Derivatives Pricing under Collateralization

Transcription

1 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Speaker, Graduae School of Economics, The Universiy of Tokyo 1 / 114 Derivaives Pricing under Collaeralizaion Masaaki Fujii, Kenichiro Shiraya, Akihiko Takahashi Seminar in Chulalongkorn Universiy February 11, 2013 This research is suppored by CARF (Cener for Advanced Research in Finance) and he global COE program The research and raining cener for new developmen in mahemaics. All he conens expressed in his research are solely hose of he auhors and do no represen he views of any insiuions. The auhors are no responsible or liable in any manner for any losses and/or damages caused by he use of any conens in his research. Graduae School of Economics, The Universiy of Tokyo Graduae School of Economics, The Universiy of Tokyo

2 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 2 / 114 Inroducion New marke realies afer he Financial Crisis Wide use of collaeralizaion in OTC Dramaic increase in recen years (ISDA Margin Survey 2011) 30%(2003) 70%(2010) in erms of rade volume for all OTC. Coverage goes up o 79% (for all OTC) and 88% (for fixed income) among major financial insiuions. More han 80% of collaeral is Cash. (Abou half of he cash collaeral is USD. ) Persisenly wide basis spreads: spread. Much more volaile Cross Currency Swap(CCS) basis Non-negligible basis spreads even in he single currency marke. (e.g. Tenor swap spread, Libor-OIS spread)

3 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 3 / 114 Source of Funding Cos Unsecured Funding and Conrac (old picure) Cash Cash=PV Loan A B Libor opion paymen Libor is unsecured offer rae in he inerbank marke. Libor discouning is appropriae for unsecured rades beween financial firms wih Libor credi qualiy.

4 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 4 / 114 Source of Funding Cos Collaeralized (Secured) Conrac (curren picure) opion paymen A cash=pv collaeral B col. rae loan No ourigh cash flow (collaeral=pv) No exernal funding is needed. Funding is deermined by over-nigh (ON) rae. Libor discouning seems inappropriae.

5 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Imporan Insrumens and Marke Realiies Overnigh Index Swap (OIS) 1 OIS rae Compounded ON Floaing side: Daily compounded ON rae Marke Quoe : fixed rae, called OIS rae 1 Usually, here is only one paymen for < 1yr. 5 / 114

6 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 6 / 114 Imporan Insrumens and Marke Realiies Hisorical behavior of IRS (1Y)-OIS (1Y) spreads (bps) Figure: Source:Bloomberg

7 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Imporan Insrumens and Marke Realiies Tenor Swap (TS) 2 Libor (shor enor) +spread Libor (long enor) Spread is quie significan and volaile since lae I is also common ha paymen of shor-enor Leg is compounded and paid a he same ime wih he oher Leg. 7 / 114

8 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 8 / 114 Imporan Insrumens and Marke Realiies Hisorical behavior of JPY TS spreads (bps) Figure: Source:Bloomberg

9 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 9 / 114 Imporan Insrumens and Marke Realiies Mark-o-Marke Cross Currency Swap (i, j) N i = f x () (i, j) f x N j 1 N i δl i (N i = N i N i δl i j) + f(i, x ) USD Libor is exchanged by Libor +spread of he oher currency. USD leg noional is rese every sar of accrual period. Spread is quie significan and volaile for long ime. (i has been changing drasically and rapidly since he financial crisis.)

10 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 10 / 114 Imporan Insrumens and Marke Realiies Hisorical behavior of USDJPY CCS spreads (bps) Figure: Source:Bloomberg

11 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 11 / 114 Imporan Insrumens and Marke Realiies Hisorical behavior of EURUSD CCS spreads (bps) Figure: Source:Bloomberg

12 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 12 / 114 Impac of Collaeralizaion Impac of collaeralizaion : Reducion of Couner pary Exposure Associaed change in CVA has been acively sudied. (e.g. CVA is charged for a conrac wih imperfec collaeralizaion.) Change of Funding Cos Require new erm srucure model o disinguish discouning and reference raes. Significan impac on derivaive pricing.

13 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Topics of his alk Valuaion framework under collaeralizaion Perfec collaeralizaion Asymmeric collaeralizaion Imperfec collaeralizaion and CVA New approximaion scheme for FBSDEs 3 (i seems useful for pricing securiies under asymmeric/imperfec collaeralizaion.) Perurbaion scheme Perurbaion wih ineracing paricle mehod Numerical example for CVA and imperfec collaeralizaion 3 forward backward sochasic differenial equaions 13 / 114

14 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 14 / 114 Simple Case: Pricing under Full Collaeralizaion Assumpion Coninuous adjusmen of collaeral amoun Symmeric/Perfec collaeralizaion by Cash Zero minimum ransfer amoun (Daily cash margin call/selemen is becoming popular.)

15 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 15 / 114 Simple Case: Pricing under Full Collaeralizaion Suppose he paymen currency of a European derivaive is i, while is collaeral currency is j. Then, under full collaeralizaion he ime- price of he derivaive wih payoff h (i) (T) a mauriy T is obained as follows: h (i) () = E Q i where [e ( T r (i) (s)ds e T ) ] y ( j) (s)ds h (i) (T), y ( j) (s) = r ( j) (s) c ( j) (s). (2.1) Q i : risk-neural measure of currency i h (i) (T): derivaive payoff a ime T in currency i collaeral is posed in currency j c ( j) (s): insananeous collaeral rae of currency j a ime s r (i) (s) (r ( j) (s)): insananeous risk-free rae of currency i ( j) a ime s

16 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 16 / 114 Simple Case: Pricing under Full Collaeralizaion Collaeral amoun in currency j a ime s is given by h(i) (s) which is invesed a he rae of y ( j) (s): h (i) () = E Q i [e T ] r (i)(s)ds h (i) (T), (i, j) f x (s) T (i, j) + f x ()E Q j e s r ( j) (u)du y ( j) h (i) (s) (s) (i, j) ds f x (s) [ = E Q i e T T r (i)(s)ds h (i) (T) + e ] s r (i)(u)du y ( j) (s)h (i) (s)ds. Noe ha X() = e 0 r(i) (s)ds h (i) () + e s 0 r(i) (u)du y ( j) (s)h (i) (s)ds 0 is a Q i -maringale. Then, he process of he derivaive value is wrien by dh (i) () = ( r (i) () y ( j) () ) h (i) ()d + dm() wih some Q i -maringale M. This esablishes he proposiion. (i, j) f x (): Foreign exchange rae a ime (he price of he uni amoun of currency j in erms of currency i ).

17 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Simple Case: Pricing under Full Collaeralizaion Special Case If paymen and collaeral currencies are he same, he opion value is given by h() = E [e Q ] T c(s)ds h(t) The discouning is deermined by collaeral rae, which is consisen wih he schemaic picure seen before. 17 / 114

18 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 18 / 114 Seup Pricing Framework ([17]) Filered probabiliy space (Ω, F, F, Q), where F conains all he marke informaion including defauls. Consider wo firms, i {1, 2}, whose defaul ime is τ i [0, ], and τ = τ 1 τ 2. τ i (and hence τ) is assumed o be oally-inaccessible F-sopping ime. (i.e. a defaul even is modeled as a jump process.) Defaul indicaor funcions: H i = 1 {τ i } (i = 1, 2), H = 1 {τ } Assume he exisence of absoluely coninuous compensaor for H i : A i = h i s 1 {τ i >s} ds, 0 0 Assume no simulaneous defauls, and hence he hazard rae of H is h = h 1 + h 2. Money marke accoun: β = exp ( 0 r udu )

19 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 19 / 114 Collaeralizaion When pary i {1, 2} has negaive mark-o-marke, i has o pos cash collaeral o pary j( i), and i is assumed o be done coninuously. collaeral coverage raio is δ i R +, and he amoun of collaeral a ime is given by δ i ( Vi ) when pary i poss collaeral. (Vi denoes he mark-o-marke value of he conrac from he view poin of pary i.) δ i effecively akes ino accoun under- as well as over-collaeralizaion. Thus, δ i < 1 and δ i > 1 are possible. pary j has o pay he collaeral rae c i on he posed cash coninuously. c i is deermined by he currency posed by pary i. marke convenion is o use overnigh (O/N) rae a ime of corresponding currency. Traded hrough OIS (overnigh index swap), which is also collaeralized. In general, c i r i. (ri is he risk-free ineres rae of he same currency.) This is necessary o explain CCS basis spread consisenly.

20 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 20 / 114 Couner pary Exposure and Recovery Scheme Couner pary exposure o pary j a ime from he view poin of pary i is given as: max(1 δ j, 0) max(vi, 0) + max(δi 1, 0) max( Vi, 0). Assume pary- j s recovery rae a ime as R j [0, 1]. Then, he recovery value a he ime of j s defaul is given as: x + max(x, 0). R j ( [1 δ j ]+ [V i ]+ + [δ i 1]+ [ V i ]+),

21 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 21 / 114 Pricing Formula Pricing from he view poin of pary 1. [ S = β E Q β 1 u 1 {τ>u}{ ddu + ( y 1 u δ1 u 1 {S u<0} + y 2 u δ2 u 1 ) {S u 0} Su du } ],T] + β 1 u 1 {τ u}( Z 1 (u, S u )dhu 1 + Z2 (u, S u )dhu 2 ) ] F ],T] D: cumulaive dividend o pary 1. Defaul payoff: Z i when pary i defauls. Z 1 (, v) = ( 1 l 1 (1 δ1 )+) v1 {v<0} + ( 1 + l 1 (δ2 1) +) v1 {v 0} Z 2 (, v) = ( 1 l 2 (1 δ2 )+) v1 {v 0} + ( 1 + l 2 (δ1 1) +) v1 {v<0}, l i (1 R i ), i = 1, 2 y i = r i ci, (i {1, 2}) denoes he insananeous reurn for j or funding cos for i a ime from he cash collaeral posed by pary i.

22 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 22 / 114 Pricing Formula (Remark) The reurn from risky invesmens, or he borrowing cos from he exernal marke can be quie differen from he risk-free rae, of course. However, if one wans o rea his fac direcly, an explici modeling of he associaed risks is required. Here, we use he risk-free rae as ne reurn/cos afer hedging hese risks. As we shall see, under full collaeralizaion he final formula does no require any knowledge of he risk-free rae, and hence here is no need of is esimaion.

23 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 4 This echnical condiion ( V τ = 0) becomes imporan when we consider credi derivaives: he condiion is violaed in general when he conagious effecs induce jumps o variables conained in pre-defaul value process. (e.g. Schönbucher(2000), Collin-Dufresne-Goldsein-Hugonnier(2004), Brigo-Capponi(2009), [19]) 23 / 114 Pricing Formula Following he mehod in Duffie&Huang (1996), he pre-defaul value of he conrac V such ha V 1 {τ>} = S is given by [ ( s V = E Q ( exp ru µ(u, V u ) ) ) ] du dd s F, T, where ],T] µ(, v) = ỹ 1 1 {v<0} + ỹ 2 1 {v 0} (adjused erm of he discoun rae) ỹ i = δ i yi (1 δi )+ (l i hi ) + (δi 1)+ (l j h j ), if some echnical condiion(so called no jump condiion for V a defaul) 4 is saisfied, which is assumed hereafer.(for insance, r, D, y i, δ i, l i and h i, (i = 1, 2) are adaped o background filraion. In paricular, we limi our aenion o he inensiies condiional on no-defaul.)

24 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 24 / 114 Symmeric Case Effecive discoun facor is non-linear: r µ(, v) = r (ỹ 1 1 {v<0} + ỹ 2 1 {v 0}), which makes he porfolio value non-addiive. If ỹ 1 = ỹ 2 = ỹ, hen we have µ(, v) = ỹ. Furher, if ỹ is no explicily dependen on V, we can recover he lineariy. [ ( s ) ] V = E Q exp (r u ỹ u )du dd s F ],T] Porfolio valuaion can be decomposed ino ha of each paymen. A good characerisic for marke benchmark price.

25 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Symmeric Perfec Collaeralizaion Special Cases Case 1 : Benchmark for single currency produc bilaeral perfec collaeralizaion (δ 1 = δ 2 = 1) boh paries use he same currency (i) as collaeral, which is also he paymen (evaluaion) currency. [ ( s ) ] V (i) = E Q(i) exp c (i) u du dd s F ],T] The valuaion mehod for single currency swap adoped by LCH Swapclear (2010) is he same wih his equaion. 5 5 See also Pierbarg (2010) for oher derivaion of his equaion. 25 / 114

26 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 26 / 114 Symmeric Perfec Collaeralizaion Special Cases Case 2 : Collaeral in a Foreign Currency bilaeral perfec collaeralizaion (δ 1 = δ 2 = 1) boh paries use he same currency (k) as collaeral, which is differen from he paymen (evaluaion) currency (i) [ ( s ) = E Q(i) ( exp c (i) u + ) ] y(i,k) u du dd s F V (i) ],T] Funding Spread beween he wo currencies y (i,k) = y (i) y (k) = ( r (i) c (i)) ( r (k) c (k)) This is necessary o explain CCS basis spreads consisenly.

27 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 27 / 114 Collaeral Rae Overnigh Index Swap (OIS) exchange fixed rae(f) wih compounded overnigh rae periodically. collaeralized by domesic currency Par rae a for T 0 (> )-sar T N -mauring OIS wih currency (i): OIS N () = F par () = D(i) (, T 0 ) D (i) (, T N ) N, n=1 nd (i) (, T n ) ( n : daycoun fracion). [ D (i) (, T) = E Q(i) e T c ] (i) u du F is a value of domesically collaeralized zero-coupon bond.

28 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 28 / 114 Funding Spread (i, j) Mark-o-Marke Cross Currency OIS: The funding spread(he difference of collaeral coss) is direcly linked o he corresponding CCOIS, hough i seems no liquid in he curren marke. compounded O/N rae of currency i is exchanged by ha of currency j wih addiional spread periodically. noional of currency j is kep consan while ha of currency i is refreshed a every rese ime wih he spo FX rae. (currency i is usually USD.) collaeralized by currency i.

29 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Funding Spread Define [ D ( j,i) (, T) = E Q( j) e T (c ( j) u y ( j,i) (, T) = T ln ET( j) [ e +y( j,i) u T )du F y ( j,i) u du F ] = D ( j) (, T)e T y ( j,i) (,s)ds. ]. (insananeous fwd rae of he funding spread) D ( j,i) (, T): he zero coupon bond of currency j collaeralized by currency i. E T( j) [ F ]: condiional expecaion under he fwd measure associaed wih D ( j) (, T). Then, under a simplifying assumpion such as independence beween c ( j) and y ( j,i)6, MMCCOIS basis spread is obained by: B N = N n=1 D( j,i) (, T n 1 ) (1 e Tn T n 1 y ( j,i) (,u)du ) N n=1 δ nd ( j,i) (, T n ) 1 T N T 0 TN T 0 y ( j,i) (, u)du. 6 The assumpion seems reasonable for he recen daa sudied in [15]. 29 / 114

30 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Modeling framework of Ineres raes Symmeric perfecly collaeralized price is becoming he marke benchmark, a leas for sandardized producs. Term srucure consrucion procedures : 7 (1), OIS c (i) (0, T)(T-mauriy insananeous fwd rae a ime 0) (2), resuls of (1) + IRS + TS B (i) (0, T; τ) (i-currency forward Libor-OIS spread wih enor τ) (3), resuls of (1),(2) +CCS y (i, j) (0, T)(funding spread) Given he iniial erm srucures, no-arbirage dynamics of c (i) (, T),B (i) (, T; τ) and y (i, j) (, T) in HJM-framework can be consruced. (For he deail, please see our paper [13], [22]. For oher approaches, see Bianchei(2010), Mercurio(2009), Morini(2009), for insance.) 7 Assume collaeralizaion in domesic currency for OIS, IRS and TS. Assume collaeralizaion in USD for CCS (USD crosses). 30 / 114

31 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 8 The shor-enor leg may be compounded and hen, addiional small correcions exis. 31 / 114 Collaeralized OIS OIS N (0) m=1 Curve Consrucion N n D(0, T n ) = D(0, T 0 ) D(0, T N ) n=1 Collaeralized IRS M M IRS M (0) m D(0, T m ) = δ m D(0, T m )E Tm [L(T m 1, T m ; τ)] Collaeralized TS 8 m=1 N δ n D(0, T n ) ( E Tn [L(T n 1, T n ; τ S )] + TS N (0) ) n=1 ( m, n, δ m, δ n : daycoun fracions) = M δ m D(0, T m )E Tm [L(T m 1, T m ; τ L )] m=1 Marke quoes of collaeralized OIS, IRS, TS, (and a proper spline mehod) allow us o deermine all he relevan {D(0, T)}, and forward Libors {E Tm [L(T m 1, T m, τ)]}.

32 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Collaeralized FX Forward: USD/JPY Curve Consrucion Suppose USD= i, JPY= j and collaeral currency is USD. Curren ime:. Mauriy: T A T, one uni of i is exchanged for K (fixed a ) unis of j. FX forward is he break-even value of K. Q( j) KE [e ] T (c ( j) s +y(i,i) s )ds = f ( j,i) x ()E [e Q(i) ] T c (i) s ds 1. 9 ( f ( j,i) x (, T; (i)) = f ( j,i) x () D(i) (, T) T D ( j) (, T) exp y ( j,i) (, T) = T ln j) ET( [e T y ( j,i) s ds ]. ) y ( j,i) (, u)du, FX Forward Forward curve of funding spread ({y ( j,i) (, T)}) CCS for longer mauriies. 9 f ( j,i) x () denoes spo FX rae a ha is, he price of he uni amoun of currency i in erms of currency j. 32 / 114

33 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 33 / 114 Curve Consrucion Remark: Consan Noional CCS vs MM-CCS (USD-LIBOR) is exchanged for (X-currency LIBOR + basis spread). Consan Noional CCS (CNCCS) Noional of boh legs are kep consan. Mark-o-Marke CCS (MMCCS) Noional of currency X is kep consan a N X. Noional of USD is readjused o f (USD;X) x N X a every sar of LIBOR accrual period.

34 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 34 / 114 Curve Consrucion Remark: he difference beween MM and Consan noional basis spreads: B M M N B CN N = N n=1 δ(i) n D(i) (0, T n )E T(i) n [( (i, j) f x (T n 1 ) (i, j) f x (0) N j) δ( n=1 n D( j,i) (0, T n ) ) ] 1 B (i) (T n 1, T n ) where B (i) (T n 1, T n ) sands for he Libor-OIS spread of he currency i a T n 1. This spread is no zero in general. For he wo USDJPY CCSs, he wo swaps should have he same basis spreads if USD LIBOR-OIS spreads are all zero. This held approximaely well before he Lehman crisis bu he spread has been far from zero since hen. If USD ineres rae level is higher han JPY, as is usually he case, he equaion ells us ha he spread for MMCCS is quie likely o be higher han ha of CNCCS, B M M > B CN. The size of N N spread may no be negligible dependen on siuaions, and hence i is worhwhile paying enough aenion o he difference in his pos crisis era.,

35 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Curve Consrucion ( ) T R y ( j, i) = 0 y j,i (0, u)du /T(funding spread curve): posing USD as collaeral ends o be expensive for collaeral payers. 35 / 114

36 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Curve Consrucion Close relaionship - CCS Basis and Funding Spread - A significan porion of CCS spreads movemen sems from he change in he funding spreads. Libor-OIS spread seems o have minor effec. 36 / 114

37 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 37 / 114 HJM-framework under full collaeralizaion ( s dc (i) (, s) = σ (i) c (, s) dy (i,k) (, s) = σ (i,k) y (, s) db (i) (, T; τ) B (i) (, T; τ) d f (i, j) x (i, j) x f () () ( s ( T = σ (i) (, T; τ) B σ (i) c (, u)du ) d + σ (i) c (, s) dwq(i) (σ (i,k) y (, u) + σ i c (, u))du ) σ (i) c (, s)ds ) = ( c (i) () c ( j) () + y (i, j) () ) d + σ d + σ (i,k) y (, s) dw Q(i) d + σ (i) (, T; τ) dwq(i) B (i, j) X () dwq(i), B (i) (, T k ; τ) = E T(i) k is forward LIBOR-OIS spread. [ L (i) (T k 1, T k ; τ) ] 1 δ (i) k ( D (i) (, T k 1 ) D (i) (, T k ) ) 1

38 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 38 / 114 Choice of Collaeral Currency Special Cases Case 3 : Muliple Eligible Collaerals bilaeral perfec collaeralizaion (δ 1 = δ 2 = 1) boh paries choose he opimal currency from he eligible collaeral se C. Currency (i) is used as he evaluaion currency. [ ( s ( = E Q(i) exp c (i) u + max u ] ) ) ] du dd s F V (i) ],T] k C [y(i,k) The pary who needs o pos collaeral has opionaliy. The cheapes collaeral currency is chosen based on CCS informaion. To choose srong currency, such as USD, is expensive for he collaeral payer.

39 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Choice of Collaeral Currency Role of y ( j,i) Opimal behavior of collaeral payer can significanly change he derivaive value. Paymen currency and USD as eligible collaeral is relaively common. Then, he effecive discouning facor becomes D ( j) j) T( (, T) E excep correlaion effecs. [e ] T max{y ( j,usd) (s),0}ds D ( j) (, T) Volailiy of y ( j,usd) is an imporan deerminan. (Embedded opion change effecive discouning facor, which crucially depends on he volailiy of funding spread.) 39 / 114

40 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Choice of Collaeral Currency vols end o be 50 bps in a calm marke, bu hey were more han a percenage poin jus afer he marke crisis, which reflecs a significan widening of he CCS basis o seek USD cash in he low liquidiy marke. 40 / 114

41 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Choice of Collaeral Currency Figure: Modificaion of JPY discouning facors based on HW model for y (J PY,USD) as of 2010/3/16. he effecive discouning rae is increased by around 50 bps annually even when he annualized vol. of y (J PY,USD) is 50 bps. 41 / 114

42 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle More generic siuaions: marginal impac of asymmery V = E Q [ ],T] ( exp µ(, v) = ỹ 1 1 {v<0} + ỹ 2 1 {v 0} s ( ru µ(u, V u ) ) ) ] du dd s F ỹ i = δ i yi (1 δi )+ (l i hi ) + (δi 1)+ (l j h j ) Make use of Gaeaux derivaive(gd) as he firs-order Approximaion 10 : lim sup ϵ 0 V ( η; η) V ( η + ϵη) V ( η) = 0, (η, η: bounded and predicable) ϵ We wan o expand he price around a symmeric benchmark price. µ(, v) = y + ỹ 1 1 {v<0} + ỹ 2 1 {v 0}, ( ỹ i = ỹ i y ) Calculae GD a symmeric µ = y poin. V (µ) V (y) + V (y, µ y) 10 Duffie&Skiadas (1994), Duffie&Huang (1996) 42 / 114

43 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 43 / 114 Asymmeric Collaeralizaion(marginal impac of asymmery) Then, V is decomposed as V = V + V, where [ ( s ) ] V = E Q exp (r u y u )du dd s F ],T] V = E Q [ T e s ( (ru yu)du V s ỹ 1 s 1 + {V ỹ2 s<0} s 1 ) ] {V s 0} ds F If y is chosen in such a way ha i reflecs he funding cos of he sandard collaeral agreemens, V urns ou o be he marke benchmark price, and V represens he correcion for i.

44 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 44 / 114 Asymmeric Collaeralizaion(marginal impac of asymmery) An example of asymmeric perfec collaeralizaion pary 1 choose opimal currency from he eligible collaeral se C, bu he pary 2 can only use currency (i) as collaeral, eiher due o he asymmeric CSA or lack of easy access o foreign currency pool. The evaluaion (paymen) currency is (i). V = E Q(i) [ ],T] [ T V = E Q(i) exp V V + V ( exp ( s s c (i) u du ) dd s F ] ) c (i) u du [ Vs ] + max k C [y(i,k) s ] ] F Expansion around he symmeric collaeralizaion wih currency (i).

45 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle OIS rae is se o make V = 0. Difference beween Receiver and Payer comes from up-ward sloping erm srucure. (he receiver s mark-o-marke value ends o be negaive in he long end of he conrac, which makes he opionaliy larger.) 11 based on he daa in early 2010, see [17] for he deail. 45 / 114 Asymmeric Collaeralizaion(marginal impac of asymmery) Numerical Example of V for JPY-OIS 11. Eligible collaeral are USD and JPY for pary-1 bu only JPY for pary-2.

46 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 46 / 114 Imperfec Collaeralizaion CVA as he Deviaion from he Perfec Collaeralizaion Assume he boh paries use he same currency for simpliciy, and hence y 1 = y 2 = y. µ(, v) = y {( (1 δ 1 )y + (1 δ 1 )+ (l 1 h1 ) (δ1 1) + (l 2 h2 )) 1 {v<0} + ( (1 δ 2 )y + (1 δ 2 )+ (l 2 h2 ) (δ2 1) + (l 1 h1 )) } 1 {v 0} GD(Gaeaux derivaive) around µ = y decomposes he price ino hree pars: Symmeric perfecly collaeralized benchmark price (1 δ i )y1 {v 0} Collaeral Cos Adjusmen (CCA) Remaining h dependen erms Credi Value Adjusmen (CVA) V V + V = V + CCA + CVA

47 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Imperfec Collaeralizaion V = E Q [ ],T] CCA = E Q [ T CVA = ( exp e s s ) ] (r u y u )du dd s F (ru yu)du y s ((1 δ 1 s )[ V s] + (1 δ 2 s )[V s] +) ] ds F [ T E Q e s (ru yu)du (l 1 s h1 s [(1 ) δ 1 s )+ [ V s ] + + (δ 2 s 1)+ [V s ] +] ds T e s (ru yu)du (l 2 s h2 s [(1 ) δ 2 s )+ [V s ] + + (δ 1 s 1)+ [ V s ] +] ] ds F The discouning rae is differen from he risk-free rae and reflecs he funding cos of collaeral, while he erms in CVA are prey similar o he usual resul of bilaeral CVA. Dependence among y, δ and oher variables such asv, h i is paricularly imporan. New ype of Wrong (Righ)-way Risk. (e.g. y is closely relaed o he CCS basis spread. Hence, y is expeced o be highly sensiive o he marke liquidiy, and is also srongly affeced by he overall marke credi condiions.) 47 / 114

48 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Collaeral Thresholds Thresholds: Γ i > 0 for pary-i: A hreshold is a level of exposure below which collaeral will no be called, and hence i represens an amoun of uncollaeralized exposure. Only he incremenal exposure will be collaeralized if he exposure is above he hreshold. Case of perfec collaeralizaion above he hresholds [ S = β E Q β 1 u 1 { {τ>u} ddu + q(u, S u )S u du } ],T] + β 1 u 1 {τ u}{ Z 1 (u, S u )dhu 1 + Z2 (u, S u )dh 2 } ] u F ],T] Γ 1 Γ 2 q(, S ) = y S 1 {S < Γ 1 } + y2 1 S 1 {S >Γ 2 } Γ Z 1 (, S ) = S l1 S 1 {S < Γ 1 } + R1 1 { Γ 1 S <0} + 1 {S 0} Γ Z 2 (, S ) = S 2 1 l2 S 1 {S Γ 2 } + R2 1 {0 S <Γ 2 } + 1 {S <0} 48 / 114

49 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Collaeral Thresholds Assume he domesic currency as collaeral y 1 = y 2 = y. [ ( s ) ] V = E Q exp c u du dd s F ],T] CCA = E Q [ T CVA = E Q [ T E Q [ T +E Q [ T e s e s e s ] c u du y s V s 1 { Γ 1 s V s<γ 2}ds F s e s c u du { y s Γ 1 s 1 {Vs< Γ 1 s } Γ2 s 1 } ] {V ds s Γ 2 s } F c u du { (l 1 s h1 s )[ V s 1 { Γ 1 s V + Γ1 s<0} s 1 ] } ] {V s< Γ 1 s } ds F c u du { (l 2 s h2 s )[ V s 1 {0<Vs Γ 2 s } + Γ2 s 1 ] } ] {V s>γ 2 s } ds F The erms in CCA reflec he fac ha no collaeral is posed in he range { Γ 1 V Γ 2 }, and ha he posed amoun of collaeral is smaller han V by he size of hreshold. The erms in CVA represen bilaeral uncollaeralized credi exposure, which is capped by each hreshold. 49 / 114

50 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 50 / 114 FBSDE Approximaion Scheme ([19]) The forward backward sochasic differenial equaions (FBSDEs) have been found paricularly relevan for various valuaion problems (e.g. pricing securiies under asymmeric/imperfec collaeralizaion, opimal porfolio and indifference pricing issues in incomplee and/or consrained markes). Their financial applicaions are discussed in deails for example, El Karoui, Peng and Quenez [1997], Ma and Yong [2000], a recen book edied by Carmona [2009], Crépey [2012(a,b)], [44], and references herein. We will presen a simple analyical approximaion wih perurbaion scheme for he non-linear FBSDEs.

51 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 51 / 114 FBSDE Approximaion Scheme - Seup- We consider he following FBSDE: dv = f(x, V, Z )d + Z dw (6.1) V T = Φ(X T ), (6.2) where V akes he value in R, W is a r-dimensional Brownian moion, and X R d is assumed o follow a diffusion which is he soluion o he (forward) SDE: dx = γ 0 (X )d + γ(x ) dw ; X 0 = x. (6.3) We assume ha he appropriae regulariy condiions are saisfied for he necessary reamens. See Takahashi-Yamada [44] for he mahemaical validiy based on Malliavin calculus.

52 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 12 Or, one can consider ϵ = 1 as simply a parameer convenien o coun he approximaion order. The acual quaniy ha should be small for he approximaion is he residual par g. 52 / 114 Perurbaive Expansion for Non-linear Generaor In order o solve he pair of (V, Z ) in erms of X, we exrac he linear erm from he generaor f and rea he residual non-linear erm as he perurbaion o he linear FBSDE. We inroduce he perurbaion parameer ϵ, and hen wrie he equaion as dv (ϵ) V (ϵ) T = c(x )V (ϵ) = Φ(X T ), d ϵ g(x, V (ϵ), Z (ϵ) )d + Z (ϵ) dw (6.4) where ϵ = 1 corresponds o he original model by 12 f(x, V, Z ) = c(x )V + g(x, V, Z ). (6.5)

53 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 53 / 114 Perurbaive Expansion for Non-linear Generaor One should choose he linear erm c(x )V (ϵ) in such a way ha he residual non-linear erm g becomes as small as possible o achieve beer convergence. Now, we are going o expand he soluion of BSDE (6.4) in erms of ϵ: ha is, suppose V (ϵ) and Z (ϵ) are expanded as V (ϵ) Z (ϵ) = V (0) = Z (0) + ϵv (1) + ϵz (1) + ϵ 2 V (2) + ϵ 2 Z (2) + (6.6) +. (6.7)

54 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 54 / 114 Perurbaive Expansion for Non-linear Generaor Once we obain he soluion up o he cerain order, say k for example, hen by puing ϵ = 1, Ṽ = k i=0 V (i), Z = k i=0 Z (i) (6.8) is expeced o provide a reasonable approximaion for he original model as long as he residual erm g is small enough o allow he perurbaive reamen. V (i) and Z (i), he correcions o each order can be calculaed recursively using he resuls of he lower order approximaions.

55 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Recursive Approximaion Zero-h Order For he zero-h order of ϵ, one can easily see he following equaion should be saisfied: dv (0) = c(x )V (0) d + Z (0) dw (6.9) V (0) T = Φ(X T ). (6.10) I can be inegraed as V (0) = E [e T ] c(x s )ds Φ(X T ) F (6.11) which is equivalen o he pricing of a sandard European coningen claim, and V (0) is a funcion of X. Applying Iô s formula (or Malliavin derivaive), we obain Z (0) a funcion of X, oo. as 55 / 114

56 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 56 / 114 Recursive Approximaion Firs Order Now, le us consider he process V (ϵ) V (0). One can see ha is dynamics is governed by d ( V (ϵ) V (ϵ) T V (0) ) V(0) T = c(x ) ( V (ϵ) V (0) ) d ϵ g(x, V (ϵ), Z (ϵ) )d + ( Z (ϵ) Z (0) ) dw = 0. (6.12) Now, by exracing he ϵ-firs order erm, we can once again recover he linear FBSDE dv (1) V (1) T which leads o V (1) = c(x )V (1) d g(x, V (0), Z (0) )d + Z (1) dw = 0, (6.13) [ T = E e u ] c(x s)ds g(x u, V (0) u, Z(0) u )du F. (6.14)

57 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 57 / 114 Recursive Approximaion Because V (0) u and Z (0) u are some funcions of X u, we obain V (1) as a funcion of X, and also Z (1) hrough Iô s formula (or Malliavin derivaive). In exacly he same way, one can derive an arbirarily higher order correcion. Due o he ϵ in fron of he non-linear erm g, he sysem remains o be linear in he every order of approximaion. e.g. dv (2) V (2) T ( = c(x )V (2) d v g(x, V (0), Z (0) )V (1) ) + z g(x, V (0), Z (0) ) Z (1) d + Z (2) dw = 0

58 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 58 / 114 Evaluaion of (V (i), Z (i) ) in erms of X Suppose we have succeeded o express backward componens (V, Z ) in erms of X up o he (i 1)-h order. Now, in order o proceed o a higher order approximaion, we have o give he following form of expressions wih some deerminisic funcion G( ) in erms of he forward componens X, in general: V (i) [ T = E e u c(x s )ds G ( ) ] X u du F (6.15)

59 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 59 / 114 Evaluaion of (V (i), Z (i) ) in erms of X Even if i is impossible o obain he exac resul, we can sill obain an analyic approximaion for (V (i), Z (i) ). For insance, an asympoic expansion mehod allows us o obain he expression. (See [29]-[30], [38] -[45] for he deail of he asympoic expansion mehod.) In fac, applying he mehod, [19] has provided some explici approximaions for V (i) and Z (i). Also, [20] has explicily derived an approximaion formula for he dynamic opimal porfolio in an incomplee marke and confirmed is accuracy comparing wih he exac resul by Cole-Hopf ransformaion. (Zariphopoulou [2001])

60 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Remark on Approximaion of Coupled FBSDEs Le us consider he following generic coupled non-linear FBSDE: dv = f(, X, V, Z )d + Z dw V T = Φ(X T ) dx = γ 0 (, X, V, Z )d + γ(, X, V, Z ) dw ; X 0 = x. We can rea his case in he similar way as before(decoupled case) by inroducing he following perurbaion o he forward process: dṽ = c(, X )Ṽ d ϵ g(, X, Ṽ, Z )d + Z dw Ṽ T = Φ( X T ) d X = ( r(, X ) + ϵµ(, X, Ṽ, Z ) ) d + ( σ(, X ) + ϵη(, X, Ṽ, Z ) ) dw We can also apply he same mehod under PDE(parial differenial equaion) formulaion based on four sep scheme (e.g. Ma-Yong [2000]). Please consul [19] for he deails. 60 / 114

61 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 61 / 114 Forward Agreemen wih Bilaeral Defaul Risk As he firs example, we consider a oy model for a forward agreemen on a sock wih bilaeral defaul risk of he conracing paries, he invesor (pary-1) and is couner pary (pary-2). The erminal payoff of he conrac from he view poin of he pary-1 is Φ(S T ) = S T K (6.16) where T is he mauriy of he conrac, and K is a consan. We assume he underlying sock follows a simple geomeric Brownian moion: ds = rs d + σs dw (6.17) where he risk-free ineres rae r and he volailiy σ are assumed o be posiive consans. The defaul inensiy of pary-i, h i is specified as h 1 = λ, h 2 = λ + h (6.18) where λ and h are also posiive consans.

62 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 62 / 114 Forward Agreemen wih Bilaeral Defaul Risk In his seup, he pre-defaul value of he conrac a ime, V follows dv = rv d h 1 max( V, 0)d + h 2 max(v, 0)d + Z dw = (r + λ)v d + h max(v, 0)d + Z dw (6.19) V T = Φ(S T ). (6.20) Now, following he previous argumens, le us inroduce he expansion parameer ϵ, and consider he following FBSDE: dv (ϵ) V (ϵ) T = µv (ϵ) d ϵ g(v (ϵ) )d + Z (ϵ) dw (6.21) = Φ(S T ) (6.22) ds = S (rd + σdw ), (6.23) where we have defined µ = r + λ and g(v) = hv1 {v 0}.

63 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 63 / 114 Forward Agreemen wih Bilaeral Defaul Risk The nex figure shows he numerical resuls of he forward conrac wih bilaeral defaul risk wih various mauriies wih he direc soluion from he PDE (as in Duffie-Huang [1996]). We have used where he srike K is chosen o make V (0) 0 r = 0.02, λ = 0.01, h = 0.03, (6.24) σ = 0.2, S 0 = 100, (6.25) = 0 for each mauriy. We have plo V (1) for he firs order, and V (1) + V (2) for he second order. (Noe ha we have pu ϵ = 1 o compare he original model.)

64 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 64 / 114 Forward Agreemen wih Bilaeral Defaul Risk Figure: Numerical Comparison o PDE

65 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Forward Agreemen wih Bilaeral Defaul Risk One can observe how he higher order correcion improves he accuracy of approximaion. In his example, he couner pary is significanly riskier han he invesor, and he underlying conrac is volaile. Even in his siuaion, he simple approximaion o he second order works quie well up o he very long mauriy. In anoher example of [19] 13, our second order approximaion has obained a fairly close value(2.953) o he one(2.95 wih sd 0.01) by a regression-based Mone Carlo simulaion of Gobe-Lemor-Warin[2005]. 13 a self-financing porfolio under he siuaion where here exiss a difference beween he lending and borrowing ineres raes 65 / 114

66 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 66 / 114 Perurbaion Technique wih Ineracing Paricle Mehod ([21], [14]) We will provide a sraighforward simulaion scheme o solve nonlinear FBSDEs a each order of perurbaive approximaion. Due o he convolued naure of he perurbaive expansion, i conains muli-dimensional ime inegraions of expecaion values, which make sandard Mone Carlo oo ime consuming. To avoid nesed simulaions, we applied he paricle represenaion inspired by he ideas of branching diffusion models(e.g. McKean (1975), Fujia (1966), Ikeda-Nagasawa-Waanabe (1965,1966,1968), Nagasawa-Sirao (1969)). Comparing wih he direc applicaion of he branching diffusion mehod, our mehod is expeced o be less numerically inensive since he ineresed sysem is already decomposed ino a se of linear problems.

67 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 67 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Again, le us inroduce he perurbaion parameer ϵ: dv (ϵ) s V (ϵ) T = ϵ f(x s, V (ϵ) s = Ψ(X T),, Z (ϵ) )ds + Z (ϵ) dw s s s (7.1) where X R d is assumed o follow a generic Markovian forward SDE dx s = γ 0 (X s )ds + γ(x s ) dw s ; X = x. (7.2) Le us fix he iniial ime as. We denoe he Malliavin derivaive of X u (u ) a ime as D X u R r d. (7.3)

68 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 68 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Is dynamics in erms of he fuure ime u is specified by sochasic flow, (Y,u ) i = j j x X i u d(y,u ) i j (Y, ) i j = k γ i 0 (X u)(y,u ) k j du + kγ i a (X u)(y,u ) k j dw a u = δ i j (7.4) where k denoes he differenial wih respec o he k-h componen of X, and δ i denoes Kronecker dela. Here, i and j run hrough {1,, d} j and {1,, r} for a. Here, we adop Einsein noaion which assumes he summaion of all he paired indexes. Then, i is well-known ha (D X i u ) a = (Y,u γ(x )) i a, where a {1,, r} is he index of r-dimensional Brownian moion.

69 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 69 / 114 Perurbaion Technique wih Ineracing Paricle Mehod ϵ-0h order: For he zeroh order, i is easy o see V (0) = E [ ] Ψ(X T ) F (7.5) = E [ i Ψ(X T )(Y T γ(x )) i ] a F. (7.6) Z a(0) I is clear ha hey can be evaluaed by sandard Mone Carlo simulaion. However, for heir use in higher order approximaion, i is crucial o obain explici approximae expressions for hese wo quaniies. (e.g. Hagan e al.[2002], asympoic expansion echnique) In he following, le us suppose we have obained he soluions up o a given order of asympoic expansion, and wrie each of hem as a funcion of x : V (0) Z (0) = v (0) (x ) = z (0) (x ). (7.7)

70 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 70 / 114 Perurbaion Technique wih Ineracing Paricle Mehod ϵ-1s order: V (1) = = T T E [ f(x u, V (0) u, Z(0) u ) ] F du E [ f ( X u, v (0) (X u ), z (0) (X u ) ) F ] du (7.8) Nex, define he new process for (s > ): ˆV (1) s = e s λ u du V (1) s, (7.9) where deerminisic posiive process λ for he simples case.). (I can be a posiive consan

71 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 71 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Then, is dynamics is given by d ˆV (1) s = λ s ˆV (1) s ds λ s ˆf s (X s, v (0) (X s ), z (0) (X s ))ds + e s λ u du Z (1) s dw s, where ˆf s (x, v (0) (x), z (0) (x)) = 1 λ s e s λ u du f(x, v (0) (x), z (0) (x)). Since we have ˆV (1) holds: V (1) = V (1), one can easily see he following relaion [ T = E e u ] λ s ds λ u ˆf u (X u, v (0) (X u ), z (0) (X u ))du F As in credi risk modeling (e.g. Bielecki-Rukowski [2002]), i is he (7.10) presen value of defaul paymen where he defaul inensiy is λ s wih he defaul payoff a s(> ) as ˆf s (X s, v (0) (X s ), z (0) (X s )). Thus, we obain he following proposiion.

72 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 72 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Proposiion The V (1) in (7.8) can be equivalenly expressed as V (1) = 1 {τ>} E [ 1 {τ<t} ˆf τ ( Xτ, v (0) (X τ ), z (0) (X τ ) ) F ]. (7.11) Here τ is he ineracion ime where he ineracion is drawn independenly from Poisson disribuion wih an arbirary deerminisic posiive inensiy process λ. ˆf is defined as ˆf s (x, v (0) (x), z (0) (x)) = 1 λ s e s λ u du f(x, v (0) (x), z (0) (x)). (7.12)

73 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Perurbaion Technique wih Ineracing Paricle Mehod Now, le us consider he componen Z (1). I can be expressed as Z (1) = T E [ D f ( X u, v (0) (X u ), z (0) (X u ) ) F ] du (7.13) Firsly, le us observe he dynamics of Malliavin derivaive of V (1) follows where d(d V (1) s ) = (D X i s ) i(x, v (0), z (0) ) f(x, v (0), z (0) ) + (D Z (1) s ) dw s ; D V (1) = Z (1), (7.14) i (x, v (0), z (0) ) i + i v (0) (x) v + i z a(0) (x) z a, (7.15) f(x, v (0), z (0) ) f(x, v (0) (x), z (0) (x)). (7.16) 73 / 114

74 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 74 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Define, for (s > ), D V (1) s = e s Then, is dynamics can be wrien as λ u du (D V (1) s ). (7.17) d( D V (1) s ) = λ s ( D V (1) s )ds λ s (D X i s ) i(x s, v (0), z (0) ) ˆf s (X s, v (0), z (0) )ds s +e λ u du (D Z (0) s ) dw s. (7.18) We again have Hence, [ T = E Z (1) e u D V (1) = Z (1). (7.19) λ s ds λ u (D X i u ) i(x u, v (0), z (0) ) ˆf u (X u, v (0), z (0) )du F ].(7.20)

75 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 75 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Thus, following he same argumen for he previous proposiion, we have he resul below: Proposiion Z (1) in (7.13) is equivalenly expressed as Z a(1) = 1 {τ>} E [ 1 {τ<t} (Y τ γ(x τ )) i a i(x τ, v (0), z (0) ) ˆf τ (X τ, v (0), z (0) ] ) F (7.21) where he definiions of random ime τ and he posiive deerminisic process λ are he same as hose in he previous proposiion.

76 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 76 / 114 Perurbaion Technique wih Ineracing Paricle Mehod Mone Carlo Mehod Now, we have a new paricle inerpreaion of (V (1), Z (1) ) as follows: V (1) = 1 {τ>} E [ ( 1 {τ<t} ˆf τ Xτ, v (0), z (0)) ] F (7.22) = 1 {τ>} E [ 1 {τ<t} (Y,τ γ(x τ )) i i (X τ, v (0), z (0) ) ˆf τ (X τ, v (0), z (0) ] ) F (7.23) Z (1) which allows efficien ime inegraion wih he following Mone Carlo scheme: Run he diffusion processes of X and Y Carry ou Poisson draw wih probabiliy λ s s a each ime s and if one is drawn, se ha ime as τ. Then sores he relevan quaniies a τ, or in he case of (τ > T) sores 0. Repea he above procedures and ake heir expecaion.

77 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 77 / 114 Z (2) The second order sochasic flow: for < s < u, (Γ,s,u ) i jk := 2 x j xk s X i u ; ( (Γ,s,s ) i jk = 0).

78 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 78 / 114

79 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Numerical Example An example for pre-defaul values wih imperfec collaeralizaion 14 : The couner pary sells OTC European opions on WTI fuures. 15 For simpliciy, we consider a unilaeral case, where couner pary is defaulable, while he invesor is defaul-free, and he collaeral is posed as he same currency as he paymen currency (ha is, he currency is USD). We consider he following imperfec collaeral cases: No collaeral Cash collaeral wih ime-lag Asse collaeral wih ime-lag 14 As for an applicaion o American opion pricing, please see [11] 15 Laer, we will see a baske opion on WTI and Bren fuures. 79 / 114

80 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 80 / 114 Model CIR model for he hazard rae process (h). SABR model for WTI fuures price process (S and ν). Log-Normal model for a collaeral asse price process ( A). dh = κ (θ h ) d + γ h c 1 dw 1 ; h 0 = h 0 (8.1) 2 ds = µ i S d + ν (S ) β ( c 2,η dw η ); S 0 = s 0, (8.2) η=1 3 dν = σ ν ν ( c 3,η dw η ); ν 0 = ν 0, (8.3) η=1 4 da = µ A A d + σ A A ( c 4,η dw η ); A 0 = a 0. (8.4) η=1

81 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle Model The dynamics of pre-defaul value V can be described by a non-linear FBSDE: dv = rv d f(h, V, Γ )d + Z dw V T = (S T K) + or (K S T ) +, (8.5) where Γ : collaeral process (e.g. cash collaeral wih a consan ime lag : Γ = V ) r(risk free rae), c(collaeral rae), l(loss rae) : nonnegaive consans for simpliciy. 16 We pu ϵ in fron of he driver, f o apply our perurbaion echnique wih ineracing paricle mehod. 16 [14] Laer, we will see a more general case, where a sochasic collaeral cos is aken ino accoun. 81 / 114

82 Inroducion Framework Symmeric Asymmeric Imperfec FBSDE Approximaion Scheme Perurbaion Technique for Non-linear FBSDEs wih Ineracing Paricle 82 / 114 Model Couner pary does no pos collaeral or poss collaeral wih he consan ime-lag ( ) by cash or an asse A. no collaeral case: ime-lag collaeral case cash collaeral: asse collaeral: f(h, V, Γ ) = lh (V ) +. (8.6) f(h, V, Γ ) = (r c)v f(h, V, Γ ) = (r c)v ( A lh (V V ) +, (8.7) A ) ( ( )) + A lh V V. (8.8) A