VALUATION OF OVER-THE-COUNTER DERIVATIVES WITH COLLATERALIZATION

Transcription

1 VALUAION OF OVER-HE-COUNER DERIVAIVES WIH COLLAERALIZAION By L. Guerrero, S. Jin, H. Koepke, E. Negahdary, B. Wang,. Zhao, Menor: Dr. Richie He IMA Preprin Series #243 (June 212) INSIUE FOR MAHEMAICS AND IS APPLICAIONS UNIVERSIY OF MINNESOA 4 Lind Hall 27 Church Sree S.E. Minneapolis, Minnesoa Phone: Fax: URL: hp://

2 Mahemaical Modeling in Indusry Workshop XVI Final Progress Repor eam 7: Valuaion of Over-he-Couner Derivaives wih Collaeralizaion L. Guerrero, S. Jin, H. Koepke, E. Negahdary, B. Wang,. Zhao Menor: Dr. Richie He (Royal Bank of Canada) June 29, 212 1

3 Absrac In his repor we ry o derive formulas for prices of financial derivaives wih collaeralizaion. We firs review approaches in 3 and 2. hen we verify ha he formula in Pierbarg 1 wih single currency and single asses. Laer we use he same framework o generalize he resuls o cross currency and muli-asses models. he repor is a resul of eam 7 in he Mahemaical Modeling in Indusry Workshop XVI held a he Universiy of Calgary in June Inroducion Collaeralized OC (over-he-couner) derivaives have grown rapidly over he las decade. According o he ISDA (Inernaional Swaps and Derivaives Associaion) Margin Survey 4, a he end of 29 7% of he rade volumn of OC derivaives were collaeralized, in comparison wih 3% in 23. he percenage is expeced o grow even faser due o he impac of he financial crisis and he growing aenion o credi risk. he collaeralizaion is usually done by a so-called credi suppor annex (CSA), which defines all deailed erms of he ransacion. Collaeralizaion significanly reduces he credi risk for he pary wih negaive derivaive value. Bu in he framework of his repor we don consider credi risk when deermine he derivaive price. We only consider he difference of he prices of collaeralized derivaive and non-collaeralized derivaive brough by cos associaed wih collaeralizaion. We assume he collaeral accoun C() is coninuously adjused. hus he cos associaed wih collaeralizaion is unknown unless C() is known. ypically C() is proporional o he value V () of he derivaive. So once V () is known, he dynamics of C() can be found. he framework we are using in he repor are adoped from Pierbarg 1, which considers consrucing a replicaing porfolio including he underlying, a collaeral accoun and a cash accoun. We ry o exend he resuls o cross currency and muli-asses models. Alhough here is a rend o use sandardized CSA agreemens, due o he OC naure of he derivaives we are considering in he paper, he specific erms of collaeralizaion may vary from case o case. he repor inends o cover a number of cases for how collaeral accouns are mainained by he wo couner paries or possibly a hird pary. he repor is organized as follows: Secion 2 gives an overview of Lieraure. Subsecion 2.1 gives an comparison of he hree paper Pierbarg 1, Casagana 2

4 3, and Fujii e al 2 (Wrien by ong). Subsecion 2.2 review he discree approach in Casagna 3 (wrien by Henrike). Subsecion 2.3 review he maringale approach in Fujii e al 2 (wrien by Henrike). Secion 3 presens our approach under Pierbarg s framework. Subsecion 3.1 review Pierbarg 1 and verifies resuls in he paper (wrien by Binbin). Subsecion 3.2 uses he same framework o obain formulas for cross currency cases (wrien by Binbin). Subsecion 3.3 deals wih muli-asses & single currency case (wrien by Shi), and Subsecion 3.4 deals wih muli-asses & cross currency case (wrien by Shi). Secion 4 gives anoher idea of pricing collaeralized derivaives (wrien by ong). 2 Overview of lieraure 2.1 Comparison of lieraure here are hree major approaches for he valuaion model of collaeralized derivaive: Vladimir Pierbarg used replicaion of porfolio o derive he mos general soluion; Anonio Casagna made he assumpion for parial collaeralized siuaion and gave he dynamic model for collaeral accoun; Massaki Fujii considered he self-financing fac and include reinvesmen facor ino his collaeral dynamic accoun. All of he above papers coincide ino several imporan resuls, for example, he equaion relaing V C Adjusmen: where V C LV A = E, V NC and LVA Liquidiy Value = V NC + LV A (1) rudu (r s c s )C s ds here r is he risk free marke rae, and c is he collaeral reurn ineres, which means ha he derivaive buyer(collaeral poser) will ge periodic ineres reurn from derivaive seller(collaeral holder) and he reurn rae is defined as c. Even hrough he hree auhors shared a lo of grea ideas, here are sill some disincions among heir specific work. We can summarize hem as follows: Pierbarg paper is he mos general paper. I conains barely no assumpion, which is favorable because of he solid cash flow replicaion: dα() = r C ()C() + r F ()(V () C()) r R () ()S() + r D () ()S()d (2) 3

5 Casagna paper is popular in financial marke because of he nice dynamic equaion for collaeral accoun. Bu i is no flawless, when applied o parial collaeral siuaions i is difficul o explain he inconsisency beween parial collaeral consrain and he differenial equaion of collaeral dynamic: dc = c C d + αdv C = αv Fujii e al paper is also a widely discussed paper, i clarifies he cash flow wihin he collaeral accoun, especially by indicaing ha he ne reurn rae for collaeral holder is equal o he difference of reinvesmen rae(risk free marke rae r ) and collaeral ineres reurn c. Bu he drawback for Fujii e al paper is ha he uses a differen collaeral dynamic from Casagna paper bu leaves a gap beween he dynamic and he conclusion. dc = (r c )C d + a()dv a() = e s (ru cu)du Laer on in our research work, we compleed he assumpion of parial collaeral and applied o Pierbarg and Fujii e al papers. he final conclusions coincide ino Casagna resul. V C = E d rs(1 α)+csα dsv 2.2 Discree Approach of Pricing OC wih Collaeralizaion Anonio Casagna presens in? an approach o price derivaives wih collaeralizaion using binomial ree models. For his approach we inroduce he following parameers: S is he underlying asse a ime. I eiher goes up o Su or goes down o Sd (noe ha u > 1 and d < 1). V C is he price of he collaeralized coningen. If he underlying asse goes up we obain he value Vu C for he price of he collaeral coningen and similarly if i goes down we obain he value V C d. 4

6 C is he value of he cash collaeral accoun. he collaeral rae is given by c (he ineres one earns on he collaeral accoun). he collaeral facor is given by γ. collaeralized. Noe if γ = 1 he derivaive is fully B is he value in he Bank accoun which earns a each period he risk-free ineres r. In order o find he arbirage free collaeralized pricing formula we will use he following replicaion sraegy: Vu C C(1 + c) = αsu + βb(1 + r) (3) Vd C C(1 + c) = αsd + βb(1 + r). (4) Wih he help of Equaion (3) and (4) we obain he following represenaion for he coefficiens α and β: α = V u C Vd C S(u d) β = uv d C dv C u (1 + c)c(u d) (u d)b(1 + r) If we are able o replicae he pay-off of he collaeralized coningen, hen a ime = we are receiving he arbirage-free price of V C. Hence: V C C = αs + βb = V u C Vd C (u d) + uv d C (5) (6) dv u C (1 + c)c(u d). (7) (u d)(1 + r) For he nex sep we assume ha we allow parial collaeralizaion, hus wih V C = γc, we can rewrie Equaion (7) as follows: V C γv C = (1 γ)v C c 1 + r γv C = ( (1 + r)(1 γ) + (1 + c)γ 1 + r (1 + r)(v C u (1 + r)(v C u ) V C = r 5 Vd C) + uv d C Vd C) + uv d C (u d)(1 + r) u 1 r u d } {{ } =p V C d dv C u (1 + c)γv C (u d) (u d)(1 + r) dv C u r d u d } {{ } =1 p V C u

7 Hence wih some simplificaion we obain he following pricing formula for collaeral coningens: V C = 1 ( pv C (1 + r)(1 γ) + (1 + c)γ u ) + (1 p)vd C. (8) he pricing formula of a non-collaeralized agreemen is given by: ( ) (1 + r)(1 γ) + (1 + c)γ V C = 1 ( ) pv C 1 + r 1 + r u + (1 p)vd C. (9) Hence by simplifying he lef hand side of Equaion (9) we have: ( ) (1 + r)(1 γ) + (1 + c)γ V NC = V C = V C γ r c 1 + r 1 + r V C. We define he las erm LV A = γ r c 1 + r V C as Liquidiy Value Adjusmen. herefore he collaeralized coningen minus he Liquidiy Value Adjusmen gives us he price of he non-collaeralized coningen. Noe ha he price of he non-collaeral coningen is he expeced reurn under he risk neural measure of an uncollaeralized derivaive. 2.3 Pricing Collaeralized Derivaives by Fujii e al Single Currency Case In his secion we are reviewing wo approaches by Fujii e al 2 for fully collaeralized derivaives. In he firs approach a new dynamic o ge he collaeralized pricing formula is inroduced, in he second approach he collaeralized price is derived from he well known LVA-formula. Fujii e al 2 inroduces he following dynamics on he cash collaeral accoun C() wih a self-financing sraegy: dc(s) = y(s)c(s)ds + a(s)dv (s), (1) where y(s) = r(s) c(s) is he difference of he risk-free ineres rae and he collaeral rae and V (s) denoes he value of a derivaive ha maures a wih cash flow V ( ) a ime s. his dynamic represens he perspecive of he borrower and he lender. he borrower of he derivaives provides he cash collaeral o he lender, he lender receives he risk-free ineres r bu has o pay he collaeral 6

8 rae c o he borrower. Hence he lender receives y(s) := r(s) c(s). By he selffinancing sraegy he lender invess he gain from he ineres raes o provide he buyer wih a(s) new derivaives. Equaion (1) can be solved by he following inegraion: dc(s) y(s)c(s)ds = a(s)dv (s) (e ) s y(u)du (dc(s) y(s)c(s)ds) = (e ) s y(u)du a(s)dv (s) d (e ) s y(u)du C(s) = (e ) s y(u)du a(s)dv (s) e y(u)du C( ) e y(u)du C() = y(u)du a(s)dv (s) C( ) = e y(u)du ( e y(u)du C() + C( ) = e y(u)du C() + e s y(u)du a(s)dv (s). We are considering he following rading sraegy: C() = V () (11) a(s) = e s y(u)du. (12) o model he amoun of addiional derivaives he borrower receives by is lender we use Equaion (12). By subsiuing Equaions (11) and (12) in we have: C( ) = e y(u)du C() + e s y(u)du a(s)dv (s) C( ) = e y(u)du V () + e s y(u)du e s y(u)du dv (s). his Equaion can be simplified by inegraion mehods as follows: C( ) = e y(u)du V () + = e = e ( y(u)du V () + e y(u)du dv (s) ) dv (s) y(u)du (V () + V ( ) V ()) = e y(u)du V ( ). ) y(u)du a(s)dv (s) 7

9 In order o obain he risk-neural collaeralized pricing formula he equaion above is discouned by he risk-free rae r: V () = E Q e r(u)du e y(u)du V ( ) = E Q e (r(s) y(s))ds V ( ) = E Q e c(s)ds V ( ). Noe ha in 2 E Q denoes he expecaion under he money-marke accoun as a numeraire. Fujii e al 2 derived wih a differen approach he same resul as saed above. his derivaion involves he LVA-Formula as saed in?: V C = V NC + LV A. (13) From arbirage heory we know ha he risk-neural pricing formula for a noncollaeral agreemen is presened by he following formula: V NC () = E Q e r(s)ds V ( ). he following LVA-formula for he coninuous case was derived in?: LV A = E Q Combining hese wo resuls we have: V () = E Q = E Q Define he following process: e r(s)ds V ( ) + E Q r(u)du y(s)v (s)ds. r(u)du y(s)v (s)ds e r(s)ds V ( ) + r(u)du y(s)v (s)ds X() = e r(s)ds V () +. r(u)du y(s)v (s)ds. (14) In order o show ha he process X() is a Q-maringale i should hold ha for τ E Q X() F τ = X(τ). 8

10 Wih he calculaion below we obain he following resul: E Q X() F τ =E Q e r(s)ds V () + r(u)du y(s)v (s)ds F τ =E Q e r(s)ds (E Q + E Q e r(s)ds V ( ) r(u)du y(s)v (s)ds F τ =E Q e r(s)ds ( e r(s)ds V ( ) + + E Q r(u)du y(s)v (s)ds F τ ( =E Q e r(s)ds V ( ) + =E Q e r(s)ds V ( ) + =E Q X( ) F τ. Similarly we can show ha + E Q ) r(u)du y(s)v (s)ds F τ ) r(u)du y(s)v (s)ds F τ ) r(u)du y(s)v (s)ds + r(u)du y(s)v (s)ds F τ E Q X( ) F τ = E Q X(τ) F τ. Hence in he end we obain ha X() is a maringale as: E Q X() F τ = E Q X( ) F τ = E Q X(τ) F τ = X(τ). By differeniaion of X() wih he help of Iô s Lemma we have: dx() = e r(s)ds ( r()v ()d + dv ()) + e r(s)ds y()v ()d dx() = e r(s)ds (y() r()) V ()d + e r(s)ds dv () Wih rearrangemens of he facors we have: dv () = (r() y()) V ()d + e r(s)ds dx(), }{{} :=dm() r(u)du y(s)v (s)ds F τ 9

11 where dm() is a maringale. herefore he dynamics of he collaeralized derivaive value can be expressed as a sochasic differenial equaion wih maringale dm(): dv () = (r() y()) V ()d + dm(). o ge he collaeralized pricing formula we use he following derivaion: dv (s) c(s)v (s)ds = dm(s) (e ) s c(u)du (dv (s) c(s)v (s)ds) = (e ) s c(u)du dm(s) d (e ) s c(u)du V (s) = (e ) s c(u)du dm(s) e c(u)du V ( ) e c(u)du V () = c(u)du dm(s) V () = e c(u)du ( e c(u)du V ( ) V () = e c(u)du V () c(u)du dm(s). ) c(u)du dm(s) Applying he condiional expecaion o he equaion above we ge he final collaeralized pricing formula: E V () = E e c(u)du V () c(u)du dm(s) V () = E e c(u)du V (). Exension o Parially Collaeralized Derivaives We will know exend Fujii e al 2 approach o parial collaeralized derivaives. We are herefore considering he well known LVA-formula in? ha allows parial collaeralizaion (noe: < γ < 1): V = E e r(u)du V ( ) + γ r(v)dv (r(s) c(s)) V (s)ds. }{{} :=y(s) Similarly as in he fully collaeral case, we define he following process: X() = e r(s)ds V () + γ 1 r(u)du y(s)v (s)ds,

12 where we show by he same argumenaion ha X() is a maringale. By differeniaion of X() using Iô s Lemma we obain: dx() = e r(s)ds r()v ()d + e r(s)ds dv () + γe r(s)ds y()v ()d. Defining dm() = e r(s)ds dx() we derive he following sochasic process: dm() = r()v ()d + dv () + γ(r() c())v ()d dv () = ((1 γ)r() + γc()) V ()d + dm() wih he maringale dm(). Inegraing his sochasic process and aking he condiional expecaion on we derive he resul: V () = E e (1 γ)r(s)+γc(s)ds V which coincides wih he resuls presened in?. Cross-Currency Case Suppose he underlying and he derivaive are in domesic currency i bu he collaeral is posed in he foreign currency j. Also suppose f x (i,j) () is he foreign exchange rae a ime represening he price of he uni amoun of currency j in erms of currency i. hen he collaeralized derivaive price wih collaeral C (i) () = V (i) () is given by 2 V (i) () =E Q i e r (i)(s)ds V (i) ( ) ( ) + f x (i,j) ()E Q j r(j) (u)du y (j) C (i) () (s) ds, f x (i,j) () where E Q i is he condiional expecaion under risk neural measure of domesic currency i and E Q j is he condiional expecaion under risk neural measure of foreign currency j. Change all he measure o Q i, Fujii e al 2 obained Define V (i) () = E Q i e r (i)(s)ds V (i) ( ) + r(i) (u)du y (j) (s)v (i) (s)ds. X() = e r(i) (s)ds V (i) () + 11 r(i) (u)du y (j) (s)v (i) (s)ds.

13 Using he argumen similar as above single currency case, we can show X() is a Q i maringale. Le < τ, hen E Q i X() F τ =E Q e r(i) (s)ds V (i) () + r(i) (u)du y (j) (s)c (i) (s)ds F τ ( =E Q i e r(i) (s)ds E Q i e r (i)(s)ds V (i) ( ) + + E Q i r(i) (u)du y (j) (s)v (i) (s)ds F τ =E Q i + E Q i =E Q i e r(i) (s)ds V (i) ( ) + r(i) (u)du y (j) (s)v (i) (s)ds F τ e r(i) (s)ds V (i) ( ) + =E Q i X( ) F τ. r(i) (u)du y (j) (s)v (i) (s)ds r(i) (u)du y (j) (s)v (i) (s)ds F τ hus, i is obvious ha E Q i X() F τ = E Q i X(τ) F τ = X(τ) and herefore X() is a Q i maringale. By Io s lemma, dx() = e r(i) (s)ds (y (j) () r (i) ())V (i) ()d + dv (i) (). Le dm() = e r(i) (s)ds dx(), hen M() is also a Q i maringale and i follows ha dv () = (r (i) () y (j) ())V (i) ()d + dm(). herefore, e s (r(i) (u) y (j)(u))du V () is also a Q i maringale. Hence, e s (r(i) (u) y (j)(u))du V (i) () = E Q i e s (r(i) (u) y (j)(u))du V (i) ( ) ) r(i) (u)du y (j) (s)v (i) (s)ds F τ hen, V (i) () = E Q i e r (i) (s)ds (e ) y (j)(s)ds V (i) ( ). 3 Our approach under Pierbarg s framework 3.1 Single currency & Single asse In his secion we review Pierbarg s 1 general seing for solving he pricing problem. he main resuls are verified wih deailed proof. he same seing are 12

14 used in he cross currency and muli-asses cases in Secion 3.2, Secion 3.3, and Secion 3.4. Le V () = V (, S ) be he ime price of a collaeralized derivaive wrien on an asse wih price dynamics ds /S = µ d + σ dw in he real world probabiliy measure. hen by Io s formula, he dynamics of V () is V dv () = V 2 σ2 S d + ()ds S 2 (15) where () = V S o replicae he derivaive we consider a self-financing porfolio wih a collaeral accoun C(), () unis of he underlying asse, and a cash accoun. Le γ() be he sum of he collaeral accoun and he cash accoun, hen he growh of γ() are deermined by each of following raes: he collaeral rae r C () for he collaeral accoun C(); he repo rae r R () used o borrow/lend ()S o purchase () unis of he underlying; he ineres rae r F () used o borrow/lend he res of he cash V () C(); he sock pays dividend a a rae r D () Remark In he view poin of a bank, say bank A, if he oher pary are required o pos collaeral o he bank, hen he bank mus pay ineres a rae rc A () o he collaeral accoun bu can also inves he collaeral C() o ge an invesmen rae rv A () (or oher discouning rae he bank use for inernal purpose). Hence he ineres rae r C () need o be replaced by rc A() ra v (). If he bank has o pos collaeral o an accoun under his own bank, he sory in he same. Bu if he oher pary is anoher bank, say bank B, hen he rae we should use is rc B() ra v (). However we don consider such complicaion in his repor. Hence he growh of γ() is dγ() = r C ()C() r R () ()S + r F ()(V () C()) + r D () ()S d (16) 13

15 he value of he porfolio ia X() = ()S + γ(), by he self-financing assumpion we have hence by Equaion (15) we have dx() = ()ds + dγ(), dγ() = Comparing equaion (16) and (17) we ge V V 2 σ2 S d (17) S 2 V σ2 S 2 V S 2 = r C()C() + r F ()(V () C()) + (r D () r R ()) V S S which can be rearranged as V + (r R() r D ()) V S S σ2 S 2 V S 2 = r F ()V () (r F () r C ())C() (18) his parial deferenial can be solved using he Feynman-Kac heorem. o remind he reader we sae he Feynman-Kac heorem. heorem 3.1 (Feynman-Kac heorem) Consider he PDE u + µ(x, ) u x σ2 (x, ) 2 u V (x, )u + f(x, ) = x2 defined for all real x and in he inerval,, subjec o he erminal condiion u(x, ) = ψ(x), where µ, σ, ψ, V are known funcions, is a parameer and u : R, R is he unknown. hen he soluion can be wrien as an condiional expecaion as follows: u(x, ) = E V (Xτ ) dτ f(x s, s)ds + e V (X τ ) dτ ψ(x ) X = x where X is an Iô process driven by he equaion dx = µ(x, ) d + σ(x, ) dw, where W () is a Brownian moion and he iniial condiion for X() is X() = x. 14

16 herefore by applying he Feynman-Kac heorem we ge he soluion of he PDE (18): V () = E Q e r F (u)du V + e u r F (v)dv (r F (u) r C (u))c(u)du where Q is he measure under which he underlying price dynamics is ds /S = (r R () r D ())d + σ d W By rearrange he erms in he PDE (18) one can ge he soluion in a differen form: V () = E Q e r C (u)du V + e u r F (v)dv (r F (u) r C (u))(v (u) C(u))du (2) Now seing C = in equaion (19) we ge he classic resul V () = E Q e r F (u)du V. wih no collaeral. Seing V = C in equaion (2) we ge he formula for he derivaive price wih full collaeral: V () = E Q e r C (u)du V. While C() usually depends on V (), equaion (19) and equaion (2) are no of pracical use when in he imperfec collaeral cases. Bu if we assume C() = αv () we can wrie he PDE (18) as (19) V + (r R() r D ()) V S S σ2 S 2 V S 2 = r F () α(r F () r C ()) V () and again by he Feynman-Kac heorem we obain an imporan formula V () = E Q e (1 α)r F (u)+αr C (u)du V which gives he derivaive price for all ype of collaeralizaion wih C() = αv () for α 1. I is consisen wih he formulas for he no collaeral case and full collaeral case. 15 (21)

17 3.2 Cross currency & Single asse In his secion we consider ha he wo couner paries are in differen counries hence use differen currency. his complicaes he problem bu i sill can be solved using he same framework in Secion 3.1. Assuming he wo currencies are he domesic currency D and foreign currency F, and pary A (domesic) and B (foreign) are involved in he derivaive rading. he exchange rae is given by r df, i.e. M f in foreign currency is worh M d = r df M f in domesic currency. he underlying is dominaed in he domesic currency. We consider he following wo cases: Case 1: he collaeral are posed in domesic currency D Case 2: he collaeral are posed in foreign currency F Case 3: he collaeral are posed in in domesic currency D when V < and in foreign currency F when V > For Case 1, sanding on he view poin of A, i is exacly he same case as a single currency case in Secion 3.1, hence he same formulas in Secion 3.1 apply. For Case 2, le C f () be he collaeral accoun. hen he collaeral accoun is worh C() = r df ()C f () in domesic currency. he growh of he cash accouns are slighly differen han he single currency case. I is deermined by he following: he collaeral rae r f C () for he collaeral accoun C f() in foreign currency, which is C() in domesic currency; he ineres rae r f F () used o creae a foreign cash accoun C f(); he res of cash is V () S, bu he porion S can be borrowed/lend a a secured repo rae r R (), and V () is borrowed a a unsecured rae r F () he sock pays dividend a a rae r D () he growh of γ() is given by dγ() = (r f C () rf F ())C() r R() ()S + r F ()V () + r D () ()S d (22) Noe ha he C() is acually r df ()C f (). he difference o he single currency case ha ha he coefficiens for C() are replaced by he foreign counerpar as i is in foreign currency. 16

18 hen we can use (22) o ge he PDE for V () and apply Feynman Kac heorem o ge he following soluion: V () = E Q e r F (u)du V + e u r F (v)dv (r f F (u) rf C (u))c(u)du he measure Q is he same measure in Secion 3.1. Again if we assume C() = αv (), we ge a formula similar o equaion 21: V () = E Q e α(rf C (u) rf F (u))+r F (u)du V (24) For Case 3 we can ge similar formulas like equaion (23) and (24) bu he r f C () has o be replaced by r C()1 V< + r f C ()1 V >. Bu since he sign of V is unknown, hese formulas are no of pracical use. 3.3 Muli-asse Model for single currency Le S = (S (1),..., S (d) ) be he d-dimensional vecor of underlyings, where he ih underlying dynamic is (23) ds (i) S (i) = µ (i) d + j=1 σ (i,j) dw (j), i = 1, 2,..., d where he measure Q such ha W (i) is Wiener process and he corresponding drif µ (i) will be specified laer for all i = 1,..., d. Denoe by V a derivaive wih underlying S. hen by Io s lemma, where L = dv = (LV )d + j=1 σ (i,j) (i) ds (i). S (i) S (j) S (i) 2 S (j) o replicae he derivaive, a ime we hold (i) unis of he i-h underlying and γ amoun of cash. he value of he replicaion porfolio Π() is hen. V () = Π() = (i) S (i) + γ 17

19 he cash amoun γ is spli among a number of accouns: adds in he growh of all cash accouns is given by: dγ = r C ()C() + r F ()(V () C()) r R () (i) (i) S (i) + r D () (i) (i) S (i) d By he self-financing condiion, hus we have ( + dγ = dv () ( rr () (i) r D () (i)) S (i) S (i) (i) ds (i) = (LV ())d j=1 σ (i,j) S (i) S (j) 2 S (i) S (j) ) V () = r F ()V () + (r C () r F ())C() he soluion is obained by Feynman-Kac formula as V () = E Q e r F (u)du V ( ) + e u r F (v)dv (r F (u) r C (u))c(u)du in he measure Q such ha he ih underlying asse has a drif µ (i) = r R () (i) r D () (i) and furhermore, he ih underlying asse has a dynamic in Q as: ds (i) S (i) = ( r R () (i) r D () (i)) d + Rewrie he above PDE in he form of ( + ( rr () (i) r D () (i)) S (i) S (i) j= σ (i,j) dw (j), i = 1, 2,..., d. j=1 σ (i,j) S (i) S (j) 2 S (i) S (j) ) V () = r C ()C() + r F ()(V () C()) and use he proof of Feynmac-Kac heorem, we can obain anoher useful formula V () =E Q e r C (u)du V ( ) E Q e u rc(v)dv (r F (u) r C (u))(v (u) C(u))du 18

20 Le C() = αv (), we obain a pracical formula: V () = E Q e r F (u)(1 α)+r C (u)αdu V ( ). 3.4 Muli-asse Model for cross currency Le S = (S (1),..., S (d) ) be he d-dimensional vecor of underlyings, where he ih underlying dynamic is ds (i) S (i) = µ (i) d + j=1 σ (i,j) dw (j), i = 1, 2,..., d where he measure Q such ha W (i) is Wiener process and he corresponding drif µ (i) will be specified laer for all i = 1,..., d. Denoe by V a derivaive wih underlying S. hen by Io s lemma, dv = (LV )d + (i) ds (i). where L = j=1 σ (i,j) S (i) S (j) S (i) 2 S (j) o replicae he derivaive, a ime we hold (i) unis of he i-h underlying and γ amoun of cash. he value of he replicaion porfolio Π() is hen. V () = Π() = (i) S (i) + γ he cash amoun γ is spli among a number of accouns: adds in he growh of all cash accouns is given by: dγ = r f C ()C() + r F ()V () r f F ()C() (r R ()) (i) (i) S (i) + (r D ()) (i) (i) S (i) d 19

21 By he self-financing condiion, hus we have ( + dγ = dv () ( (rr ()) (i) (r D ()) (i)) S (i) (i) ds (i) = (LV ())d S (i) he soluion is obained by Feynman-Kac formula as e r F (u)du + V () = E Q i j=1 σ (i,j) S (i) S (j) 2 S (i) S (j) ) V () = r f C ()C() + r F ()V () r f F ()C() e u r F (v)dv (r f F (u) rf C ())C(u)du in he measure Q such ha he ih underlying asse has a drif µ (i) = r R () (i) r D () (i) and furhermore, he ih underlying asse has a dynamic in Q as: ds (i) S (i) = ( r R () (i) r D () (i)) d + j=1 σ (i,j) dw (j), i = 1, 2,..., d. Le C() = αv (), we obain a pracical formula: V () = E Q e r F (u) (r f F (u) rf C (u))αdu V ( ). 4 Ye anoher idea Here we provide anoher approach o he final valuaion model. o clarify he dynamic and eliminae he possibiliy of inroducing any conflicion, we use he porfolio replicaion framework and only one collaeral dynamic modified from Fujii e al s paper. For convenience we will no consider he underlying dividend in his case. ds() = r S()d + σs()dz (25) dv = L r V d + σs() V dz (26) S() 2

22 where LȧV = V V + ȧs S + σ2 S 2 2 V 2 S 2 and on he oher hand, for he replicaed porfolio: dc = αdv + (r c )C d (27) db = rb d (28) se hence specifically, V C = αs() + B (29) dv dc = αds() + db (3) RHS = αr S()d + ασs()dz + r B d LHS = (1 α)dv (r c )C d o eliminae dz, se = (1 α)l r V d + (1 α)σs() V S ()dz (r c )C d he he PDE urns ou o be, afer simplificaion By Feynman-Kac formula, L r V + α = (1 α) V () S (31) B = V C αs() (32) c 1 α C r 1 α V = V C = E ru 1 α du c s 1 α C sds + e ru 1 α du V (33) If we define he discouned reurn: r = r 1 α 21

23 c = x 1 α hen he above resul coincide Paper one resul = E r udu c s C s ds + e V C r udu V (34) On he oher hand, we can achieve he expression of collaeral from is dynamic. By moving (r c)cd ogeher wih dc, we can achieve dc s (r s c s )C s ds = αdv s We are going o muliply he discoun facor wih above equaion, hen we will have de s (rτ cτ )dτ C s = (rτ cτ )dτ αdv s hen inegrae boh sides, firsly from o. e (rτ cτ )dτ C e (rτ cτ )dτ C = (rτ cτ )dτ αdv s References C = e (rτ cτ )dτ C + (rτ cτ )dτ αdv s 1 V. Pierbarg, Funding beyond discouning: collaeral agreemens and derivaive pricing, Risk, 97-12, 21 2 Masaaki Fujii, Yasufumi Shimada, Akihiko akahashi, A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaerol, FSA Research Review Vol. 6, March A. Casagna, Pricing of Derivaives Conracs under Collaeral Agreemens: Liquidiy and Funding Value Adjusmens, iason Preprin, ISDA Margin Survey hp: 22