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 Absrac: The main objecive is o 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 [EKQ97] and El Karoui e al. [EKPQ97]. 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 Regulaory Capial 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 Arbirage-Free Trading Models 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 Replicaion on [, T ] and he Gained Value Marke Regulariy on [, T ] Replicable Conracs Non-replicable Conracs Pricing in Regular Models Replicaion and Marke Regulariy on [, T ] Hedger s Ex-dividend Price a Time Marked-o-Marke Value Offseing Price A BSDE Approach o Nonlinear Pricing BSDE for he Gained Value BSDE for he Ex-dividend Price BSDE for he CCR Price Appendix: Examples of Non-Regular Models Model wih Trading Consrains Model Wihou Trading Consrains Appendix: Local and Global Pricing Problems 42

3 Derivaives Pricing in Nonlinear Models 3 1 Inroducion The presen work conribues o he nonlinear pricing heory, which arises in a naural way when accouning for salien feaures of real-world rades such as: rading consrains, differenial funding coss, collaeralizaion, counerpary credi risk and capial requiremens. To be more specific, he aim of our sudy is o exend in several respecs he hedging and pricing approach in nonlinear marke models developed in El Karoui and Quenez [EKQ97] and El Karoui e al. [EKPQ97] see also [CK93, DQS14, DQS15, KK96, KK98] for hedging and pricing wih consrained porfolios by accouning for he complexiy of over-he-couner financial derivaives and specific feaures of he rading environmen afer he global financial crisis. Le us briefly summarize he main conribuions of his work: The firs goal is o discuss he rading sraegies in he presence of differenial funding raes and adjusmen processes. We sress ha he need for a more general approach arises due o he fac ha we sudy general conracs wih cash flow sreams, raher han simple coningen claims wih a single payoff a mauriy or upon exercise. Second, we examine in deail he issues of he 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 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. Third, we propose and analyze he concep of a regular marke model, which exends he concep of a nonlinear pricing sysem inroduced in El Karoui and Quenez [EKQ97]. The goal is o idenify a class of nonlinear marke models, which are arbirage-free for he rading desk and, in addiion, enjoy he desirable propery ha for conracs ha can be replicaed, he cos of replicaion is also he fair price for he hedger. Nex, we focus on replicaion of a conrac in a regular marke model and we discuss he BSDE approach o he valuaion and hedging of conracs in a model wih differenial funding raes, he counerpary credi risk and rading adjusmens. We propose wo main definiions of no-arbirage prices, namely, he gained value and he ex-dividend price, and we show ha hey in fac coincide, under suiable echnical condiions, when he pricing problem under consideraion is local. I is worh sressing ha in he case of a global pricing problem he wo above-menioned definiions will ypically yield differen pricing resuls for he hedger. To complee our sudy, we also examine he marked-o-marke valuaion of a conrac and he problem of unwinding and offseing for an exising conrac. We conclude he paper by briefly addressing he issue of fair valuaion and hedging of a counerpary risky conrac in a nonlinear marke model. I should be acknowledged ha we focus on fair unilaeral pricing from he perspecive of he hedger, alhough i is clear ha he same definiions and pricing mehods are applicable o his counerpary as well. Hence, in principle, i is also possible o use our resuls in order o examine he inerval of fair bilaeral prices in a regular marke model. Paricular insances of bilaeral pricing problems were sudied in [NR15, NR16a, NR16c] where i was shown ha a non-empy inerval of fair bilaeral prices or bilaerally profiable prices can be obained in some nonlinear models for

4 4 T.R. Bielecki, I. Cialenco and M. Rukowski conracs wih eiher an exogenous or an endogenous collaeralizaion. For more pracical sudies of pricing and hedging subjec o differenial funding coss and he counerpary credi risk o which our general heory can be applied, he reader is referred o [BCS15, BP14, BK11, BK13, Cré15a, Cré15b, PPB12a, PPB12b, Pi1]. 2 Nonlinear Marke Model In his secion, we exend a generic marke model inroduced in [BR15]. Throughou he paper, we fix a finie rading horizon dae T > for our model of he financial marke. Le Ω, G, G, P be a filered probabiliy space saisfying he usual condiions of righ-coninuiy and compleeness, where he filraion G = G [,T ] models he flow of informaion available o all raders. For convenience, we assume ha he iniial σ-field G is rivial. Moreover, all processes inroduced in wha follows are implicily assumed o be G-adaped and, as usual, any semimaringale is assumed o be càdlàg. Also, we will assume ha any process Y saisfies Y := Y Y =. 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 is aimed o represen he ex-dividend price of any raded securiy, such as, sock, sovereign or corporae bond, sock opion, ineres rae swap, currency opion or swap, CDS, CDO, ec. 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. 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 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. For breviy, 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. 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 mauriy dae T. The process A is assumed o model he cumulaive cash flows of a given conrac, which are eiher paid ou from he hedger s wealh or added o he his wealh via he value process of his porfolio of raded asses including cash. Noe ha he price of he conrac exchanged a is iniiaion will no be included in A. For example, if a conrac sipulaes ha he hedger will receive he possibly random cash flows a 1, a 2,..., a m a imes 1, 2,..., m, T ], hen A = m 1 [l, a l. l=1

5 Derivaives Pricing in Nonlinear Models 5 Le A, denoe a conrac iniiaed a ime wih X =. Then he only cash flow exchanged beween paries a ime is he price of he conrac, denoed as p, and hus he remaining cash flows of A, are given as A u := A u A for u [, T ]. In paricular, he equaliy A = is valid for any 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, bu he price p needs o be firs properly defined and nex compued using some paricular marke model. As an example, consider he siuaion where he hedger sells a ime a European call opion on he risky asse S i. Then m = 1, 1 = T, and he erminal payoff from he perspecive of he issuer equals a 1 = ST i K+. Consequenly, for every [, T, he process A saisfies A u = ST i K+ 1 [T, u for all u [, T ]. Obviously, he price p of he opion depends boh on a marke model and a pricing mehod. To accoun for marke fricions, we posulae ha he cash flows A resp. A of a conrac are complemened by rading adjusmens, which are formally represened by he process X resp. X given as X = X 1,..., X n ; α 1,..., α n ; β 1,..., β n. Is role is o describe addiional rading covenans associaed wih a given conrac, such as collaeralizaion and regulaory capial, as well as he adjusmens due o he counerpary credi risk. For each adjusmen process X k, he auxiliary process α k X k represens addiional incoming or ougoing cash flows for he hedger, which are eiher sipulaed in he clauses of a conrac e.g., he credi suppor annex or imposed by a hird pary for insance, he regulaor. In addiion, each process X k, k = 1, 2,..., n is complemened by he corresponding remuneraion process β k, which is used o deermine 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. 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. Valuaion or pricing of a given conrac means, in paricular, o find he range of fair prices p a any dae from he viewpoin of eiher he hedger or he counerpary. Alhough i will be posulaed ha boh paries 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 fair unilaeral prices for a given bilaeral conrac. The discrepancy in pricing for he wo paries is a consequence of he nonlineariy of he wealh dynamics in rading sraegies, so ha i may occur even wihin he framework a complee marke model where he perfec replicaion of any conrac can be achieved by boh paries. 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 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 denoe by B j, for j =, 1,..., d, he corresponding cash or funding accoun and we denoe ψ j = ψ j,l + ψ j,b. I is also assumed hroughou ha he processes ξ 1,..., ξ d are G-predicable.

6 6 T.R. Bielecki, I. Cialenco and M. Rukowski We say ha a porfolio ϕ is consrained if a leas one of he componens of he process ϕ is assumed o saisfy some addiional consrains. 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. We are now in a posiion o sae some fairly sandard echnical assumpions underpinning our furher developmens. Assumpion 2.1. We work hroughou under he following sanding assumpions: 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 is iniiaed and 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, ha is, N C 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 a conrac C. 2.4

7 Derivaives Pricing in Nonlinear Models 7 When sudying 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 prices p for C. For insance, 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 in he definiion of he marke model we do no assume ha he quaniy x is predeermined. 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. Noe ha, in principle, he marke model defined above is nonlinear, 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 boh. Therefore, we refer o his model 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 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 here are no rading consrains. 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. 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 2.3 yields he following equaliies for any rading sraegy x, p, ϕ, C Φ,x C V p x, p, ϕ, C = ξs i i + 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 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 ψ i,l u Bi,l u 1 db,l Xu k + βu k,l 1 d β u k,l ξ i u ds i u + dd i u + u + Bu,l d B i,l u X k u β k,b u α k X k + A Xu k + Bu,l 1 dbu,l Xu k Bu,b 1 dbu,b + ψ i,b u Bi,b u 1 db,b 1 d β k,b, u u + Bu,b d i,b B u

8 8 T.R. Bielecki, I. Cialenco and M. Rukowski 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 ψ i,l u Bi,l u 1 db,l Xu k + βu 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 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,..., 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 as a collaeral or hold as a regulaory capial when α k X k <. Le us illusrae alernaive inerpreaions of cash adjusmens given by 2.6. We denoe 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 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.

9 Derivaives Pricing in Nonlinear Models 9 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. As was argued above, in mos pracical applicaions, he cash adjusmen process can be represened as follows n 1 n 1 ϖ = X k + β k u d X k u 1 +n 2 k=n 1 +1 X k u dβ k u + n 1 +n 2 +n 3 k=n 1 +n 2 +1 where he non-negaive inegers n 1, n 2, n 3 are assumed o saisfy n 1 + n 2 + n 3 = n. 2.5 Wealh Process X k, 2.8 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 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.

10 1 T.R. Bielecki, I. Cialenco and M. Rukowski Definiion 2.5. The wealh process of a self-financing rading sraegy x, p, ϕ, C 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 s and corresponding processes α k s. 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 2.5, 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. We obain in his 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 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 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

11 Derivaives Pricing in Nonlinear Models 11 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 + ξ 1 u ds 1 u + dd 1 u + ψu,l dbu,l + ψu,b dbu,b + A More pracical shor selling convenions 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. For example, if we assume ha he risky asse is purchased using cash as in Secion 2.6.1, we ge he following equaliy, which is a sligh exension of equaliy 2.2, whereas equaion 2.3 becomes V p V p x, p, ϕ, C = x, p, ϕ, C = x + p + ξu 1 dsu 1 + ddu ξ 1 + S i + ψ,l B,l + ψ,b B,b 2.13 β 1 u δ 1 uξ 1 u S 1 u dβ 1 u ψu,l dbu,l + ψu,b dbu,b + A 2.14 β 2 u 1 δ 1 uξ 1 u S 1 u dβ 2 u. If, however, a specific ineres rae for remuneraion of an addiional collaeral is no specified, hen we se X i+d = and hus he las erm in 2.14 should be omied 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 The ineresed reader may consul he web pages hp:// and hps: // for more deails on he mechanics of shor-sales. 2 We refer o hps:// for a deailed descripion of mechanics of repo rading.

12 12 T.R. Bielecki, I. Cialenco and M. Rukowski 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 2.16 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,l = B i,b. In ha case, we may and do se B i := B i,l = B i,b and ψ i = 1 h i B i 1 ξ i S i, so ha equaions 2.15 and 2.16 reduce o jus one equaion 1 h i ξ i S i + ψ i B i = According o his inerpreaion of B i, equaliy 2.18 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 collaeral rae equals 1. Under 2.18, equaion 2.17 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 2.19 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 2, 2.2 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 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., [BR15]. According o ISDA Margin Survey 214, abou 75% of non-cleared 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 2.6. The G-adaped collaeral amoun process C saisfies C T =.

13 Derivaives Pricing in Nonlinear Models 13 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 he 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 he closeou payoff 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 [BR15] and [CBB14] 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, we have ϖ = α 1 C + α 2 C β 1 u 1 C + u dβ 1 u + β 2 u 1 C u dβ 2 u, 2.21 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 2.21 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 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 2 X 1 u dβ 1 u + β 2 u d X 2 u. 2.23

14 14 T.R. Bielecki, I. Cialenco and M. Rukowski Iniial and Variaion Margins 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, a leas wih respec o he conrac under sudy. Hence he counerpary risk is a relevan aspec of our model 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 aking ino accoun 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 defauled prior o or a he conrac s mauriy. We will someimes refer o A as o he counerpary risk-free cash flows or counerpary clean cash flows and we will call he conrac wih cash flows A he counerpary risk-free conrac or he counerpary clean conrac. The key concep in he conex of counerpary risk is he counerpary risky conrac Closeou Payoff On he even {τ < }, we define he random variable Υ as Υ = Q τ + A τ C τ, 2.24 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 bulle dividend A τ o be received/paid a ime τ, less he collaeral amoun C τ since i is already held by he hedger if C τ > or by he counerpary if C τ <. We refer he reader o [CBB14, Secion 3.1.3] for more deails regarding he financial inerpreaion of Υ. One of he key financial aspecs of he counerpary risky conrac 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 firs-pary-defaul ime. The following definiion of he closeou payoff, as seen from he perspecive of he hedger, is aken from [CBB14]. The random variables R c and R h, which ake values in [, 1], represen he recovery raes of he counerpary and he hedger, respecively.

15 Derivaives Pricing in Nonlinear Models 15 Definiion 2.7. 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 legal ile o he collaeral amoun comes ino force only a he ime of he firs defaul. The following hree erms correspond o he CSA convenion ha, in principle, he nominal cash flow a he firs defaul from he perspecive of he hedger is given as 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 2.8. 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 clean ha is, 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 clean 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 [CBB14, Secion 3.2.3] 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. Definiion 2.9. 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 and he replacemen process is given by CG = 1 { τ} 1 {τ=τ h }1 R h Υ, CR = 1 { τ} A τ A + Q τ. The CCR cash flow is given by A CCR = CL + CG + CR.

16 16 T.R. Bielecki, I. Cialenco and M. Rukowski I is worh noing ha he CCR cash flows depend on processes A, C and Q. The nex proposiion shows ha we may inerpre he counerpary risky conrac as he clean 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 2.1. 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 2.24 in he las equaliy. Therefore, from 2.26 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 2.9. More generally, given wo conracs, say A i, X i for i = 1, 2, we are ineresed in pricing and hedging issues for a compound conrac A, X where A = fa 1, A 2 in relaion o pricing and hedging of is componens A 1, X 1 and A 2, X 2. To be more specific, we wish o find ou wheher individual no-arbirage pricing for A 1, X 1 and A 2, X 2 leads o, a leas approximae, fair valuaion for he conrac A, X. We need o sress ha here may be a feedback effec involved beween he compound conrac and is componens. For insance, i follows from Proposiion 2.1 ha he counerpary risky conrac can be decomposed ino he clean componen A 1, X 1 = A, X and he CCR componen A 2, X 2 = A CCR,. In bank s pracice, he exi price for a counerpary risky conrac is he combinaion of he clean price of he conrac and he price of he counerpary credi risk, which is referred o as he CCR price in wha follows. The clean price and he corresponding hedge are esablished by he rading desk, whereas he price and hedge for he counerpary credi risk are deal wih by he dedicaed CVA desk. To sum up, he ypical procedure used in indusry o derive he exi price of he conrac A, X is based on he following addiive decomposiion price A, X = price A, X + price A CCR, = clean price + CCR price I is unclear under which condiions his procedure resuls in an overall arbirage-free valuaion and hedging of he counerpary risky conrac, in general, since he implicily assumed addiiviy of pricing does no necessarily hold under marke fricions. In he exising lieraure, he counerpar of he above relaionship is usually represened by he equaliy counerpary risky price = clean price TVA where TVA sands for he oal valuaion adjusmen. This requires wo commens. Firs, he TVA erm accouns for several adjusmens, and no only he counerpary credi risk, ypically represened by he CVA and DVA. In paricular, i may accoun for he funding valuaion adjusmen FVA. In our approach, he funding adjusmen resuls from he funding coss aribued o

17 Derivaives Pricing in Nonlinear Models 17 hedging he wo componens of A, X. Second, in our formula we have he sum, raher han he difference, in he righ-hand side of This discrepancy is simply due o our definiion of he adjusmens CL, CG and CR, which are se o be negaives of heir counerpars encounered in oher papers. 2.9 Regulaory Capial The case of he regulaory capial can be formally covered by adding he process X k = K, where K is a non-negaive sochasic process and by seing α k = 1 for all [, T ]. If he regulaory capial is remuneraed, hen we also need o specify he corresponding remuneraion process β k. Of course, an imporan issue of explici specificaions of hese processes will arise when he general heory is implemened. A pracical approach o he capial valuaion adjusmen associaed wih he regulaory capial was developed in a recen work by Albanese e al. [ACC16]. 3 Arbirage-Free Trading Models The analysis of he self-financing propery of a rading sraegy should be complemened by he sudy of some kind of a no-arbirage propery for he adoped marke model. Due o he nonlineariy of a marke model wih differenial funding raes, he concep of no-arbirage is nonrivial, even when no rading adjusmens are presen. We will argue ha i can be effecively deal wih using a reasonably general definiion of an arbirage opporuniy associaed wih rading. Le us sress ha we only consider here a nonlinear exension of he classical concep of an arbirage opporuniy as opposed o oher relaed conceps, such as: NFLVR, NUPBR, ec No-arbirage Pricing Principles Le us firs describe very briefly he commonly adoped pricing paradigm for financial derivaives. In essence, a general approach o no-arbirage pricing hinges on he following seps: Sep L.1. One firs checks wheher a marke model wih predeermined rading rules and primary raded asses is arbirage-free, where he definiion of an arbirage opporuniy is a mahemaical formalizaion of he real-world concep of a risk-free profiable rading opporuniy. Sep L.2. Given a financial derivaive for which he price is ye unspecified, one proposes a price no necessarily unique and checks wheher he exended model ha is, he model where he financial derivaive is posulaed o be an addiional raded asse is also arbirage-free in he sense made precise in Sep L.1. The valuaion procedure oulined above can be referred o as he no-arbirage pricing paradigm. In any linear marke model see he commens afer Definiion 2.3, he unique price given by replicaion or he range of no-arbirage prices obained using he concep of superhedging sraegies in he case of an incomplee marke is consisen wih he no-arbirage pricing paradigm L.1 L.2, alhough o esablish his propery in a coninuous-ime framework, one needs also o inroduce he concep of admissibiliy of a rading sraegy. This is feasible since he sric comparison propery of linear BSDEs can be employed o show ha replicaion or superhedging will indeed yield prices for derivaives ha are consisen wih he no-arbirage pricing paradigm. Alernaively, he fundamenal heorem of asse pricing FTAP can be used o show ha he discouned prices defined hrough admissible rading sraegies are local maringales in fac, supermaringales under an equivalen local maringale measure. The laer propery is a well known fundamenal feaure of sochasic inegraion, so i covers all linear marke models.

18 18 T.R. Bielecki, I. Cialenco and M. Rukowski Le us now commen on he exising approaches o nonlinear pricing of derivaives. The mos common approach o he pricing problem in a nonlinear framework seems o hinge, a leas implicily, on he following seps in which i is usually assumed ha he hedger s iniial endowmen is null. In fac, Sep N.1 was explicily addressed only in some works, whereas mos papers in he exising lieraure were concerned wih finding a replicaing sraegy menioned in Sep N.2. Also, o he bes of our knowledge, he imporan issue oulined in Sep N.3 was up o now compleely ignored. Sep N.1. The sric comparison argumen for he BSDE associaed wih he wealh dynamics is used o show ha i is no possible o consruc an admissible rading sraegy wih he null iniial wealh and he erminal wealh, which is non-negaive almos surely and sricly posiive wih a posiive probabiliy. Sep N.2. The price for a European coningen claim is defined using eiher he cos of replicaion or he minimal cos of superhedging. A suiable version of he sric comparison propery for he wealh dynamics can be used o show ha in some marke models referred o as regular models in his work he wo pricing approaches yield he same value for any European claim ha can be replicaed. Sep N.3. I remains o check if he prices given by he cos of replicaion or seleced o be below he upper bound given by he minimal cos of superhedging comply wih some form of he no-arbirage principle. The problem wheher he exended nonlinear marke model is sill arbirage-free in some sense is much harder o ackle han i was he case for he linear framework, since rading in derivaives may essenially change he properies of he marke. However, in he case of a regular model, his sep is relaively easy o handle due o he posulaed regulariy condiions see, in paricular, Definiion 4.6 and Proposiion Discouned Wealh and Admissible Sraegies To deal wih he issue of no-arbirage, we need o inroduce he discouned wealh process and properly define he concep of admissibiliy of rading sraegies. Le us denoe B x := 1 {x } B,l + 1 {x<} B,b. 3.1 Noe ha if B,l = B,b, hen Bx = B = B. Furhermore, if x =, hen xb,b T = xb,l T = and hus he choice of eiher B,l or B,b in he righ-hand side of 3.1 is immaerial. I is naural o posulae ha he iniial endowmen x resp. x < has he fuure value xb,l resp. xb,b a ime [, T ] when invesed in he cash accoun B,l resp. B,b. We hus henceforh work under he following assumpion. Assumpion 3.1. We posulae ha: i for any iniial endowmen x R of he hedger, he null conrac N =, belongs o C, ii for any x R, he rading sraegy x,, ϕ, N where ϕ has all componens null excep for eiher ψ,l if x or ψ,b if x < belongs o Φ,x C and V p x,, ϕ, N = V x,, ϕ, N = xb x for all [, T ]. Assumpion 3.1 may look redundan a he firs glance, bu i is neverheless needed and useful in derivaion of basic properies of fair prices. Condiion i is a raher obvious requiremen. Noe ha condiion ii canno be deduced direcly from he self-financing condiion, since i hinges on he addiional posulae ha here are no rading adjusmens such as: axes, ransacions coss,