi risk is sill presen, despie he posulae of he full recovery, unless he legal value Q τ perfecly maches he counerpary risk-free ex-dividend price P e τ. Obviously, he counerpary credi risk vanishes when R c = R h = 1 and Q τ = P e τ, since in ha case he so-called exposure a defaul see Secion in Crépey e al. [19] is null Counerpary Credi Risk Decomposiion To effecively deal wih he closeou payoff in our general framework, we now define he counerpary credi risk CCR cash flows, which are someimes called CCR exposures. Noe ha he evens {τ = τ h } = {τ h τ c } and {τ = τ c } = {τ c τ h } may overlap.

17 Derivaives Pricing in Nonlinear Models 17 Definiion 2.1. By he CCR processes, we mean he processes CL, CG and RP where he credi loss CL equals CL = 1 { τ} 1 {τ=τ c }1 R c Υ +, he credi gain CG equals CG = 1 { τ} 1 {τ=τ h }1 R h Υ, and he replacemen process is given by CR = 1 { τ} A τ A + Q τ. The CCR cash flow is given by A CCR = CL + CG + CR. I is worh noing ha he CCR cash flows depend on he processes A, C and Q. The nex proposiion shows ha we may inerpre he counerpary risky conrac as he basic conrac A, which is complemened by he collaeral adjusmen process X = X 1, X 2 = C +, C and he CCR cash flow A CCR. In view of his resul, he counerpary risky conrac A, X admis he following formal decomposiions A, X = A, X + A CCR, and A, X = A, + A CCR, X. Proposiion The equaliy A = A + A CCR holds for all [, T ]. Proof. We firs noe ha K = C τ + 1 {τ c <τ h }R c Υ + Υ + 1 {τ h <τ c }Υ + R h Υ + 1 {τ h =τ c }R c Υ + R h Υ = C τ 1 {τ c τ h }1 R c Υ {τ h τ c }1 R h Υ + Υ = Q τ + A τ 1 {τ c τ h }1 R c Υ {τ h τ c }1 R h Υ, where we used 2.23 in he las equaliy. Therefore, from 2.25 we obain A = 1 {<τ}a + 1 { τ} A τ + K = 1 {<τ} A + 1 { τ} A τ A τ + K = A τ + 1 { τ} K A τ = A + A τ A + 1 { τ} K A τ = A + 1 { τ} Aτ A + Q τ 1 {τ c τ h }1 R c Υ {τ h τ c }1 R h Υ, which is he desired equaliy in view of Definiion 2.1. Proposiion 2.11 shows ha cash flows of he counerpary risky conrac can be formally decomposed ino he counerpary risk-free componen A 1, X 1 = A, X and he CCR componen A 2, X 2 = A CCR,. This addiive decomposiion of he conrac s cash flows may be employed in pricing of a counerpary risky conrac. For insance, one could aemp o compue he price of he conrac A, X using he following enaive decomposiion price A, X = price A, X + price A CCR, = counerpary risk-free price + CCR price. I is unlikely ha his procedure would resul in an overall arbirage-free valuaion of he counerpary risky conrac in a nonlinear framework since, as we argue in Secion 6, he addiiviy of ex-dividend prices obained by solving nonlinear BSDEs fails o hold, in general.

18 18 T.R. Bielecki, I. Cialenco and M. Rukowski 2.9 Local and Global Valuaion Problems Marke adjusmens, represened in our framework by he process X, may in fac depend boh on he cash flow process A and he rading sraegy ϕ. By he same oken, he rading sraegy ϕ will ypically depend on he rading adjusmens. So, a feedback effec beween ϕ and X is poenially presen in our rading universe and, of course, his feaure should be properly accouned for in valuaion and hedging. Furhermore, i is imporan o disinguish beween he case where he above-menioned dependence is only on he curren composiion of he hedging porfolio and/or he curren level of he wealh process and he case, where his dependence exends o he hisory of a hedging sraegy. If a conrac A, X, he cash and funding accouns, and he prices of risky asses do no depend on he sric hisory i.e., he hisory no including he curren values of processes of ineres of a hedger s rading sraegy ϕ and he wealh process V ϕ, hen we say ha a valuaion problem is local; oherwise, i is referred o as a global valuaion problem. In view of 2.11, he disincion beween local and global valuaion problems can be formalized as follows. Definiion A valuaion problem is said o be local if X k = v k, V ϕ, ϕ and dβ k = w k, V ϕ, ϕ d for some G-progressively measurable mappings v k, w k : Ω [, T ] R 3d+1 R for every k = 1, 2,..., n. A valuaion problem is said o be global if X k = v k, V ϕ, ϕ and dβ k = w k, V ϕ, ϕ d for some G-non-anicipaive funcionals v k, w k : Ω [, T ] D[, T ], R 3d+1 R for every k = 1, 2,..., n where D[, T ], R 3d+1 is he space of R 3d+1 -valued, G-adaped, càdlàg processes on [, T ]. As one migh guess, soluions o he wo valuaion problems will always coincide a ime bu, in general, hey may have very differen properies a any dae, T. In paricular, hey will ypically correspond o differen classes of BSDEs: local problems correspond o classical BSDEs, whereas global ones can be deal wih hrough generalized BSDEs, which were inroduced in he recen work by Cheridio and Nam [16] see also Zheng and Zong [51]. I is imporan o sress ha he disincion beween he local and global problems is no relaed o he concep of pahindependen coningen claims or he Markov propery of he underlying model for primary risky asses. I is only due o he above-menioned eiher local or global feedback effec beween he hedger s rading decisions and he marke condiions inclusive of paricular adjusmens for he conrac a hand. Example As a sylized example of a global valuaion problem, le us consider a conrac, which lass for wo monhs for concreeness, assume ha i is a simple combinaion of he pu and he call on he sock S 1 wih mauriies equal o one monh and wo monhs, respecively. The borrowing rae for he hedger is se o be 5% per annum, rising o 6% afer one monh if he hedger borrows any cash during he firs monh and i will say a 5% if he does no. Similarly, he lending rae iniially equals 3% per annum and drops o 2% if he hedger borrows any cash during he firs monh. I is inuiively clear ha he valuaion problem is here global, since is soluion on [, T ] will depend on he sric hisory of rading. In conras, if he rading model has possibly differen, bu fixed, borrowing and lending raes, hen he valuaion problem for any conrac will be local, in he sense inroduced above, of course, unless some oher rading adjusmens will depend on he sric hisory of rading. For insance, if he only adjusmen is he variable margin accoun deermined by he hedger s valuaion and wih a consan remuneraion rae, hen he hedger s valuaion problem is local. Noe ha he valuaion problem described above can be inherenly global even when he sock price is governed under he real-world probabiliy measure by Markovian dynamics and he conrac under sudy is a sandard call or pu opion or any oher pah-independen coningen claim.