RISK-NEUTRAL VALUATION UNDER FUNDING COSTS AND COLLATERALIZATION

RISK-NEUTRAL VALUATION UNDER FUNDING COSTS AND COLLATERALIZATION Damiano Brigo Dep. of Mahemaics Imperial College London Andrea Pallavicini Dep. of Mahemaics Imperial College London Crisin Buescu Dep. of Mahemaics King s College London Marco Francischello Dep. of Mahemaics Imperial College London Marek Rukowski School of Mahemaics and Saisics Universiy of Sydney June 25, 216 Absrac The expeced cash flows approach leading o an exended version of he classical risk-neural valuaion formula was proposed in he recen works by Pallavicini e al. [6] and Brigo e al. [3] who sudied he problem of valuaion of conracs under differenial funding coss and collaeralizaion. The crucial difference beween he approach developed in [3, 6] and he presen noe is ha he pricing formula via condiional expecaion of discouned adjused cash flows is posulaed in [3, 6] and i is jusified using financial argumens, whereas in his noe he formula is esablished in more general se-up and i is shown o be a consequence of he sandard replicaion-based argumens. Keywords: risk-neural valuaion, hedging, funding coss, collaeral, margin agreemen Mahemaics Subjecs Classificaion 21: 91G4, 6J28 1

2 Brigo Buescu Francischello Pallavicini Rukowski 1 Simple Trading Model We firs provide an informal explanaion how he cash flows adjusmens are moivaed in [3, 6]. To show how he adjused cash flows originae, we assume ha he hedger buys a call opion on an equiy asse S T wih srike K. In oher words, he eners ino a conrac A, C where A = 1 {=T } S T K +. We assume ha collaeral is re-hypohecaed and, as in [6], we denoe he cash in he collaeral accoun a ime by C. When C > hen he cash collaeral is received as a guaranee and remuneraed by he hedger a he rae c b and when C is negaive, hen he cash collaeral is posed by he hedger and remuneraed by he counerpary a he rae c l. We analyze he hedger s operaions wih he reasury, he repo marke and he counerpary, in order o fund his rade. The following seps in each small inerval [, + d] are presened from he poin of view of he hedger buying he opion. Le us firs describe he hedger s iniial rades a ime : 1. The hedger wishes o buy a call opion wih mauriy T whose curren price is V <. The price is negaive for he hedger since he is he buyer and hus has o pay he price a ime. He borrows V cash from he reasury and buys he call opion. 2. To hedge he call opion he bough, he akes he synheic shor posiion ξ < in he sock in he repo marke. Specifically, he borrows H = ξ S from he reasury and lends cash a repo marke and ges ξ > shares of he sock as collaeral. 3. He immediaely sells he sock jus obained from he repo o he marke, geing back H in cash and gives H back o he reasury. Hence his ousanding deb a ime o he reasury is V. 4. He poss when C = C < or receives when C = C + > he cash collaeral, which can be rehypohecaed in oher ransacions. We now proceed o he descripion of he hedger s erminal rades a ime + d: 1. To close he repo, he hedger needs o buy and deliver ξ sock. For his rade, he needs ξ S +d cash and hus he borrows ha amoun of cash from he reasury. He buys ξ sock and gives he sock back o close he repo. Hence he ges back he cash H deposied a ime wih ineres h H d, ha is, 1 + h dh cash. Thus he ne value of hese rades is 1 + h dh + ξ S +d = 1 + h dξ S + ξ S +d = ξ ds h ξ S d. 1.1 2. He sells he call opion in he marke and hus ges V +d cash recall ha V +d < for he buyer of he opion. He pays back his ousanding deb V o he reasury plus ineres f V d using he cash V +d jus obained, he ne effec being V +d + V 1 + f d = dv + f V d. 1.2 3. He eiher pays he ineres c b C + d or receives he ineres c l C d on he collaeral amoun C and pays he ineres f C d o he reasury or receives he ineres f C + d from he reasury. Hence he ne cash flow a ime + d due o he margin accoun equals dĉ := c l f C d c b f C + d. 1.3 4. The oal value of cash flows from equaions 1.1 1.3 equals ξ ds h S d dv + f V d + dĉ = 1.4 where he equaliy is a consequence of he self-financing propery, which underpins he hedger s rades. Equaliies 1.3 and 1.4 will be formally esablished in Lemma 2.1.

Risk-neural valuaion under funding coss and collaeralizaion 3 Assume ha Q h is a probabiliy measure such ha E Q hds h S d F = or, more formally, ha he process S := B h 1 S is a local maringale under Q h where db h = h B h d wih B h = 1. Then we obain an explici represenaion under Q h for he price V for < T V = B f E Q h B f T 1 S T K + Bu f 1 dĉu F where db f = f B f d wih B f = 1. I is naural o refer o 1.5 as he exended risk-neural valuaion formula wih funding coss. To derive anoher version of he risk-neural valuaion formula, we now assume ha here exiss a risk-neural measure Q r associaed wih a locally risk-free bank accoun numeraire wih he risk-free ineres rae r, so ha E Q rds r S d F =. This is a formal posulae, which is saisfied in a ypical model, and i does no mean ha funding from he risk-free bank accoun is available o he hedger or ha he model a hand is arbirage-free in any sense. Then, using 1.4, we obain dv r V d = f r V d + ξ ds r S d h r H d + dĉ. Hence he price V admis also he following implici represenaion under Q r called he risk-naural valuaion formula wih adjused cash flows V =B r E Q r B r T 1 S T K + B r u 1 dĉu + + B r E Q r Bu r 1 h u r u H u du F where B r = r B r d wih B r = 1 and Ĉ equals see 1.3 Ĉ = c l u f u C u du Bu r 1 r u f u V u du F 1.5 1.6 c b u f u C + u du. 1.7 The formal derivaion of 1.5 and 1.6 is given in Proposiion 2.1. I is clear ha 1.5 and 1.6 are equivalen and hey reduce o he classic risk-neural valuaion formula when f = h = r. Noe ha 1.6 coincides wih he resul given earlier in [6]. I will be more formally derived in Corollary 2.1 in a more general framework. Observe ha he financial inerpreaion of he process r was employed in he derivaion of 1.6 given in [6], bu in fac i is no relevan a all for he validiy of 1.6. 2 Exended Risk-Neural Valuaion Formulae We fix a finie rading horizon dae T > for our model of he financial marke. Le Ω, F, F, P be a filered probabiliy space saisfying he usual condiions of righ-coninuiy and compleeness, where he filraion F = F [,T ] models he flow of informaion available o all raders. For convenience, we assume ha he iniial σ-field F is rivial. Moreover, all processes inroduced in wha follows are implicily assumed o be F-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. Traded risky asses. We denoe by S 1,..., S d he collecion of he prices of a family of d risky asses, which do no pay dividends. We assume ha he processes S 1, S 2,..., S d are coninuous semimaringalea.

4 Brigo Buescu Francischello Pallavicini Rukowski Treasury raes. The lending respecively, borrowing cash accoun B l respecively, B b is used for unsecured lending respecively, borrowing of cash from he reasury. When he borrowing and lending cash raes are equal, we denoe he single cash accoun by B f. We assume ha db l = fb l l d, db b = f b B b d and db f = f B f d. Repo marke. We denoe by B i,l respecively B i,b he lending respecively borrowing repo accoun associaed wih he ih risky asse. In he special case when B i,l = B i,b, we will use he noaion B i. We assume ha db i,l = h i,l B i,l d, db i,b = h i,b B i,b d and db i = h i B i d. 2.1 Valuaion in a Linear Model wih Funding Coss We will now examine a special case of he linear model wih funding coss inroduced in [1]. We assume ha we have he cash funding accoun B f and d risky asses raded in he repo marke wih he asse prices S i and he corresponding as funding accoun B i for i = 1, 2,..., d. Recall from [1] ha he rading consrain ψb i i + ξs i i = means ha he posiions in sock are funded using exclusively he accoun B i wih he repo rae h i. The value process of a rading sraegy ϕ hus equals ϕ := ψ f B f + ψb i i + ξs i i = F + where F sands for he funding par and H = d Hi represens he hedging par of he value process ϕ, specifically, F := ψ f B f + H i ψb i, i H i := ξs i i = ψb i. i Under he posulae ha ψ i B i + ξ i S i =, we also have ϕ = ψ f B f and hus ϕ coincides wih he cash funding process F f := ψ f B f. Assuming ha A represens he cash flows sream also known as he dividends sream of a given conrac, he self-financing condiion inclusive of he sream of cash flows A reads ϕ = ψ f db f + = ψ f db f ψ i db i + ξ i ds i + da ξs i B i i 1 db i + where he second equaliy is he consequence of he rading consrain. ξ i ds i + da Le us now consider he case of a collaeralized conrac A, C. The margin accoun C is assumed o be any adaped process of finie variaion such ha C T =. We do no consider C o be a par of he hedger s rading sraegy, bu raher as a par of cash flows of a conrac. This is moivaed by wo reasons. Firs, he margin accoun C will always be presen no maer wheher a given conrac is hedged or no. Second, he process C is assumed here o be exogenously given, so i does no depend on he hedger s rading sraegy. Therefore, he value process of a rading sraegy ϕ is sill defined as he sum F + H, raher han F + H + C. A he same ime, we assume ha he cash collaeral C is rehypohecaed, ha is, i is used for he hedger s rading purposes and hus i is implici in F and H hrough he self-financing condiion

Risk-neural valuaion under funding coss and collaeralizaion 5 saed in Definiion 2.1. As in [1], he process ϕ := F + H is he value process of he hedger s rading sraegy, whereas he process V ϕ := ϕ C = F +H C represens he hedger s wealh under rehypohecaion alhough he symbols F and H were no used in [1]. Due o he erminal condiion C T =, we always have ha V T ϕ = T ϕ, bu V ϕ ϕ for < T, in general. In he case of a collaeralized conrac wih he margin process C and remuneraion raes c l and c b for he margin accoun, o compue he price and hedge for a collaeralized conrac, i suffices o replace he cash flow sream A by he process A C given by he following expression A C := A + C + = A + C + Cu Bu c,l 1 dbu c,l c l uc u du C u + Bu c,b 1 dbu c,b c b uc + u du. 2.1 Hence he conrac A, C can be formally idenified wih he cash flows sream A C. 2.1.1 Dynamics of he Value Process of a Trading Sraegy We will now examine he dynamics of he value process of a self-financing rading sraegy inclusive of he cash flows sream A C and he concep of replicaion of he conrac A C. The following definiion summarizes hese noions. Definiion 2.1 Assume ha a conrac has cash flows given by a process A C of finie variaion and a replicaing sraegy ϕ for he cash flow sream A C exiss, meaning ha here exiss a rading sraegy ϕ such ha V T ϕ = T ϕ = where V ϕ = ϕ C and he process ϕ saisfies he self-financing condiion ϕ = ψ f db f ξs i B i i 1 db i + ξ i ds i + da C 2.2 Then he ex-dividend price equals π A C = V ϕ = ϕ C for all [, T ]. Since db i = h i B i d, he self-financing condiion can be represened as follows ϕ = f ϕ d + ξds i i h i S i d + da C. 2.3 We are in a posiion o formally esablish equaions 1.3 and 1.4, which were informally posulaed in Secion 1 Lemma 2.1 Assume ha a single sock S is raded and he conrac A, C has null cash flows A before ime T. Then he self-financing condiion reduces o he following equaliy for he hedger s wealh process V ϕ dv ϕ = f V ϕ d + ξ ds h S d + dĉ 2.4 where Ĉ is given by 1.3. Proof. Equaliy 2.4 is an immediae consequence of equaions 2.1 and 2.3 and he relaionship V ϕ = ϕ C. Remark 2.1 We have implicily assumed ha he iniial endowmen of he hedger equals zero. This assumpion can be made here wihou loss of generaliy, since we deal wih a linear model, bu i is no longer rue when dealing wih a non-linear se-up examined in Secion 2.2. Neverheless, for simpliciy of presenaion, we will sill assume in Secion 2.2 ha he iniial endowmen of he hedger equals zero.

6 Brigo Buescu Francischello Pallavicini Rukowski 2.1.2 Linear BSDE Recall ha replicaion of a conrac wih cash flows A C by a self-financing rading sraegy ϕ means ha V T ϕ = T ϕ =. Moreover, we assume ha ha he hedger s iniial endowmen is null. Using 2.3, we hus obain he following linear BSDE for he porfolio s value and hedging sraegy dy = f Y Zh i i S i d + Z i ds i + da C 2.5 wih he erminal condiion Y T =. Under mild echnical assumpions, he unique soluion o his BSDE exiss and i is given by an explici formula of course, provided ha he dynamics of S i are known. We hus henceforh assume ha he processes V ϕ and ϕ are well defined and we will examine heir properies. 2.1.3 Absrac Risk-Neural Valuaion Formulae Le us define noe ha, by convenion, we se B γi = 1 B γi := exp γu i du where γ i is an arbirary adaped and inegrable process. Le γ = γ 1, γ 2,..., γ d and le Q γ be a probabiliy measure such ha he process S i = B γi 1 S i is a Q γ local maringale. This is equivalen o he propery ha he process S i γ i us i u du is a Q γ local maringale for i = 1, 2,..., d, meaning ha Q γ is an equivalen local maringale measure ELMM for he asse price S i discouned wih he process B γi. Le ϕ be a self-financing rading sraegy, in he sense of Definiion 2.1, and le η be an arbirary adaped and inegrable process. We define B η := exp η u du. Lemma 2.2 Le η be an arbirary F-adaped process and le he process V η ϕ be given by V η ϕ := ϕ+b η α u F f u B η u 1 du+ B η β i uh i ub η u 1 du B η,] B η u 1 da C u. 2.6 If α = η f, β i = h i γ i, 2.7 hen he process V η := B η 1 V η is a local maringale under Q γ Proof. Equaion 2.6 implies ha dv η ϕ = ϕ + α F f d + β i H i d + V η η d da C.

Risk-neural valuaion under funding coss and collaeralizaion 7 Since = ψ f B f = F f, we obain dv η η V η d = ψ f db f = α + f η F f d + ξs i B i i 1 db i + β i h i + γh i i d + ξ i ds i + α F f d + ξ i ds i γs i i d. I is now clear ha if α and β saisfy 2.7, hen V η is a local maringale under Q γ. βh i i d η F f d The following resul is a simple consequence of Lemma 2.2. We sress ha he financial inerpreaion of processes η and γ i for i = 1, 2,..., d, if any, is no relevan in he derivaion of an absrac risk-neural valuaion formulae 2.8 2.9. Proposiion 2.1 Assume ha a conrac A, C can be replicaed by a rading sraegy ϕ and he associaed process V η ϕ is a rue maringale under Q γ. Then he ex-dividend price of A, C saisfies π A, C = V ϕ = ϕ C where ϕ = B η E Q γ Bu η 1 da C u + η u f u Fu f Bu η 1 du F 2.8 + B η E Q γ,t ] h i u γuh i ub i u η 1 du F or, equivalenly, V ϕ = B η E Q γ Bu η 1 da u + c u f u C u Bu η 1 du F,T ] + B η E Q γ η u f u V u ϕbu η 1 du + h i u γ u HuB i u η 1 du F where we denoe 2.9 c := c l 1 {C<} + c b 1 {C }. 2.1 Proof. Equaliy 2.8 is an immediae consequence of he maringale propery of V η ϕ and he equaliies T ϕ = and ϕ = F f. To derive 2.9 from 2.8, we noe ha F C = c u C u du and we apply he inegraion by pars formula and he equaliy C T =, o ge Bu η 1 dc u = C T B η T 1 C B η 1 C u dbu η 1,T ] This complees he derivaion of 2.9. = C B η 1 + η u Bu η 1 C u du. As special cases, we obain equaions 1.5 and 1.6. To ge 1.5, i suffices o se η = f and γ i = h i for i = 1, 2,..., d so ha ϕ = B f E Q h Bu f 1 da C u F.,T ]

8 Brigo Buescu Francischello Pallavicini Rukowski Similarly, upon seing η = γ i = r, we obain he following formula ϕ = B r E Q r Bu η 1 da C u + r u f u Fu f Bu r 1 du F,T ] + B r E Q r h i u r u HuB i u r 1 du F, which can be referred o as he risk-neural valuaion wih adjused cash flows. 2.2 Valuaion in a Non-Linear Model wih Funding Coss A non-linear exension of he framework inroduced in he preceding secion is obained when a single cash rae f is replaced by he lending and borrowing raes, denoed as f l and f b and, similarly, by inroducing differen repo raes for he long and shor posiions in sock, denoed as h i,l and h i,b, respecively. Then we have he following generic decomposiion of he value process ϕ of a rading sraegy ϕ where in urn ϕ = ψ l B l + ψ b B i,b + F := ψ l B l + ψ b B b + ψ i,l B i,l ψ i,l + ψ i,b B i,b + ξs i i = F + B i,l H i = F + H + ψ i,b B i,b, H i := ξs i. i Moreover, we posulae ha ψ l, ψ b and ψψ l b = for all. Finally, we assume ha ψ i,l, ψ i,b, ψ i,l ψ i,b = and he following repo funding condiion holds ψ i,l B i,l + ψ i,b B i,b + ξs i i =. Consequenly, ϕ = F f := ψ l B l + ψ b B b. As before, we se V ϕ = ϕ C where C is he margin accoun. 2.2.1 Dynamics of he Value Process of a Trading Sraegy We are now in a posiion o derive he non-linear dynamics of he value process and hus also obain a generic non-linear pricing BSDE for he conrac A, C. Lemma 2.3 We have ψ l = B l 1 ϕ +, ψ b = B b 1 ϕ and ψ i,l Proof. Noe ha = B i,l 1 ξs i i = B i,l 1 H i, ψ i,b ψ l B l + ψ b B b = ϕ, ψ i,l B i,l = B i,b 1 ξs i i + = B i,b 1 H i +. + ψ i,b B i,b = ξs i. i I hus suffices o use he posulaed naural condiions ψ l, ψ b, ψψ l b = and ψ i,l, ψ i,b, ψ i,l ψ i,b =.

Risk-neural valuaion under funding coss and collaeralizaion 9 Using Lemma 2.3, we obain he unique dynamics for he value process of a self-financing rading sraegy ϕ inclusive of cash flows sream A C. We posulae ha and hus ϕ = ψ l db l + ψ b db b + ψ i,l db i,l ϕ = B l 1 ϕ + db l B b 1 ϕ db b + B i,b 1 ξ i S i + db i,b + + ψ i,b db i,b + ξ i ds i + da C 2.11 ξ i ds i + da C. B i,l 1 ξ i S i db i,l When all accoun processes are absoluely coninuous, we obain he following lemma. Lemma 2.4 The value process of a self-financing rading sraegy ϕ saisfies ϕ = f l ϕ + d f b ϕ d+ where H i = ξ i S i. ξ i ds i +h i,l H i d h i,b H i + d +da C. 2.12 To formally linearize he problem, we will now inroduce he effecive raes f and h i, which depend on he hedger s rading sraegy. Lemma 2.5 The value process ϕ saisfies ϕ = f F f d + where he effecive raes f and h i are given by ξ i ds i h i S i d + da C 2.13 f := f l 1 {F f } + f b 1 {F f ϕ<} = f l 1 {Vϕ+C } + f b 1 {Vϕ+C <} 2.14 and h i := h i,l 1 {H i } + h i,b 1 {H i ϕ>}. 2.15 Proof. From 2.12, we obain ϕ = f l ϕ + d f b ϕ d + = f F f ϕ d + ξ i ds i + h i,l H i d h i,b H i + d + da C ξ i ds i h i S i d + da C 2.16 where f and h i are given by 2.14 and 2.15, respecively.

1 Brigo Buescu Francischello Pallavicini Rukowski 2.2.2 Nonlinear BSDE Under he sanding assumpion ha he iniial endowmen of he hedger is null, equaion 2.16 leads o he following nonlinear BSDE for he porfolio s value and he hedging sraegy dy = f Y h i ZS i i d + wih he erminal condiion Y T = where from 2.14 2.15 Z i ds i + da C 2.17 f := f l 1 {Y } + f b 1 {Y<} and h i := h i,l 1 {Z i S i } + h i,b 1 {Z i S i>}. We henceforh assume ha BSDE 2.17 has a unique soluion in a suiable space of sochasic processes. 2.2.3 Absrac Risk-Neural Valuaion Formulae Recall ha and B η := exp η u du B γi := exp γu i du where η and γ i for i = 1, 2,..., d are arbirary adaped and inegrable processes. Le Q γ be a probabiliy measure such ha he processes B γi 1 S i, i = 1, 2,..., d are Q γ -local maringales. Then Lemmas 2.2 and 2.5 yield he following resul, which exends Proposiion 2.1 and agrees wih he pricing formula posulaed in [3]. For he validiy of his resul, one needs o impose some mild inegrabiliy assumpions. Proposiion 2.2 Assume ha a collaeralized conrac A, C can be replicaed by a rading sraegy ϕ. Then is ex-dividend price a ime equals V ϕ = ϕ C where ϕ = B η E Q γ Bu η 1 da C u + η u f u Fu f Bu η 1 du F + B η E Q γ,t ] h i u γuh i ub i u η 1 du F or, equivalenly, V ϕ = B η E Q γ Bu η 1 da u + c u f u C u Bu η 1 du F 2.18,T ] + B η E Q γ η u f u V u ϕbu η 1 du + h i u γuh i ub i u η 1 du F where c := c l 1 {C<} + c b 1 {C }. Proof. The firs assered formula follows from 2.16. The second is easy consequences of he firs one.

Risk-neural valuaion under funding coss and collaeralizaion 11 References [1] Bielecki, T. R. and Rukowski, M.: Valuaion and hedging of conracs wih funding coss and collaeralizaion. SIAM Journal on Financial Mahemaics 6 215, 594 655. [2] Brigo, D., Liu, Q., Pallavicini, A., and Sloh, D.: Nonlineariy valuaion adjusmen. In: Grbac, Z., Glau, K, Scherer, M., and Zags, R. Eds, Innovaion in Derivaives Markes Fixed Income Modeling, Valuaion Adjusmens, Risk Managemen, and Regulaion. Springer, 216. [3] Brigo, D., Francischello, M., and Pallavicini, A.: Analysis of nonlinear valuaion equaions under credi and funding effecs. In: Grbac, Z., Glau, K, Scherer, M., and Zags, R. Eds, Innovaions in Derivaives Markes Fixed Income Modeling, Valuaion Adjusmens, Risk Managemen, and Regulaion. Springer, Heidelberg, 216. [4] Brigo, D. and Pallavicini A.: Nonlinear consisen valuaion of CCP cleared or CSA bilaeral rades wih iniial margins under credi, funding and wrong-way risks. Journal of Financial Engineering 11 214, 1 6. [5] Burgard, C. and Kjaer, M.: Parial differenial equaion represenaions of derivaives wih bilaeral counerpary risk and funding coss. The Journal of Credi Risk 73 211, 75 93. [6] Pallavicini, A., Perini, D. and Brigo, D.: Funding, collaeral and hedging: uncovering he mechanics and he subleies of funding valuaion adjusmens. Working paper, December 13, 212.